Algebra Practice Questions — GRE General Test — Page 13

For all real numbers

x

, the function

f

is defined by

f(x) = x^2 - k

, where

k

is a constant. The custom operation

\star

is defined for all real numbers

a

and

b

by

a \star b = f(a + b) - f(a) - f(b)

If

5 \star (-3) = -12

, what is the value of

k

?

18

12

-12

0

-42

Show answer & explanation

Answer: 18

Answer

The value of

k

is 18.

Evaluating

f(2) = 4 - k

,

f(5) = 25 - k

, and

f(-3) = 9 - k

yields

5 \star (-3) = (4 - k) - (34 - 2k) = -30 + k

. Setting

-30 + k = -12

correctly gives

k = 18

.

Step-by-Step Solution

1

Evaluate

f(a+b)

for

a=5

and

b=-3

a + b = 5 + (-3) = 2

, so

f(2) = 2^2 - k = 4 - k

.

Substitute the input

a+b = 2

into the function definition

f(x) = x^2 - k

.

2

Evaluate

f(a)

and

f(b)

individually

f(5) = 5^2 - k = 25 - k

and

f(-3) = (-3)^2 - k = 9 - k

.

Apply the function rule to inputs 5 and -3.

3

Substitute the evaluated expressions into the custom operation definition

5 \star (-3) = (4 - k) - [(25 - k) + (9 - k)] = (4 - k) - (34 - 2k) = -30 + k

.

Simplify the algebraic expression by combining like terms and distributing the negative sign.

4

Solve for

k

using the given equation

5 \star (-3) = -12

-30 + k = -12 \implies k = 18

.

Add 30 to both sides of the equation to isolate

k

.

Key Concept

Evaluating custom binary operations by substituting function definitions and simplifying algebraic expressions.

Estimated Time:1m 30s

Question 242Question →

The quadratic equation $x^2 - 2mx + (m^2 - 4m + 12) = 0$ , where $m$ is a real constant, has two distinct real roots $x_1$ and $x_2$ . If the distance between the two roots on the real number line is $4\sqrt{3}$ , what is the value of $x_1^2 + x_2^2$ ?

48

96

120

144

192

Show answer & explanation

Answer: 96

Answer

96

By Vieta's formulas, the sum of the roots is

x_1 + x_2 = 2m

and the product of the roots is

x_1 x_2 = m^2 - 4m + 12

. The distance between the roots is given by

|x_1 - x_2| = \sqrt{(x_1 + x_2)^2 - 4x_1 x_2}

. Squaring both sides gives

(4\sqrt{3})^2 = 48 = (2m)^2 - 4(m^2 - 4m + 12) = 16m - 48

. Solving

16m - 48 = 48

yields

m = 6

. Substituting

m = 6

into the expressions for the sum and product of the roots gives

x_1 + x_2 = 12

and

x_1 x_2 = 24

. Finally, applying the identity

x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1 x_2

yields

12^2 - 2(24) = 144 - 48 = 96

.

Step-by-Step Solution

1

Apply Vieta's formulas to determine the sum and product of the roots in terms of

m

.

For

x^2 - 2mx + (m^2 - 4m + 12) = 0

, the sum of roots is

x_1 + x_2 = 2m

and the product of roots is

x_1 x_2 = m^2 - 4m + 12

.

Vieta's relations relate quadratic coefficients directly to root sums and products.

2

Express the distance between the roots

|x_1 - x_2|

in terms of

m

and solve for

m

.

|x_1 - x_2|^2 = (x_1 + x_2)^2 - 4x_1 x_2 = (2m)^2 - 4(m^2 - 4m + 12) = 16m - 48

. Given

|x_1 - x_2| = 4\sqrt{3}

, we have

16m - 48 = (4\sqrt{3})^2 = 48 \implies 16m = 96 \implies m = 6

.

The difference between two roots of a quadratic is linked to the discriminant via

|x_1 - x_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}

.

3

Calculate the numerical values of the root sum

x_1 + x_2

and root product

x_1 x_2

using

m = 6

.

x_1 + x_2 = 2(6) = 12

and

x_1 x_2 = 6^2 - 4(6) + 12 = 24

.

Substituting

m = 6

gives the exact values needed for algebraic evaluation.

4

Compute

x_1^2 + x_2^2

using the identity

x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1 x_2

.

x_1^2 + x_2^2 = 12^2 - 2(24) = 144 - 48 = 96

.

The sum of squares identity isolates

x_1^2 + x_2^2

without requiring explicit calculation of individual root values.

Key Concept

Quadratic Root Relationships and Vieta's Formulas

Estimated Time:2m 30s

Question 243Question →

For all real numbers $x$ , the function $f$ is defined by $f(x) = 3 - x^2$ . For all real numbers $x \neq -2$ , the function $g$ is defined by $g(x) = \frac{2x - 1}{x + 2}$ . What is the value of $g(f(-3))$ ?

\frac{13}{4}

\frac{23}{14}

-46

-42

-\frac{1}{2}

Show answer & explanation

Answer: $\frac{13}{4}$

Answer

The correct answer is

\frac{13}{4}

.

First, evaluate the inner expression

f(-3) = 3 - (-3)^2 = 3 - 9 = -6

. Next, evaluate the outer expression at this value:

g(-6) = \frac{2(-6) - 1}{-6 + 2} = \frac{-13}{-4} = \frac{13}{4}

.

Step-by-Step Solution

1

Evaluate the inner function

f(-3)

f(-3) = 3 - (-3)^2 = 3 - 9 = -6

Applying the definition of

f(x)

where the negative base is squared to yield positive

9

.

2

Substitute the result into the outer function

g(x)

g(-6) = \frac{2(-6) - 1}{-6 + 2} = \frac{-12 - 1}{-4} = \frac{-13}{-4} = \frac{13}{4}

Evaluating

g(x)

at

x = -6

and simplifying the fraction.

