All practice questions

Question 1Question →

When the polynomial $g(x) = x^4 - 2x^3 + ax^2 - 8$ is divided by $x - 3$ , the remainder is $37$ , where $a$ is a constant. What is the value of $a$ ?

Show answer & explanation

Answer: 2

Answer

2

According to the Polynomial Remainder Theorem, dividing a polynomial

g(x)

by a linear divisor

x - c

yields a remainder equal to

g(c)

. In this problem, the divisor is

x - 3

, so we evaluate the polynomial at

x = 3

and set it equal to the given remainder of

37

. Substituting

3

for

x

in

g(x) = x^4 - 2x^3 + ax^2 - 8

yields

3^4 - 2(3)^3 + a(3)^2 - 8 = 37

. Simplifying the numerical expressions gives

81 - 54 + 9a - 8 = 37

, which simplifies to

19 + 9a = 37

. Subtracting

19

from both sides results in

9a = 18

. Dividing both sides by

9

gives the value of the constant

a

as

2

.

Step-by-Step Solution

1

Apply the Polynomial Remainder Theorem

g(3) = 37

The Remainder Theorem states that when a polynomial

g(x)

is divided by

x - c

, the remainder is

g(c)

.

2

Substitute

x = 3

into the polynomial

g(x)

3^4 - 2(3)^3 + a(3)^2 - 8 = 37

This sets the value of the polynomial evaluated at

x = 3

equal to the given remainder of

37

.

3

Simplify the constant terms

19 + 9a = 37

Evaluating the exponents and multiplying:

3^4 = 81

,

2(3^3) = 54

, and

a(3^2) = 9a

. Combining the constant terms gives

81 - 54 - 8 = 19

.

4

Solve the linear equation for

a

a = 2

Subtracting

19

from both sides gives

9a = 18

, and dividing both sides by

9

gives

a = 2

.

Key Concept

Polynomial Remainder Theorem

Question 2Question →

A laboratory dispensing pump releases a liquid solution at a constant rate of $0.75\text{ deciliters per minute}$ . If $1\text{ deciliter} = 100\text{ milliliters}$ , what is the pump's dispensing rate, in milliliters per second?

Show answer & explanation

Answer: 1.25

Answer

The correct answer is 1.25. The pump dispenses 1.25 milliliters per second.

To find the dispensing rate in milliliters per second, we first convert the volume from deciliters to milliliters. Multiplying

0.75\text{ deciliters}

by

100\text{ milliliters per deciliter}

gives

75\text{ milliliters per minute}

. Next, to convert minutes to seconds, we divide the rate by

60

(since

1\text{ minute} = 60\text{ seconds}

), which yields

\frac{75}{60} = 1.25\text{ milliliters per second}

.

Step-by-Step Solution

1

Convert the dispensing rate from deciliters per minute to milliliters per minute using the conversion factor

1\text{ deciliter} = 100\text{ milliliters}

.

0.75\text{ deciliters/minute} \times 100\text{ milliliters/deciliter} = 75\text{ milliliters/minute}

This converts the volume unit from deciliters to milliliters.

2

Convert the rate from milliliters per minute to milliliters per second by dividing by

60

, since there are

60\text{ seconds}

in

1\text{ minute}

.

75\text{ milliliters/minute} \div 60\text{ seconds/minute} = 1.25\text{ milliliters/second}

This converts the time unit from minutes to seconds, yielding the final rate.

Key Concept

Unit Conversions

Alternative Method

Alternatively, you can convert the units in a single dimensional analysis step:

0.75 \text{ dL/min} \times \frac{100 \text{ mL}}{1 \text{ dL}} \times \frac{1 \text{ min}}{60 \text{ s}} = \frac{75}{60} \text{ mL/s} = 1.25 \text{ mL/s}

.

Estimated Time:1m 30s

Question 3Question →

A florist sells carnations for $1.50$ each and roses for $3.00$ each. The equation $1.50c + 3.00r = 45.00$ represents the possible number of carnations, $c$ , and roses, $r$ , that a customer can buy for exactly $45.00$ . If the customer buys 8 carnations, how many roses can they buy?

Show answer & explanation

Answer: 11

Answer

11

To find the number of roses the customer can buy, substitute the number of carnations bought, which is 8, for

c

in the given equation. This yields

1.50(8) + 3.00r = 45.00

. Simplifying the product gives

12.00 + 3.00r = 45.00

. Subtracting 12.00 from both sides of the equation gives

3.00r = 33.00

. Finally, dividing both sides by 3.00 yields

r = 11

. Thus, the customer can buy 11 roses.

Step-by-Step Solution

1

Substitute the given value for carnations,

c = 8

, into the linear equation.

1.50(8) + 3.00r = 45.00

We are given that the customer buys 8 carnations, so

c

is replaced by 8 to solve for the number of roses,

r

.

2

Multiply

1.50

by

8

to simplify the term.