Key Concept

Nested Function Evaluation

g(f(x))

Estimated Time:1m 30s

Question 244Question →

In the $xy$ -plane, line $L_1$ passes through the points $(2, 5)$ and $(6, -3)$ . Line $L_2$ is perpendicular to line $L_1$ and passes through the point $(1, 2)$ . Line $L_3$ is parallel to line $L_2$ and has a $y$ -intercept that is $5$ units greater than the $y$ -intercept of line $L_2$ . What is the $x$ -intercept of line $L_3$ ?

-13

-\frac{5}{2}

\frac{9}{2}

13

15

Show answer & explanation

Answer: $-13$

Answer

The

x

-intercept of line

L_3

is

-13

.

The slope of line

L_1

is

\frac{-3 - 5}{6 - 2} = -2

. The negative reciprocal slope for line

L_2

is

\frac{1}{2}

. Using point-slope form with point

(1,2)

, line

L_2

is

y = \frac{1}{2}x + \frac{3}{2}

, giving a

y

-intercept of

\frac{3}{2}

. Line

L_3

, being parallel to

L_2

, shares slope

\frac{1}{2}

and has

y

-intercept

\frac{3}{2} + 5 = \frac{13}{2}

. Finding the

x

-intercept of

y = \frac{1}{2}x + \frac{13}{2}

by setting

y=0

yields

x = -13

.

Step-by-Step Solution

1

Calculate the slope of line

L_1

The slope

m_1 = \frac{-3 - 5}{6 - 2} = \frac{-8}{4} = -2

.

The slope formula is

m = \frac{y_2 - y_1}{x_2 - x_1}

.

2

Determine the slope and equation of perpendicular line

L_2

The slope

m_2 = -\frac{1}{-2} = \frac{1}{2}

. Since

L_2

passes through

(1, 2)

, its equation is

y - 2 = \frac{1}{2}(x - 1)

, which simplifies to

y = \frac{1}{2}x + \frac{3}{2}

.

Perpendicular lines have negative reciprocal slopes.

3

Determine the equation of line

L_3

The

y

-intercept of

L_2

is

\frac{3}{2}

. The

y

-intercept of

L_3

is

\frac{3}{2} + 5 = \frac{13}{2}

. Since

L_3

is parallel to

L_2

, its slope is

m_3 = \frac{1}{2}

. Thus, the equation of

L_3

is

y = \frac{1}{2}x + \frac{13}{2}

.

Parallel lines have equal slopes.

4

Find the

x

-intercept of line

L_3

Setting

y = 0

gives

0 = \frac{1}{2}x + \frac{13}{2} \implies \frac{1}{2}x = -\frac{13}{2} \implies x = -13

.

The

x

-intercept is the value of

x

when

y = 0

.

Key Concept

Slope of parallel and perpendicular lines, line equations, and intercept calculations

Question 245Question →

In the $xy$ -plane, line $K$ is defined by the equation $3x - 4y = 12$ . Line $M$ is perpendicular to line $K$ and passes through the point $(6, -1)$ . Which of the following statements regarding line $M$ must be true? Select all that apply.

Select all that apply

Line

M

has a

y

-intercept of

(0, 7)

.

Line

M

passes through the point

(-3, 11)

.

Line

M

has an

x

-intercept of

(7, 0)

.

Line

M

passes through Quadrant III.

The point of intersection between line

K

and line

M

lies in Quadrant IV.

Show answer & explanation

Answer: Line $M$ has a $y$ -intercept of $(0, 7)$ .; Line $M$ passes through the point $(-3, 11)$ .

Answer

The statements asserting that Line

M

has a

y

-intercept of

(0, 7)

and that Line

M

passes through the point

(-3, 11)

are correct.

Line

K

has a slope of

\frac{3}{4}

, making the perpendicular slope of line

M

equal to

-\frac{4}{3}

. Using the point

(6, -1)

, the equation of line

M

is

y = -\frac{4}{3}x + 7

. Evaluating the options: setting

x = 0

gives

y = 7

, confirming the

y

-intercept is

(0, 7)

; substituting

x = -3

yields

y = 11

, confirming

(-3, 11)

lies on line

M

. Both of these statements are true.

Step-by-Step Solution

1

Determine the slope of line

K

and the perpendicular slope of line

M

.

Line

K

in slope-intercept form is

y = \frac{3}{4}x - 3

, so its slope is

m_K = \frac{3}{4}

. The perpendicular slope for line

M

is the negative reciprocal:

m_M = -\frac{4}{3}

.

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other.

2

Find the equation of line

M

using point-slope form with point

(6, -1)

.

y - (-1) = -\frac{4}{3}(x - 6) \implies y + 1 = -\frac{4}{3}x + 8 \implies y = -\frac{4}{3}x + 7

.

Knowing the slope and a point on the line allows determination of the line's exact linear equation.

3

Evaluate the statements using the equation of line

M

.

1.

y

-intercept: set

x = 0 \implies y = 7

, so

(0,7)

is correct.
2. Point

(-3, 11)

:

y = -\frac{4}{3}(-3) + 7 = 4 + 7 = 11

, so

(-3, 11)

is on the line.
3.

x

-intercept: set

y = 0 \implies -\frac{4}{3}x + 7 = 0 \implies x = \frac{21}{4} = 5.25

, so

(7,0)

is incorrect.
4. Quadrants: A line with negative slope and positive

y

-intercept covers Quadrants I, II, and IV only, so passing through Quadrant III is false.
5. Intersection with line

K

: set

\frac{3}{4}x - 3 = -\frac{4}{3}x + 7 \implies \frac{25}{12}x = 10 \implies x = 4.8

,

y = 0.6

, which is in Quadrant I, not Quadrant IV.

Direct algebraic verification confirms which geometric properties hold true for line

M

.

Key Concept

Perpendicular Slopes and Linear Properties in Coordinate Geometry

Question 246Question →

A logistics company operates a delivery van and a cargo drone along a straight route of $270$ kilometers connecting Hub X and Hub Y. The delivery van departs from Hub X toward Hub Y at a constant speed of $60$ kilometers per hour. Exactly $2$ hours later, the cargo drone departs from Hub Y toward Hub X along the same route at a constant speed of $90$ kilometers per hour. How many hours after the delivery van departs will the van and the cargo drone meet?