12.00 + 3.00r = 45.00

Simplifying the constant term on the left side of the equation.

3

Subtract

12.00

from both sides of the equation to isolate the term with

r

.

3.00r = 33.00

To solve for

r

, we must get the term

3.00r

by itself on one side.

4

Divide both sides of the equation by

3.00

to solve for

r

.

r = 11

Dividing isolates the variable

r

.

Key Concept

Solving a linear equation in two variables by substitution when one variable's value is known.

Question 4Question →

In the quadratic equation $2x^2 - 16x + c = 0$ , $c$ is a constant. If one of the solutions to the equation is $4 - \sqrt{3}$ , what is the value of $c$ ?

Show answer & explanation

Answer: 26

Answer

26

Since the coefficients of the quadratic equation are real, any irrational roots must occur in conjugate pairs. Therefore, if one solution is

4 - \sqrt{3}

, the other solution must be

4 + \sqrt{3}

. According to Vieta's formulas, the product of the roots

r_1

and

r_2

for a quadratic equation

ax^2 + bx + c = 0

is

\frac{c}{a}

. In this equation,

a = 2

, so the product of the roots is

\frac{c}{2}

. Multiplying the two solutions gives

(4 - \sqrt{3})(4 + \sqrt{3}) = 16 - 3 = 13

. Setting

\frac{c}{2} = 13

and solving for

c

yields

c = 26

.

Step-by-Step Solution

1

Determine the second root of the quadratic equation.

The second root is

4 + \sqrt{3}

.

Since the quadratic equation has real coefficients, the irrational roots must be conjugate pairs.

2

Express the product of the roots using Vieta's formulas.

The product of the roots is

\frac{c}{2}

.

For any quadratic equation

ax^2 + bx + c = 0

, the product of the roots is equal to

\frac{c}{a}

. Here,

a = 2

.

3

Calculate the product of the two roots.

(4 - \sqrt{3})(4 + \sqrt{3}) = 16 - 3 = 13

.

Using the difference of squares formula,

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

.

4

Equate the product of the roots to the expression from step 2 and solve for

c

.

\frac{c}{2} = 13 \Rightarrow c = 26

.

By substituting the calculated product into the formula for the product of the roots, we find the value of the constant.

Key Concept

Using Vieta's formulas and the conjugate root theorem to solve for coefficients of a quadratic equation.

Alternative Method

Alternatively, substitute the given solution

x = 4 - \sqrt{3}

directly into the equation

2x^2 - 16x + c = 0

and solve for

c

. First, calculate

x^2 = (4 - \sqrt{3})^2 = 16 - 8\sqrt{3} + 3 = 19 - 8\sqrt{3}

. Then substitute this into the equation:

2(19 - 8\sqrt{3}) - 16(4 - \sqrt{3}) + c = 0

. Simplifying this yields

38 - 16\sqrt{3} - 64 + 16\sqrt{3} + c = 0 \Rightarrow -26 + c = 0 \Rightarrow c = 26

.

Estimated Time:1m 30s

Question 5Question →

If $5^{x+2} - 5^x = 120\sqrt{5}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 1.5

Answer

1.5 (or 3/2)

By factoring out the common term

5^x

from the left side of the equation, we get

5^x(5^2 - 1) = 120\sqrt{5}

, which simplifies to

24 \cdot 5^x = 120\sqrt{5}

. Dividing both sides by 24 isolates the exponential term:

5^x = 5\sqrt{5}

. Since

5\sqrt{5}

can be written as

5^1 \cdot 5^{0.5} = 5^{1.5}

, we have

5^x = 5^{1.5}

. Equating the exponents gives

x = 1.5

(or

3/2

).

Step-by-Step Solution

1

Factor out

5^x

from the left side of the equation.

5^x(5^2 - 1) = 120\sqrt{5}

To apply exponent rules to rewrite

5^{x+2}

as

5^x \cdot 5^2

and then factor out the common term

5^x

.

2

Simplify the constant term inside the parentheses.

24 \cdot 5^x = 120\sqrt{5}

Evaluating

5^2 - 1 = 25 - 1 = 24

simplifies the coefficient of the exponential expression.

3

Divide both sides of the equation by 24.

5^x = 5\sqrt{5}

To isolate the exponential term

5^x

on one side of the equation.

4

Express the right side as a single power of 5.

5^x = 5^{1.5}

Using exponent rules where

5\sqrt{5} = 5^1 \cdot 5^{0.5} = 5^{1.5}

so that both sides have the same base.

5

Equate the exponents of the common base 5.

x = 1.5

Since the bases on both sides of the equation are equal and positive, their exponents must be equal.

Key Concept

Solving exponential equations by factoring and rewriting terms using a common base.

Estimated Time:1m 30s

Question 6Question →

If $27^x = 9^{x + 1}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 2

Answer

The value of

x

is

2

.

By rewriting

27

as

3^3

and

9

as

3^2

, the equation becomes

(3^3)^x = (3^2)^{x+1}

. Applying the power rule of exponents, this simplifies to

3^{3x} = 3^{2x+2}

. Since the bases are now the same, the exponents can be set equal to each other:

3x = 2x + 2

. Solving for

x

gives

x = 2

.

Step-by-Step Solution

1

Rewrite

27

and

9

as powers of

3

.

(3^3)^x = (3^2)^{x + 1}

To solve exponential equations with different bases, it is helpful to express them using a common base.

2

Apply the power of a power rule

(a^m)^n = a^{mn}

to simplify the exponents.

3^{3x} = 3^{2x + 2}

This simplifies the exponential expressions on both sides of the equation.

3

Set the exponents equal to each other.

3x = 2x + 2

Since the bases are equal (

3 = 3

), the exponents must also be equal for the equation to hold.

4

Solve the linear equation for

x

.

x = 2

Subtracting

2x

from both sides isolates the variable

x

.

Key Concept

Solving exponential equations by expressing both sides with a common base and equating exponents.

Estimated Time:45s

Question 7Question →

A solar energy farm has two arrays of solar panels, Array A and Array B. At the beginning of 2024 ( $t = 0$ ), the generating capacity of Array A is $50$ megawatts and increases by $11$ megawatts each year. At the beginning of 2024, the generating capacity of Array B is $32$ megawatts and increases exponentially by a constant percent each year. If the generating capacities of the two arrays are equal at the beginning of 2026 ( $t = 2$ ), what is the generating capacity, in megawatts, of Array B at the beginning of 2027 ( $t = 3$ )?

Show answer & explanation

Answer: 108

Answer

The correct answer is 108.

The capacity of Array A at

t = 2

is calculated using the linear growth model

A(t) = 50 + 11t

, which yields

A(2) = 50 + 11(2) = 72

megawatts. Since the capacity of Array B grows exponentially and starts at

32

megawatts, its model is

B(t) = 32 \cdot b^t

, where

b

is the annual growth factor. At

t = 2

, their capacities are equal, so

32 \cdot b^2 = 72

. Dividing both sides by

32

yields

b^2 = 2.25

. Taking the positive square root gives

b = 1.5

. To find the capacity of Array B at

t = 3

, we compute

B(3) = 32 \cdot (1.5)^3 = 108

megawatts.

Step-by-Step Solution

1

Write the linear model for the capacity of Array A.

A(t) = 50 + 11t

, where

t

is the number of years since the beginning of 2024.

Array A increases by a constant number of megawatts each year, which represents linear growth.

2

Calculate the capacity of Array A at

t = 2

.

A(2) = 50 + 11(2) = 72

megawatts.

We need to find the capacity of Array A at the beginning of 2026 to determine the capacity of Array B at that same time.

3

Represent the exponential model for the capacity of Array B.

B(t) = 32 \cdot b^t

, where

b

is the constant annual growth factor.

Array B increases exponentially by a constant percent each year, which is represented by a geometric sequence or exponential function.

4

Solve for the growth factor

b

using the capacity at

t = 2

.

32 \cdot b^2 = 72 \implies b^2 = 2.25 \implies b = 1.5

.

Since the capacities of both arrays are equal at the beginning of 2026 (

t = 2

), we set

B(2) = A(2) = 72

to find the positive base

b

.

5

Find the capacity of Array B at

t = 3

.

B(3) = 32 \cdot (1.5)^3 = 108

megawatts.

We substitute

t = 3

into the exponential model for Array B to find its capacity at the beginning of 2027.

Key Concept

Modeling linear and exponential growth, finding parameters of functions, and calculating values at specific time steps.

Alternative Method

Once the growth factor

b = 1.5

is found, the capacity of Array B at

t = 3

can be calculated by multiplying the capacity at

t = 2

by the growth factor:

B(3) = B(2) \cdot b = 72 \cdot 1.5 = 108

megawatts.

Estimated Time:1m 30s

Question 8Question →

At the beginning of a school year, a bookstore had $250$ copies of a textbook in stock. During the first week of classes, the bookstore sold $30\%$ of the textbooks in stock. During the second week, a new shipment increased the number of textbooks in stock by $40\%$ of the number of textbooks in stock at the end of the first week. How many copies of the textbook were in stock at the end of the second week?

Show answer & explanation

Answer: 245

Answer

245

To find the final quantity, we must calculate the successive percentage changes sequentially. First, a

30\%

decrease on the initial

250

textbooks leaves

70\%

of the stock:

250 \times 0.70 = 175

. Second, a

40\%

increase on this new amount of

175

textbooks results in

175 \times 1.40 = 245

textbooks in stock at the end of the second week.

Step-by-Step Solution

1

Find the number of textbooks remaining after the first week's sales.

175

Selling

30\%

of the

250

textbooks leaves

70\%

of them:

250 \times (1 - 0.30) = 175

.