1.8 hours

2.6 hours

3.0 hours

4.5 hours

5.0 hours

Show answer & explanation

Answer: 3.0 hours

Answer

3.0 hours

The option specifying '3.0 hours' is correct. During the 2 hours before the drone departs, the van covers

60 \text{ km/h} \times 2 \text{ h} = 120 \text{ km}

. The remaining distance between the vehicles is

270 \text{ km} - 120 \text{ km} = 150 \text{ km}

. Once the drone departs, the two vehicles close the gap at a combined rate of

60 + 90 = 150 \text{ km/h}

. The time needed to cover the remaining

150 \text{ km}

is

\frac{150 \text{ km}}{150 \text{ km/h}} = 1 \text{ hour}

. Adding the van's initial

2

hours gives a total travel time of

3.0

hours.

Step-by-Step Solution

1

Define variables for travel time and compute distance covered by the van before the drone departs.

In

2

hours at

60

km/h, the van covers

60 \times 2 = 120

kilometers.

The van travels alone for the first two hours.

2

Determine the remaining distance separating the two vehicles when the drone starts moving.

Remaining distance =

270 - 120 = 150

kilometers.

The total distance between the hubs is

270

kilometers.

3

Calculate the relative speed of approach and the additional time required to meet.

Combined speed =

60 + 90 = 150

km/h. Additional time =

\frac{150}{150} = 1

hour.

Since the vehicles travel toward each other, their speeds add up.

4

Find the total time elapsed since the van departed.

Total time =

2 + 1 = 3.0

hours.

The question asks for total hours since the van's departure.

Key Concept

Distance-Rate-Time linear modeling with staggered start times and opposing directions

Estimated Time:1m 45s

Question 247Question →

An investment firm allocated a total principal of $\$120,000$ between two accounts, Account X and Account Y. Account X earned a simple annual interest rate of $6\%$ , and Account Y earned a simple annual interest rate of $10\%$ . At the end of one year, the combined total interest earned from both accounts was $\$9,200$ . An annual administrative fee was then deducted from the interest: a $25\%$ fee on the interest earned from Account X, and a $10\%$ fee on the interest earned from Account Y. What was the net amount of interest, in dollars, remaining after these administrative fees were deducted?

\$7,530

\$7,650

\$7,750

\$8,775

\$10,750

Show answer & explanation

Answer: $\$7,650$

Answer

The net amount of interest remaining after administrative fees were deducted is

\$7,650

.

The correct answer is

\$7,650

. Solving the system

x + y = 120,000

and

0.06x + 0.10y = 9,200

gives

\$70,000

invested in Account X and

\$50,000

in Account Y. Account X produces

\$4,200

in interest, from which a

25\%

fee (

\$1,050

) is deducted. Account Y produces

\$5,000

in interest, from which a

10\%

fee (

\$500

) is deducted. Subtracting total fees of

\$1,550

from total interest of

\$9,200

yields

\$7,650

.

Step-by-Step Solution

1

Set up a system of linear equations for the principal amounts invested in Account X and Account Y.

Let

x

be the amount invested in Account X and

y

be the amount invested in Account Y. The total principal is

x + y = 120,000

. The total interest earned is

0.06x + 0.10y = 9,200

.

Modeling the word problem using two variables captures both total principal and total combined interest.

2

Solve the system of equations for

x

and

y

.

Multiplying

x + y = 120,000

by

0.06

gives

0.06x + 0.06y = 7,200

. Subtracting this equation from

0.06x + 0.10y = 9,200

yields

0.04y = 2,000

, so

y = 50,000

. Substituting back gives

x = 70,000

.

Determining individual account principals allows calculation of interest generated by each account.

3

Calculate the interest earned from each account and the corresponding administrative fees.

Interest from X =

0.06 \times 70,000 = \$4,200

. Interest from Y =

0.10 \times 50,000 = \$5,000

. Fee for X =

0.25 \times 4,200 = \$1,050

. Fee for Y =

0.10 \times 5,000 = \$500

. Total fees =

1,050 + 500 = \$1,550

.

Each fee rate must be multiplied by the specific interest amount earned by that account.

4

Subtract the total administrative fees from the total interest earned.

Net Interest =

\$9,200 - \$1,550 = \$7,650

.

Deducting fees yields the net interest remaining.

Key Concept

System of Linear Equations for Mixture and Interest Word Problems

Estimated Time:2m 0s

Question 248Question →

A commercial print shop uses two high-speed printing presses, Press Alpha and Press Beta. Press Alpha operates at a constant rate of $120$ pages per minute, while Press Beta operates at a constant rate of $180$ pages per minute. Press Alpha begins printing a job of $15,000$ pages at 9:00 AM. At 9:15 AM, Press Beta is turned on to assist Press Alpha, and both presses continue printing simultaneously at their respective constant rates until the job is completed. How many total minutes after 9:00 AM will the entire $15,000$ -page job be finished?

Show answer & explanation

Answer: 59

Answer

The entire 15,000-page job will be finished 59 minutes after 9:00 AM.

Press Alpha operates alone for the first 15 minutes, completing

15 \times 120 = 1,800

pages. That leaves

15,000 - 1,800 = 13,200

pages. Once Press Beta joins at 9:15 AM, the combined rate becomes

120 + 180 = 300

pages per minute. The remaining pages require

13,200 / 300 = 44

minutes. Summing the 15-minute initial period and the 44-minute joint period gives a total of 59 minutes after 9:00 AM.

Step-by-Step Solution

1

Find the work completed by Press Alpha during the 15-minute staggered start period.

1,800 pages completed.

Press Alpha ran alone for 15 minutes at 120 pages per minute.

2

Determine the remaining work to be done after 9:15 AM.

13,200 pages remaining.

Subtract the completed pages from the total batch size of 15,000 pages.

3

Calculate the combined work rate of Press Alpha and Press Beta.

300 pages per minute.

When working together, rates add linearly: 120 + 180 = 300.

4

Calculate time needed to complete the remaining pages.

44 minutes.

Divide remaining work (13,200 pages) by combined rate (300 pages/min).

5

Calculate total elapsed time from 9:00 AM.

59 minutes.

Combine the 15 initial minutes with the 44 subsequent minutes.

Key Concept

Linear work-rate equations with staggered initial start times

Estimated Time:1m 30s

Question 249Question →

What is the sum of all real solutions to the equation

(x^2 - 3x)^2 - 8(x^2 - 3x) + 12 = 0

?

3

6

8

12

-6

Show answer & explanation

Answer: $6$

Answer

The sum of all real solutions to the given equation is 6.

By substituting

u = x^2 - 3x

, the quartic equation reduces to the quadratic equation

u^2 - 8u + 12 = 0

, which factors as

(u - 2)(u - 6) = 0

. Setting

x^2 - 3x = 2

and

x^2 - 3x = 6

yields two distinct quadratic equations:

x^2 - 3x - 2 = 0

and

x^2 - 3x - 6 = 0

. Since both discriminants (

17

and

33

) are strictly positive, each equation has two distinct real solutions. By Vieta's formulas, the sum of the roots for a quadratic equation

ax^2 + bx + c = 0

is

-b/a

. Thus, the sum of the roots for

x^2 - 3x - 2 = 0

is

3

, and the sum of the roots for

x^2 - 3x - 6 = 0

is

3

. Adding these together gives a total sum of

3 + 3 = 6

.

Step-by-Step Solution

1

Apply algebraic substitution to reduce the equation to a standard quadratic form.

Let

u = x^2 - 3x

. The original equation becomes

u^2 - 8u + 12 = 0

.

Recognizing repeated quadratic expressions allows for simplification into a single quadratic in terms of

u

.

2

Factor the quadratic equation in

u

to find its roots.

(u - 2)(u - 6) = 0

, which gives

u = 2

and

u = 6

.

The factors of

12

that sum to

-8

are

-2

and

-6

.

3

Substitute

x^2 - 3x

back for

u

and check the discriminant of each resulting quadratic equation.

For

u = 2

:

x^2 - 3x - 2 = 0

has discriminant

\Delta_1 = (-3)^2 - 4(1)(-2) = 17 > 0

(2 distinct real roots).
For

u = 6

:

x^2 - 3x - 6 = 0

has discriminant

\Delta_2 = (-3)^2 - 4(1)(-6) = 33 > 0

(2 distinct real roots).

Verifying that the discriminant is positive ensures that all four roots are real numbers.

4

Calculate the sum of the roots for each quadratic equation using Vieta's formulas.

For

x^2 - 3x - 2 = 0

, the sum of roots is

-\frac{-3}{1} = 3

.
For

x^2 - 3x - 6 = 0

, the sum of roots is

-\frac{-3}{1} = 3

.

According to Vieta's formulas, for a quadratic

ax^2 + bx + c = 0

, the sum of the roots is

-b/a

.

5

Add the sums of the roots from both equations to find the total sum of all real solutions.

Total sum

= 3 + 3 = 6

.

Since all four roots are real and distinct, the overall sum is the sum of the roots of the two constituent quadratics.

Key Concept

Quadratic Substitution and Vieta's Formulas for Root Sums

Question 250Question →

For all non-zero real numbers $a$ and $b$ , the custom binary operation $\diamondsuit$ is defined by $a \diamondsuit b = \frac{a^2 - b^2}{ab}$ . If the function $f$ is defined for all $x \neq 0$ by $f(x) = x \diamondsuit 2$ , what is the value of $f(4) - f(1)$ ?

Show answer & explanation

Answer: 3

Answer

3

Evaluating

f(4) = 4 \diamondsuit 2 = \frac{16 - 4}{8} = 1.5

and

f(1) = 1 \diamondsuit 2 = \frac{1 - 4}{2} = -1.5

, the required difference is

f(4) - f(1) = 1.5 - (-1.5) = 3

.

Step-by-Step Solution

1

Evaluate

f(4)

using the custom operation definition

f(4) = \frac{4^2 - 2^2}{4 \cdot 2} = \frac{12}{8} = 1.5

Substitute

a = 4

and

b = 2

into

a \diamondsuit b = \frac{a^2 - b^2}{ab}

.

2

Evaluate

f(1)

using the custom operation definition

f(1) = \frac{1^2 - 2^2}{1 \cdot 2} = -1.5

Substitute

a = 1

and

b = 2

into

a \diamondsuit b = \frac{a^2 - b^2}{ab}

.

3

Compute the difference

f(4) - f(1)

3

Subtracting

-1.5

from

1.5

yields

1.5 - (-1.5) = 1.5 + 1.5 = 3

.

Key Concept

Custom Symbol Operations and Function Evaluation

Estimated Time:1m 30s

Question 251Question →

In the $xy$ -plane, line $k$ passes through the points $(-3, 5)$ and $(1, -3)$ . Line $m$ is perpendicular to line $k$ at line $k$ 's $x$ -intercept. What is the $y$ -intercept of line $m$ ?

\frac{1}{4}

-\frac{1}{4}

10

-1

-4

Show answer & explanation

Answer: $\frac{1}{4}$

Answer

The

y

-intercept of line

m

is

\frac{1}{4}

.

The line

k

has a slope of

-2

and an

x

-intercept of

\left(-\frac{1}{2}, 0\right)

. A line perpendicular to line

k

must have a slope of

\frac{1}{2}

. Substituting the point

\left(-\frac{1}{2}, 0\right)

into the line equation yields

y = \frac{1}{2}x + \frac{1}{4}

, so the

y

-intercept is

\frac{1}{4}

.

Step-by-Step Solution

1

Calculate the slope of line

k

Slope

m_k = \frac{-3 - 5}{1 - (-3)} = \frac{-8}{4} = -2

.

The slope formula is

m = \frac{y_2 - y_1}{x_2 - x_1}

.

2

Find the equation of line

k

and determine its

x

-intercept

Line

k

equation:

y - (-3) = -2(x - 1) \implies y = -2x - 1

. Setting

y = 0

gives

0 = -2x - 1 \implies x = -\frac{1}{2}

. The

x

-intercept is

\left(-\frac{1}{2}, 0\right)

.