2

Find the total number of textbooks in stock after the second week's shipment increase.

245

An increase of

40\%

on the

175

remaining textbooks results in

140\%

of that amount:

175 \times (1 + 0.40) = 245

.

Key Concept

Applying consecutive percent decreases and increases to changing base values.

Question 9Question →

If $4^{x+2} \cdot 8^{x-1} = 16^{x+1}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 3

Answer

3

To solve the equation

4^{x+2} \cdot 8^{x-1} = 16^{x+1}

, rewrite all bases in terms of base 2:

(2^2)^{x+2} \cdot (2^3)^{x-1} = (2^4)^{x+1}

. Simplifying using the power rule yields

2^{2x+4} \cdot 2^{3x-3} = 2^{4x+4}

. Applying the product rule on the left side gives

2^{(2x+4)+(3x-3)} = 2^{5x+1}

. Equating the exponents gives

5x + 1 = 4x + 4

. Solving for

x

results in

x = 3

.

Step-by-Step Solution

1

Express the bases 4, 8, and 16 as powers of 2.

4^{x+2} = (2^2)^{x+2} = 2^{2x+4}

,

8^{x-1} = (2^3)^{x-1} = 2^{3x-3}

, and

16^{x+1} = (2^4)^{x+1} = 2^{4x+4}

Writing all parts of the equation with a common base allows the exponents to be compared directly.

2

Combine the terms on the left side by adding their exponents.

2^{2x+4} \cdot 2^{3x-3} = 2^{(2x+4) + (3x-3)} = 2^{5x+1}

According to the product rule of exponents,

b^m \cdot b^n = b^{m+n}

when the bases are the same.

3

Equate the exponents from both sides of the equation.

5x + 1 = 4x + 4

If two exponential expressions with the same positive base (other than 1) are equal, their exponents must be equal.

4

Solve the linear equation for

x

.

x = 3

Subtract

4x

and

1

from both sides to isolate the variable

x

.

Key Concept

Solving exponential equations by finding a common base and applying exponent laws.

Question 10Question →

Two lines, $L_1$ and $L_2$ , are graphed in the $xy$ -plane. Line $L_1$ passes through the points $(2, 11)$ and $(6, 23)$ . If line $L_2$ is perpendicular to line $L_1$ and contains the point $(3, 10)$ , what is the $x$ -coordinate of the $x$ -intercept of line $L_2$ ?

Show answer & explanation

Answer: 33

Answer

33

To find the

x

-coordinate of the

x

-intercept of line

L_2

, first determine the slope of line

L_1

from the given points

(2, 11)

and

(6, 23)

using the formula

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

, which results in

m_1 = \frac{23 - 11}{6 - 2} = 3

. Since line

L_2

is perpendicular to line

L_1

, its slope must be the negative reciprocal of 3, which is

-\frac{1}{3}

. Using the point-slope form with the point

(3, 10)

, the equation of line

L_2

is

y - 10 = -\frac{1}{3}(x - 3)

, which simplifies to

y = -\frac{1}{3}x + 11

. Setting

y = 0

to find the

x

-intercept yields

0 = -\frac{1}{3}x + 11

, which simplifies to

x = 33

.

Step-by-Step Solution

1

Calculate the slope of line

L_1

using the points

(2, 11)

and

(6, 23)

with the slope formula

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

.

m_1 = \frac{23 - 11}{6 - 2} = \frac{12}{4} = 3

The slope of a line represents its rate of change and is required to find the relationship with perpendicular lines.

2

Find the slope of line

L_2

which is perpendicular to

L_1

.

m_2 = -\frac{1}{3}

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

3

Write the linear equation for line

L_2

using the point-slope formula

y - y_1 = m(x - x_1)

with the point

(3, 10)

and slope

m_2 = -\frac{1}{3}

.

y - 10 = -\frac{1}{3}(x - 3) \implies y = -\frac{1}{3}x + 11

Defining the equation of the line allows us to find the coordinates of any of its intercepts.

4

Set

y = 0

in the equation for line

L_2

to find the

x

-coordinate of the

x

-intercept.

0 = -\frac{1}{3}x + 11 \implies \frac{1}{3}x = 11 \implies x = 33

The

x

-intercept is the point where the line crosses the

x

-axis, which mathematically corresponds to

y = 0

.

Key Concept

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

x

-intercept of a linear graph represents the value of

x

when

y = 0

.

Question 11Question →

A digital marketing firm allocates its monthly advertising budget between search engine campaigns and social media campaigns. Let $x$ represent the amount, in thousands of dollars, spent on search engine campaigns, and let $y$ represent the amount, in thousands of dollars, spent on social media campaigns. The system of inequalities below represents the firm's monthly constraints:

\begin{aligned} x &\ge 2y \\ x + y &\le 25 \\ 3x + 4y &\le 80 \\ y &\ge 3 \end{aligned}

Based on these constraints, what is the maximum possible amount, in thousands of dollars, the firm can spend on social media campaigns?