The

x

-intercept is the point where the line crosses the

x

-axis (

y = 0

).

3

Find the slope of line

m

Slope

m_m = -\frac{1}{m_k} = -\frac{1}{-2} = \frac{1}{2}

.

Perpendicular lines have slopes that are negative reciprocals of each other.

4

Determine the equation of line

m

and its

y

-intercept

Using point-slope form with

\left(-\frac{1}{2}, 0\right)

and slope

\frac{1}{2}

:

y - 0 = \frac{1}{2}\left(x - \left(-\frac{1}{2}\right)\right) \implies y = \frac{1}{2}x + \frac{1}{4}

. Setting

x = 0

gives

y = \frac{1}{4}

.

The

y

-intercept is the constant term

b

when written in slope-intercept form

y = mx + b

.

Key Concept

Perpendicular line slopes and intercept calculations

Estimated Time:2m 0s

Question 252Question →

The quadratic equation $(x - a)(x - b) = c$ , where $a$ , $b$ , and $c$ are real constants with $a < b$ and $c > 0$ , has two real roots $x_1$ and $x_2$ such that $x_1 < x_2$ . Which of the following statements must be true? Indicate all such statements.

Select all that apply

x_1 < a

x_2 > b

The distance between the roots,

x_2 - x_1

, is strictly greater than

b - a

x_1 + x_2 > a + b

The product of the roots,

x_1 x_2

, must be negative

Show answer & explanation

Answer: $x_1 < a$ ; $x_2 > b$ ; The distance between the roots, $x_2 - x_1$ , is strictly greater than $b - a$

Answer

The correct statements are that the smaller root is less than

a

(

x_1 < a

), the larger root is greater than

b

(

x_2 > b

), and the distance between the roots is greater than

b - a

(

x_2 - x_1 > b - a

).

Evaluating

f(x) = (x-a)(x-b) - c

at

x = a

and

x = b

yields negative values (

-c

). Because the parabola opens upward, the graph must cross the x-axis to the left of

a

and to the right of

b

. This establishes

x_1 < a

and

x_2 > b

. Combining these inequalities shows that the distance between the roots

x_2 - x_1

must exceed

b - a

.

Step-by-Step Solution

1

Formulate the quadratic function and analyze its values at key points

Let

f(x) = (x - a)(x - b) - c = 0

. Evaluating at

x = a

and

x = b

gives

f(a) = (0)(a-b) - c = -c < 0

and

f(b) = (b-a)(0) - c = -c < 0

.

Evaluating at

a

and

b

reveals the sign of the function inside the interval

[a, b]

.

2

Use the sign of the leading coefficient and intermediate value properties to locate the roots

The coefficient of

x^2

is

1 > 0

, so the parabola opens upward. Since

f(a) < 0

and

f(b) < 0

, and

f(x) \to +\infty

as

x \to \pm\infty

, the two real roots

x_1

and

x_2

must satisfy

x_1 < a < b < x_2

.

A continuous upward-opening parabola must cross the x-axis outside any interval where its values are negative.

3

Analyze the distance between the roots

Subtracting

x_1 < a

from

x_2 > b

gives

x_2 - x_1 > b - a

.

Since

a

and

b

fall strictly between

x_1

and

x_2

, the distance between

x_1

and

x_2

exceeds the distance between

a

and

b

.

4

Check the remaining statements using Vieta's formulas

Expanding

f(x) = x^2 - (a+b)x + (ab - c) = 0

yields sum of roots

x_1 + x_2 = a + b

and product of roots

x_1 x_2 = ab - c

. Thus

x_1 + x_2 > a+b

is false, and

x_1 x_2 < 0

is not necessarily true.

Vieta's formulas give exact values for the sum and product of roots in terms of coefficients.

Key Concept

Geometric interpretation of quadratic functions and root locations relative to evaluated points

Question 253Question →

Two industrial processing units, Unit X and Unit Y, process liquid solution at constant rates of $40$ liters per hour and $60$ liters per hour, respectively. Unit X begins processing a $1,100$ -liter batch of solution alone. After $5$ hours, Unit Y is activated and joins Unit X, working simultaneously at their respective constant rates until the entire batch is completely processed. What is the total time, in hours, from when Unit X started processing until the entire batch was completely processed?

9

11

14

16

20

Show answer & explanation

Answer: 14

Answer

14 hours

Unit X works alone for 5 hours, processing

40 \times 5 = 200

liters. This leaves

1,100 - 200 = 900

liters of solution remaining. When Unit Y joins, the units process solution at a joint rate of

40 + 60 = 100

liters per hour. The remaining 900 liters require

\frac{900}{100} = 9

hours of combined work. Summing the 5 hours of solo work and 9 hours of combined work gives a total time of 14 hours.

Step-by-Step Solution

1

Calculate the volume processed by Unit X during its solo operation

40 \text{ liters/hour} \times 5 \text{ hours} = 200 \text{ liters}

Unit X operates alone for the first 5 hours at a constant rate of 40 liters per hour.

2

Determine the remaining batch volume to be processed

1,100 \text{ liters} - 200 \text{ liters} = 900 \text{ liters}

Subtract the volume already completed from the total batch size.

3

Calculate the combined processing rate of Unit X and Unit Y

40 \text{ liters/hour} + 60 \text{ liters/hour} = 100 \text{ liters/hour}

Both units work together simultaneously after the first 5 hours.

4

Find the time needed for both units to process the remaining volume

\frac{900 \text{ liters}}{100 \text{ liters/hour}} = 9 \text{ hours}

Divide the remaining volume by the combined rate.

5

Calculate the total elapsed time from the start

5 \text{ hours (solo)} + 9 \text{ hours (combined)} = 14 \text{ hours}

Add the initial solo work time to the combined work time.