Show answer & explanation

Answer: 8

Answer

8

To find the maximum possible value of

y

, we analyze the boundaries of the feasible region. The boundary lines of the system are

x = 2y

,

x + y = 25

,

3x + 4y = 80

, and

y = 3

. We can find the upper limit of

y

by combining the inequalities

x \ge 2y

and

3x + 4y \le 80

. Multiplying the first inequality by

3

gives

3x \ge 6y

. Adding

4y

to both sides yields

3x + 4y \ge 10y

. Because

3x + 4y \le 80

, it follows that

10y \le 80

, which simplifies to

y \le 8

. We verify that the point

(16, 8)

satisfies the remaining inequalities:

16 + 8 = 24 \le 25

and

8 \ge 3

, which are both true. Thus, the maximum possible value of

y

is

8

.

Step-by-Step Solution

1

Relate the variables using the constraints to establish an upper bound for

y

.

Since

x \ge 2y

, multiplying both sides by

3

gives

3x \ge 6y

.

This allows us to express the

3x

term in the cost inequality in terms of

y

to determine the maximum boundary.

2

Substitute

3x \ge 6y

into the inequality

3x + 4y \le 80

.

We get

6y + 4y \le 3x + 4y \le 80

, which simplifies to

10y \le 80

.

This establishes a direct upper limit for

y

based on the intersection of the two active boundary lines.

3

Solve the inequality

10y \le 80

for

y

.

y \le 8

.

This determines that the maximum possible value for

y

under these constraints is

8

.

4

Verify that the point corresponding to

y = 8

satisfies all other constraints in the system.

When

y = 8

, the boundary

x = 2y

gives

x = 16

. Checking

(16, 8)

against all inequalities:
-

16 \ge 2(8) \implies 16 \ge 16

(True)
-

16 + 8 \le 25 \implies 24 \le 25

(True)
-

3(16) + 4(8) \le 80 \implies 80 \le 80

(True)
-

8 \ge 3

(True)

We must verify that the optimal vertex lies within the feasible region defined by all four inequalities.

Key Concept

Maximizing a coordinate value within a bounded feasible region defined by a system of linear inequalities.

Question 12Question →

\begin{aligned} y &= x^2 - 8x + c \\ y &= 2x - 5 \end{aligned}

In the system of equations above, $c$ is a constant. If the system has two real solutions, $(x_1, y_1)$ and $(x_2, y_2)$ , such that the product of the $y$ -coordinates of the solutions, $y_1 y_2$ , is equal to $5$ , what is the value of $c$ ?

Show answer & explanation

Answer: 15

Answer

The value of the constant

c

is

15

.

Substituting the expression for

y

from the linear equation into the quadratic equation yields the single variable quadratic equation

x^2 - 10x + (c+5) = 0

. Using Vieta's formulas, the sum of the roots is

x_1 + x_2 = 10

and the product of the roots is

x_1 x_2 = c+5

. Substituting these relationships into the expanded product of the

y

-coordinates,

y_1 y_2 = (2x_1 - 5)(2x_2 - 5) = 4x_1 x_2 - 10(x_1 + x_2) + 25

, allows us to set up the equation

5 = 4(c+5) - 100 + 25

. Solving for

c

yields

15

. Checking the discriminant of the quadratic equation at

c=15

gives

100 - 80 = 20

, which is positive, confirming the existence of two distinct real solutions.

Step-by-Step Solution

1

Substitute the expression for

y

from the second equation into the first equation.

2x - 5 = x^2 - 8x + c \implies x^2 - 10x + (c+5) = 0

This substitution reduces the system to a single quadratic equation whose roots,

x_1

and

x_2

, represent the

x

-coordinates of the intersection points.

2

Apply Vieta's formulas to the resulting quadratic equation.

x_1 + x_2 = 10

and

x_1 x_2 = c+5

Vieta's formulas state that for a quadratic equation

ax^2 + bx + d = 0

, the sum of the roots is

-\frac{b}{a}

and the product of the roots is \frac{d}{a}.

3

Express the product of the

y

-coordinates,

y_1 y_2

, in terms of

x_1

and

x_2

using the linear relationship.

y_1 y_2 = (2x_1 - 5)(2x_2 - 5) = 4x_1 x_2 - 10(x_1 + x_2) + 25

Since both intersection points lie on the line

y = 2x - 5

, we can substitute

y_1 = 2x_1 - 5

and

y_2 = 2x_2 - 5

and expand.

4

Substitute the Vieta's formulas relations into the product equation and solve for

c

.

5 = 4(c + 5) - 10(10) + 25 \implies 5 = 4c + 20 - 100 + 25 \implies 5 = 4c - 55 \implies 60 = 4c \implies c = 15

By substituting the known values of

(x_1 + x_2)

and

(x_1 x_2)

and setting the product

y_1 y_2

to

5

, we obtain a linear equation in terms of

c

that we can solve directly.

Key Concept

Solving systems of linear-quadratic equations using algebraic substitution and Vieta's formulas.

Question 13Question →

A quadratic function $g$ is defined by $g(x) = 3(x - 5)^2 - 4$ . If the graph of $y = g(x)$ in the $xy$ -plane is translated $6$ units up to produce the graph of $y = f(x)$ , what is the $y$ -coordinate of the vertex of the graph of $y = f(x)$ ?

Show answer & explanation

Answer: 2

Answer

The correct answer is 2.

The original function

g(x) = 3(x - 5)^2 - 4

is in vertex form

y = a(x - h)^2 + k

, where the vertex is

(h, k)

. Therefore, the vertex of the graph of

g

is

(5, -4)

. Translating a graph upward by

6

units is represented by adding

6

to the function, so

f(x) = g(x) + 6

. This transformation shifts the vertex from

(5, -4)

to

(5, -4 + 6)

, which is

(5, 2)

. The

y

-coordinate of this new vertex is

2

.

Step-by-Step Solution

1

Identify the vertex of the original quadratic function.

The vertex of the graph of

g(x) = 3(x - 5)^2 - 4

is

(5, -4)

.

A quadratic function written in vertex form

y = a(x - h)^2 + k

has its vertex at

(h, k)

.

2

Determine the vertex of the translated function.

The vertex of the graph of

y = f(x)

is

(5, 2)

.

Translating a graph

6

units up increases the

y

-coordinate of all points, including the vertex, by

6

.

Key Concept

Vertex form of a quadratic function and vertical translation of graphs.

Estimated Time:45s

Question 14Question →

A commercial agricultural irrigation system distributes liquid fertilizer from a storage tank. The volume of fertilizer remaining in the tank, $V$ , in liters, $m$ minutes after the system is turned on is modeled by the equation:

V = 1,800 - 15(m - 3c)

where $c$ is the number of times the system's nozzles are cleaned during the irrigation process. During each cleaning cycle, the flow of fertilizer is completely paused, and no fertilizer is distributed. Based on the model, what is the duration, in minutes, of a single cleaning cycle?

Show answer & explanation

Answer: 3

Answer

The duration of a single cleaning cycle is 3 minutes.

In the model

V = 1,800 - 15(m - 3c)

, the term

(m - 3c)

represents the total time, in minutes, that the irrigation system is actively distributing fertilizer. Since the total elapsed time is

m

minutes and the system pauses during cleaning cycles, the term

3c

represents the total paused time for

c

cleanings. Therefore, the duration of a single cleaning cycle is

\frac{3c}{c} = 3

minutes.

Step-by-Step Solution

1

Analyze the structure of the equation to identify the meaning of each term.

The coefficient 15 is the active flow rate in liters per minute, and

(m - 3c)

is the active distribution time in minutes.

To understand how the time variables affect the volume of fertilizer remaining.

2

Relate the total elapsed time to the active time and the paused time.

The total elapsed time is

m

minutes, and the active time is

m - 3c

minutes, meaning the system is paused for a total of

3c

minutes.

To find the expression for the total duration of all cleaning cycles.

3

Determine the duration of a single cleaning cycle.

Since

c

cleaning cycles result in a total pause of

3c

minutes, each cycle lasts

\frac{3c}{c} = 3

minutes.

To calculate the duration of one individual cleaning cycle.

Key Concept

Interpreting Linear Relationships in Context

Question 15Question →

If $\frac{8}{x - 3} = 2$ , what is the value of $x$ ?

Show answer & explanation

Answer: 7

Answer

7

To solve the equation

\frac{8}{x - 3} = 2

, we clear the denominator by multiplying both sides of the equation by

x - 3

. This gives

8 = 2(x - 3)

. Distributing the

2

results in

8 = 2x - 6

. Adding

6

to both sides yields

14 = 2x

. Dividing both sides by

2

gives the solution

x = 7

.

Step-by-Step Solution

1

Multiply both sides of the equation by

x - 3

to eliminate the fraction.

8 = 2(x - 3)

To clear the denominator and simplify the equation.

2

Distribute the 2 to the terms inside the parentheses.

8 = 2x - 6

To expand the right-hand side of the equation.

3

Add 6 to both sides of the equation.

14 = 2x

To isolate the term with the variable

x

.

4

Divide both sides by 2 to solve for

x

.

x = 7

To find the final value of the variable.

Key Concept

Solving a rational equation that simplifies to a linear equation by multiplying by the common denominator.

Question 16Question →

A sample of a radioactive isotope decays such that its mass, in grams, is modeled by the function $M(t) = A \cdot d^{\frac{t}{12}}$ , where $t$ is the time in hours since the measurement began, and $A$ and $d$ are positive constants. The mass of the sample decreases by $75\%$ every $24$ hours. If the mass of the sample is $15$ grams when $t = 36$ , what was the initial mass, in grams, of the sample?