Key Concept

Algebraic modeling of combined work rates with staggered start times

Estimated Time:1m 30s

Question 254Question →

The function $f$ is defined by $f(t) = 2^t$ for all real numbers $t$ , and the binary operation $\odot$ is defined for all non-zero real numbers $m$ and $n$ by $m \odot n = \frac{m}{n} - \frac{n}{m}$ . What is the value of $f(2) \odot f(-2)$ ?

\frac{255}{16}

\frac{15}{16}

\frac{15}{4}

0

-\frac{255}{16}

Show answer & explanation

Answer: $\frac{255}{16}$

Answer

\frac{255}{16}

Evaluating

f(2)

yields

2^2 = 4

, and evaluating

f(-2)

yields

2^{-2} = \frac{1}{4}

. Substituting these values into the binary operation rule

m \odot n = \frac{m}{n} - \frac{n}{m}

gives

\frac{4}{1/4} - \frac{1/4}{4} = 16 - \frac{1}{16} = \frac{255}{16}

.

Step-by-Step Solution

1

Evaluate the function

f(t) = 2^t

at

t = 2

and

t = -2

.

f(2) = 2^2 = 4

and

f(-2) = 2^{-2} = \frac{1}{2^2} = \frac{1}{4}

.

Negative exponents follow the rule

x^{-a} = \frac{1}{x^a}

.

2

Substitute

m = f(2) = 4

and

n = f(-2) = \frac{1}{4}

into the custom operation definition

m \odot n = \frac{m}{n} - \frac{n}{m}

.

4 \odot \frac{1}{4} = \frac{4}{\frac{1}{4}} - \frac{\frac{1}{4}}{4}

.

Apply the defined binary operation rule.

3

Simplify the complex fraction terms.

\frac{4}{\frac{1}{4}} = 4 \times 4 = 16

and

\frac{\frac{1}{4}}{4} = \frac{1}{4 \times 4} = \frac{1}{16}

.

Dividing by a fraction is equivalent to multiplying by its reciprocal.

4

Subtract the two simplified terms.

16 - \frac{1}{16} = \frac{256}{16} - \frac{1}{16} = \frac{255}{16}

.

Find a common denominator to compute the final value.

Key Concept

Function evaluation with exponent rules combined with custom binary symbol operations

Question 255Question →

For a nonzero real constant $m$ , the quadratic equation $x^2 - \left(m + \frac{6}{m}\right)x + 6 = 0$ has real roots $a$ and $b$ . If $a^2 + b^2 = 25$ , what is the sum of all possible values of $m^2$ ?

13

25

37

49

5

Show answer & explanation

Answer: $25$

Answer

The sum of all possible values of

m^2

is

25

.

Using Vieta's formulas on

x^2 - \left(m + \frac{6}{m}\right)x + 6 = 0

, the sum of the roots is

a + b = m + \frac{6}{m}

and the product is

ab = 6

. Substituting these into the identity

a^2 + b^2 = (a+b)^2 - 2ab = 25

yields

\left(m + \frac{6}{m}\right)^2 - 12 = 25

, which simplifies to

m^2 + \frac{36}{m^2} = 25

. Setting

u = m^2

gives

u^2 - 25u + 36 = 0

. Since this quadratic has two distinct positive real roots for

u

, the sum of all possible values of

m^2

is given by Vieta's formula as

25

.

Step-by-Step Solution

1

Apply Vieta's formulas to relate the roots

a

and

b

to the coefficients of the given quadratic equation.

a + b = m + \frac{6}{m}

and

ab = 6

.

For any quadratic equation

x^2 - Sx + P = 0

, the sum of the roots is

S

and the product of the roots is

P

.

2

Express

a^2 + b^2

using the algebraic identity for the sum of squares.

a^2 + b^2 = (a + b)^2 - 2ab = \left(m + \frac{6}{m}\right)^2 - 2(6) = \left(m + \frac{6}{m}\right)^2 - 12

.

Rewriting the sum of squares allows substitution of the known sum and product of the roots.

3

Set the expression equal to the given value

25

and expand the square.

m^2 + 2(m)\left(\frac{6}{m}\right) + \frac{36}{m^2} - 12 = 25 \implies m^2 + 12 + \frac{36}{m^2} - 12 = 25 \implies m^2 + \frac{36}{m^2} = 25

.

Expanding

\left(m + \frac{6}{m}\right)^2

yields

m^2 + 12 + \frac{36}{m^2}

, which simplifies directly after subtracting

12

.

4

Formulate a quadratic equation in terms of

u = m^2

.

u + \frac{36}{u} = 25 \implies u^2 - 25u + 36 = 0

.

Multiplying through by

u = m^2

transforms the rational equation into a standard polynomial quadratic equation.

5

Find the sum of all valid solutions for

u = m^2

using Vieta's formulas.

The discriminant of

u^2 - 25u + 36 = 0

is

\Delta = (-25)^2 - 4(1)(36) = 625 - 144 = 481 > 0

, and the product of the roots is

36 > 0

, so both roots

u_1, u_2

are positive real numbers. By Vieta's formulas,

u_1 + u_2 = 25

.

Since both roots

u_1

and

u_2

are positive real numbers, they correspond to valid real values of

m^2

, making their sum equal to

25

.

Key Concept

Quadratic Equations, Vieta's Formulas, and Factoring Substitution

Estimated Time:2m 0s

Question 256Question →

In the $xy$ -plane, line $k$ is defined by the equation $3x - 4y = 12$ . Line $m$ is parallel to line $k$ , and the perpendicular distance between line $k$ and line $m$ is $5$ units. If the $y$ -intercept of line $m$ is greater than the $y$ -intercept of line $k$ , what is the $y$ -intercept of line $m$ ?

Show answer & explanation

Answer: 3.25

Answer

3.25

Rewriting line

k

as

3x - 4y - 12 = 0

shows its

y

-intercept is

-3

. Line

m

is parallel, so its equation is

3x - 4y + C = 0

. Using the formula for perpendicular distance between parallel lines

d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}

, we have

\frac{|-12 - C|}{\sqrt{3^2 + (-4)^2}} = 5

. This simplifies to

|-12 - C| = 25

, giving

C = 13

or

C = -37

. Setting

x = 0

for line

m

gives

y = -\frac{C}{-4} = \frac{C}{4}

. For

C = 13

, the

y

-intercept is

\frac{13}{4} = 3.25

. Since

3.25 > -3

, this meets all criteria.

Step-by-Step Solution

1

Find the

y

-intercept of line

k

Line

k

has a

y

-intercept at

(0, -3)

.

Setting

x = 0

in

3x - 4y = 12

gives

-4y = 12 \implies y = -3

.

2

Formulate the general equation for line

m

Line

m

has the equation

3x - 4y + C = 0

.

Parallel lines share the same linear coefficients

A = 3

and

B = -4

.

3

Set up the distance formula between parallel lines

\frac{|-12 - C|}{5} = 5

The distance between

Ax + By + C_1 = 0

and

Ax + By + C_2 = 0

is

d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}

.

4

Solve for constant

C

C = 13

or

C = -37

|-12 - C| = 25

yields

-12 - C = 25 \implies C = -37

and

-12 - C = -25 \implies C = 13

.

5

Determine the required

y

-intercept

y = 3.25

For

C = 13

, the

y

-intercept is

\frac{13}{4} = 3.25

, which is greater than

-3

.

Key Concept

Perpendicular distance between parallel lines and line intercept calculation

Estimated Time:2m 30s

Question 257Question →

For all non-zero real numbers $x$ and $y$ , the custom binary operation $\star$ is defined by $x \star y = \frac{x}{y} - \frac{y}{x}$ . Which of the following statements must be true for all non-zero real numbers $a$ , $b$ , and $c$ ? Select all that apply.

Select all that apply

a \star b = -(b \star a)

a \star a = 0

(a \star b)^2 = \frac{a^2}{b^2} + \frac{b^2}{a^2} - 2

a \star (b \star c) = (a \star b) \star c

a \star (b + c) = (a \star b) + (a \star c)

Show answer & explanation

Answer: $a \star b = -(b \star a)$ ; $a \star a = 0$ ; $(a \star b)^2 = \frac{a^2}{b^2} + \frac{b^2}{a^2} - 2$

Answer

The statements

a \star b = -(b \star a)

,

a \star a = 0

, and

(a \star b)^2 = \frac{a^2}{b^2} + \frac{b^2}{a^2} - 2

must be true for all non-zero real numbers

a

,

b

, and

c

.

The statements showing anti-commutativity, zero self-operation, and the expanded squared expression are all mathematically valid for all non-zero real numbers upon expanding their definitions using algebra.

Step-by-Step Solution

1

Evaluate

a \star b = -(b \star a)

a \star b = \frac{a}{b} - \frac{b}{a}

and

-(b \star a) = -\left(\frac{b}{a} - \frac{a}{b}\right) = \frac{a}{b} - \frac{b}{a}

.

Verify anti-commutativity property by direct substitution.

2

Evaluate

a \star a = 0

a \star a = \frac{a}{a} - \frac{a}{a} = 1 - 1 = 0

.

Verify identity property for identical inputs.

3

Evaluate

(a \star b)^2

(a \star b)^2 = \left(\frac{a}{b} - \frac{b}{a}\right)^2 = \left(\frac{a}{b}\right)^2 - 2\left(\frac{a}{b}\right)\left(\frac{b}{a}\right) + \left(\frac{b}{a}\right)^2 = \frac{a^2}{b^2} - 2(1) + \frac{b^2}{a^2} = \frac{a^2}{b^2} + \frac{b^2}{a^2} - 2

.

Apply binomial expansion to the squared custom operation.

4

Test associativity statement

a \star (b \star c) = (a \star b) \star c

with a counterexample

Let

a = 4, b = 2, c = 1

. Then

b \star c = \frac{2}{1} - \frac{1}{2} = \frac{3}{2}

.

a \star (b \star c) = 4 \star \frac{3}{2} = \frac{4}{3/2} - \frac{3/2}{4} = \frac{8}{3} - \frac{3}{8} = \frac{55}{24}

. Meanwhile,

a \star b = 4 \star 2 = \frac{4}{2} - \frac{2}{4} = \frac{3}{2}

, and

(a \star b) \star c = \frac{3}{2} \star 1 = \frac{3/2}{1} - \frac{1}{3/2} = \frac{3}{2} - \frac{2}{3} = \frac{5}{6}

. Since

\frac{55}{24} \neq \frac{5}{6}

, associativity fails.

A single counterexample disproves a general identity statement.

5

Test distributivity statement

a \star (b + c) = (a \star b) + (a \star c)

with a counterexample

Let

a = 1, b = 1, c = 1

. Then

a \star (b + c) = 1 \star 2 = \frac{1}{2} - \frac{2}{1} = -\frac{3}{2}

. Meanwhile,

(a \star b) + (a \star c) = (1 \star 1) + (1 \star 1) = 0 + 0 = 0

. Since

-\frac{3}{2} \neq 0

, distributivity fails.

A single counterexample disproves distributivity over addition.

Key Concept

Evaluating algebraic properties and identity statements for custom defined binary operations.

Question 258Question →

A cyclist travels from City $X$ to City $Y$ at a constant speed of $24$ miles per hour. On the return trip from City $Y$ to City $X$ along the exact same route, adverse weather conditions reduce the cyclist's average speed by $25\%$ . If the total time for the entire round trip is $7$ hours, what is the distance, in miles, between City $X$ and City $Y$ ?

73.5

96

72

126

144

Show answer & explanation

Answer: $72$

Answer

The distance between City

X

and City

Y

is

72

miles.

To find the one-way distance

d

, first calculate the return speed:

24 \times 0.75 = 18

mph. Express the total time spent traveling as the sum of the time for each leg:

\frac{d}{24} + \frac{d}{18} = 7

. Combining the fractions over a common denominator of

72

gives

\frac{7d}{72} = 7

, which simplifies to

d = 72

miles.

Step-by-Step Solution

1

Determine the return speed

Return speed =

24 \times (1 - 0.25) = 18

miles per hour.