Show answer & explanation

Answer: 120

Answer

The initial mass of the sample was 120 grams.

The correct answer is 120. A decrease of

75\%

over 24 hours means the mass at

t+24

is

0.25

times the mass at

t

. According to the function,

M(t+24) = A \cdot d^{(t+24)/12} = A \cdot d^{t/12} \cdot d^2 = M(t) \cdot d^2

. Equating the two yields

d^2 = 0.25

, so

d = 0.5

. Substituting

t = 36

and

M(36) = 15

gives

15 = A \cdot (0.5)^{36/12} = A \cdot (0.5)^3 = 0.125A

. Solving for

A

gives

A = 120

.

Step-by-Step Solution

1

Relate the 24-hour decay rate to the exponent in the function to set up an equation for

d

.

d^2 = 0.25

Every 24 hours (

t

increases by 24), the mass decreases by

75\%

, so it becomes

25\%

(

0.25

) of its previous value. The exponent increases by

24/12 = 2

, multiplying the mass by

d^2

.

2

Solve for the decay base

d

.

d = 0.5

Since

d

is a positive constant, taking the square root of

0.25

gives

0.5

.

3

Substitute the given mass at

t = 36

into the model to solve for the initial mass

A

.

A = 120

Plugging

t = 36

and

M(36) = 15

into

M(t) = A \cdot (0.5)^{t/12}

gives

15 = A \cdot (0.5)^3

, which simplifies to

15 = 0.125A

.

Key Concept

Interpreting and solving exponential decay functions with fractional exponents

Alternative Method

Instead of solving for

d

first, recognize that

36

hours is exactly

1.5

intervals of

24

hours. Since the mass is multiplied by

0.25

(or

\frac{1}{4}

) every

24

hours, after

36

hours it will be multiplied by

(\frac{1}{4})^{1.5} = (\frac{1}{4})^{3/2} = \frac{1}{8}

of its initial value. Therefore,

15 = A \cdot \frac{1}{8} \implies A = 120

.

Estimated Time:3m 0s

Question 17Question →

If the expression $\frac{3x^2 - bx + 10}{x - 2}$ is equivalent to $3x - c - \frac{12}{x - 2}$ for all $x \neq 2$ , where $b$ and $c$ are positive constants, what is the value of $b$ ?

Show answer & explanation

Answer: 17

Answer

17

Multiplying both sides of the equivalence by

x - 2

results in

3x^2 - bx + 10 = (3x - c)(x - 2) - 12

. Expanding the right side gives

3x^2 - (6 + c)x + (2c - 12)

. By equating the coefficients of corresponding terms, we get

2c - 12 = 10

, which solves to

c = 11

. Substituting this into the

x

-coefficient equivalence

-b = -(6 + c)

gives

b = 6 + 11 = 17

.

Step-by-Step Solution

1

Multiply both sides of the expression by the denominator

x - 2

to equate the numerators.

3x^2 - bx + 10 = (3x - c)(x - 2) - 12

Since the rational expressions are equivalent for all

x \neq 2

, their numerators must be equal when written over a common denominator.

2

Expand and simplify the right side of the equation.

3x^2 - bx + 10 = 3x^2 - (6 + c)x + (2c - 12)

Expanding

(3x - c)(x - 2)

yields

3x^2 - 6x - cx + 2c

, and subtracting

12

gives the simplified polynomial expression.

3

Equate the constant terms to solve for

c

.

10 = 2c - 12 \Rightarrow c = 11

For the polynomials to be equivalent, their constant terms must be equal.

4

Equate the coefficients of the

x

terms to solve for

b

.

b = 6 + c \Rightarrow b = 6 + 11 = 17

Equating the coefficients of

x

gives

-b = -(6 + c)

, which simplifies to

b = 6 + c

.

Key Concept

Equivalence of rational and polynomial expressions via coefficient comparison

Alternative Method

Alternatively, you can evaluate the equivalence at a convenient value of

x

. For instance, substituting

x = 0

into the expression gives

-5 = -c + 6

, yielding

c = 11

. Then, evaluating the equation at another convenient value such as

x = 1

allows you to solve for

b

directly using the now-known value of

c

.

Estimated Time:1m 30s

Question 18Question →

If $x > 2$ and satisfies the equation

x - \frac{8}{x^2 - 2x} = 5 - \frac{4}{x - 2}

what is the value of $x$ ?

Show answer & explanation

Answer: 4

Answer

The correct answer is 4.

To solve the rational equation, we first determine the domain restrictions:

x \neq 0

and

x \neq 2

. We then clear the denominators by multiplying the entire equation by the least common denominator

x(x - 2)

, which simplifies the equation to the cubic form

x^3 - 7x^2 + 14x - 8 = 0

. Factoring the cubic polynomial yields

(x - 1)(x - 2)(x - 4) = 0

. The potential solutions are

x = 1

,

x = 2

, and

x = 4

. Since

x = 2

is an extraneous solution and the constraint requires

x > 2

,

x = 4

is the only valid solution.