Adverse weather reduces the outbound speed of

24

mph by

25\%

.

2

Formulate the total time equation in terms of distance

d

\frac{d}{24} + \frac{d}{18} = 7

Time equals distance divided by rate. The sum of the outbound time and return time is

7

hours.

3

Solve the algebraic equation for

d

\frac{3d + 4d}{72} = 7 \implies \frac{7d}{72} = 7 \implies d = 72

Finding a common denominator of

72

allows combining the fractional time expressions.

Key Concept

Distance, Rate, and Time Word Problems

Estimated Time:1m 30s

Question 259Question →

In the $xy$ -plane, line $L$ passes through the point $(4, 2)$ and has a negative slope $m$ . Line $L$ intersects the positive $x$ -axis at point $P$ and the positive $y$ -axis at point $Q$ . If the area of triangle $OPQ$ , where $O$ is the origin, is equal to $16$ , what is the slope of line $L$ ?

-2

-1

-\frac{1}{2}

-\frac{1}{4}

\frac{1}{2}

Show answer & explanation

Answer: $-\frac{1}{2}$

Answer

The slope of line

L

is

-\frac{1}{2}

.

The line equation in point-slope form is

y - 2 = m(x - 4)

. Setting

y = 0

yields the

x

-intercept

P\left(4 - \frac{2}{m}, 0\right)

, and setting

x = 0

yields the

y

-intercept

Q(0, 2 - 4m)

. The area of right triangle

OPQ

is

\frac{1}{2}\left(4 - \frac{2}{m}\right)(2 - 4m) = 16

. Expanding this equation leads to

4m^2 + 4m + 1 = 0

, which factors as

(2m + 1)^2 = 0

, giving

m = -\frac{1}{2}

.

Step-by-Step Solution

1

Write the point-slope form of line

L

using point

(4, 2)

and slope

m

.

The equation of line

L

is

y - 2 = m(x - 4)

.

Point-slope form allows expressing intercepts directly in terms of the unknown slope

m

.

2

Find the coordinates of the

x

-intercept

P

and

y

-intercept

Q

.

Setting

y = 0

gives

x_P = 4 - \frac{2}{m}

. Setting

x = 0

gives

y_Q = 2 - 4m

.

Intersections with the axes define the base and height of right triangle

OPQ

.

3

Set up the area equation for triangle

OPQ

.

\text{Area} = \frac{1}{2} \cdot x_P \cdot y_Q = \frac{1}{2} \left(4 - \frac{2}{m}\right)(2 - 4m) = 16.

Triangle

OPQ

is a right triangle with base along the

x

-axis and height along the

y

-axis.

4

Solve the algebraic equation for slope

m

.

Expanding gives

\frac{1}{2}\left(8 - 16m - \frac{4}{m} + 8\right) = 16 \implies 16 - 16m - \frac{4}{m} = 32 \implies -16m - 16 - \frac{4}{m} = 0

. Dividing by

-4

and multiplying by

m

yields

4m^2 + 4m + 1 = 0 \implies (2m + 1)^2 = 0 \implies m = -\frac{1}{2}

.

Factoring the perfect square quadratic equation yields the single real value for

m

.

Key Concept

Coordinate Geometry: Intercepts and Area of Triangles

Alternative Method

Use the intercept form of a line

\frac{x}{a} + \frac{y}{b} = 1

. The area condition gives

ab = 32

. Substituting

(4, 2)

gives

\frac{4}{a} + \frac{2}{b} = 1

. Substituting

b = \frac{32}{a}

leads to

\frac{4}{a} + \frac{2a}{32} = 1 \implies \frac{4}{a} + \frac{a}{16} = 1 \implies a^2 - 16a + 64 = 0 \implies (a-8)^2 = 0 \implies a = 8

. Then

b = 4

, and the slope is

m = -\frac{b}{a} = -\frac{4}{8} = -\frac{1}{2}

.

Estimated Time:2m 0s

Question 260Question →

The quadratic function $f(x) = -2x^2 + kx - 18$ has a maximum value of $14$ , where $k$ is a positive constant. What is the value of $k$ ?

Show answer & explanation

Answer: 16

Answer

The value of

k

is

16

.

The vertex of the parabola

f(x) = -2x^2 + kx - 18

is located at

x = \frac{k}{4}

. Evaluating

f\left(\frac{k}{4}\right)

gives the maximum value

\frac{k^2}{8} - 18

. Setting this expression equal to

14

leads to

\frac{k^2}{8} = 32

, so

k^2 = 256

. Taking the positive root as required by the problem statement yields

k = 16

.

Step-by-Step Solution

1

Find the

x

-coordinate of the vertex of the quadratic function.

For

f(x) = -2x^2 + kx - 18

, we have

a = -2

,

b = k

, and

c = -18

. The vertex occurs at

x = -\frac{b}{2a} = -\frac{k}{2(-2)} = \frac{k}{4}

.

The maximum or minimum of any quadratic function

ax^2 + bx + c

occurs at its vertex, where

x = -\frac{b}{2a}

.

2

Evaluate the function at the vertex to determine the maximum value in terms of

k

.

f\left(\frac{k}{4}\right) = -2\left(\frac{k}{4}\right)^2 + k\left(\frac{k}{4}\right) - 18 = -2\left(\frac{k^2}{16}\right) + \frac{k^2}{4} - 18 = -\frac{k^2}{8} + \frac{k^2}{4} - 18 = \frac{k^2}{8} - 18

.

Substituting the vertex

x

-coordinate into

f(x)

yields the maximum value of the downward-opening parabola.

3

Set the maximum value expression equal to

14

and solve for

k^2

.

\frac{k^2}{8} - 18 = 14 \implies \frac{k^2}{8} = 32 \implies k^2 = 256.

The problem states that the maximum value of

f(x)

is

14

.

4

Solve for the positive constant

k

.

k = \sqrt{256} = 16.

Taking the square root of

256

gives

k = 16

or

k = -16

. Since

k

is given as a positive constant,

k = 16

.

Key Concept

Finding the extreme value of a quadratic function by locating its vertex

Algebra

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

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