Step-by-Step Solution

1

Determine the domain restrictions of the rational terms in the equation.

x \neq 0

and

x \neq 2

Division by zero is undefined, so the denominators

x(x-2)

and

x-2

cannot be zero.

2

Multiply the entire equation by the least common denominator

x(x - 2)

to eliminate the fractions.

x^2(x - 2) - 8 = 5x(x - 2) - 4x

Multiplying by the common denominator converts the rational equation into an equivalent polynomial equation within the defined domain.

3

Expand the terms and collect them on one side to form a standard polynomial equation.

x^3 - 7x^2 + 14x - 8 = 0

Simplifying the expressions allows us to find the roots of the polynomial.

4

Factor the cubic polynomial using the rational root theorem or synthetic division.

(x - 1)(x - 2)(x - 4) = 0

Since

x = 1

makes the polynomial zero, we factor out

(x-1)

to get the remaining quadratic factor

x^2 - 6x + 8

, which factors into

(x - 2)(x - 4)

.

5

Identify the valid solution based on the domain restriction and the given inequality constraint.

x = 4

The value

x = 2

is extraneous because it makes the original equation undefined. The value

x = 1

is rejected because the problem specifies

x > 2

.

Key Concept

Solving rational equations by clearing denominators, factoring polynomials, identifying extraneous solutions, and applying inequality constraints.

Question 19Question →

What is the value of the real solution to the equation below?

\sqrt{3x - 2} - \sqrt{x + 3} = 1

Show answer & explanation

Answer: 6

Answer

The only real solution to the equation is 6.

Substituting

x = 6

into the original equation yields

\sqrt{3(6) - 2} - \sqrt{6 + 3} = \sqrt{16} - \sqrt{9} = 4 - 3 = 1

, which is a true statement. Therefore, the only real solution is 6.

Step-by-Step Solution

1

Isolate one of the radical terms

\sqrt{3x - 2} = \sqrt{x + 3} + 1

Preparing the equation to square both sides.

2

Square both sides of the equation

3x - 2 = x + 4 + 2\sqrt{x + 3}

Eliminating one of the square root radicals.

3

Isolate the remaining radical term and simplify

x - 3 = \sqrt{x + 3}

Simplifying the equation by isolating the second radical and dividing both sides by 2.

4

Square both sides again to eliminate the second radical

x^2 - 6x + 9 = x + 3

Converting the radical equation into a polynomial equation.

5

Write in standard quadratic form and factor

(x - 6)(x - 1) = 0

Setting the quadratic equation to zero and factoring to find potential solutions.

6

Verify solutions in the original equation to check for extraneous solutions

x = 6

is valid;

x = 1

is extraneous

Squaring both sides of an equation can introduce extraneous solutions that do not satisfy the original equation.

Key Concept

Solving equations containing radical expressions and identifying extraneous solutions.

Estimated Time:2m 30s

Question 20Question →

In 2024, a city recycled a certain number of tons of plastic. In 2025, the number of tons of plastic recycled was $15\%$ greater than the number of tons recycled in 2024. In 2026, the number of tons of plastic recycled was $20\%$ less than the number of tons recycled in 2025. If the city recycled $5,520$ tons of plastic in 2026, how many tons of plastic did the city recycle in 2024?

Show answer & explanation

Answer: 6000

Answer

The city recycled

6,000

tons of plastic in 2024.

To find the number of tons of plastic recycled in 2024, we must work backward from the 2026 value of 5,520 tons. Let

x

represent the tons recycled in 2025. Since the 2026 value represents a

20\%

decrease from 2025, we have

0.80x = 5,520

, which simplifies to

x = 6,900

tons. Next, let

y

represent the tons recycled in 2024. Since the 2025 value represents a

15\%

increase from 2024, we have

1.15y = 6,900

. Solving for

y

yields

y = \frac{6,900}{1.15} = 6,000

tons.

Step-by-Step Solution

1

Set up an equation to find the amount of plastic recycled in 2025 using the 2026 amount of 5,520 tons.

The amount of plastic recycled in 2025 was

6,900

tons.

The 2026 amount is a

20\%

decrease from the 2025 amount, so

5,520

is equal to

80\%

of the 2025 amount. Solving

0.80x = 5,520

gives

x = 6,900

.

2

Set up an equation to find the amount of plastic recycled in 2024 using the 2025 amount of 6,900 tons.

The amount of plastic recycled in 2024 was

6,000

tons.

The 2025 amount is a

15\%

increase from the 2024 amount, so

6,900

is equal to

115\%

of the 2024 amount. Solving

1.15y = 6,900

gives

y = 6,000

.

Key Concept

Working backward through multiple sequential percentage changes.

Answer

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