All practice questions

Question 1701Question →

In the $xy$ -plane, a line passes through the point $(5, -1)$ and has a slope of $\frac{2}{5}$ . If the line also passes through the point $(15, p)$ , what is the value of $p$ ?

Show answer & explanation

Answer: 3

Answer

The value of

p

is

3

.

To find the value of

p

, the equation of the line can be established using the point-slope form:

y - y_1 = m(x - x_1)

. Substituting the given slope

m = \frac{2}{5}

and the point

(5, -1)

gives

y - (-1) = \frac{2}{5}(x - 5)

. Simplifying this yields

y + 1 = \frac{2}{5}x - 2

, which reduces to

y = \frac{2}{5}x - 3

. Substituting the point

(15, p)

into this equation gives

p = \frac{2}{5}(15) - 3 = 6 - 3 = 3

.

Step-by-Step Solution

1

Determine the equation of the line using point-slope form.

y = \frac{2}{5}x - 3

The equation of a line with slope

m

passing through a point

(x_1, y_1)

is

y - y_1 = m(x - x_1)

. Substituting

m = \frac{2}{5}

and the point

(5, -1)

gives

y - (-1) = \frac{2}{5}(x - 5)

. Simplifying this equation results in

y + 1 = \frac{2}{5}x - 2

, which becomes

y = \frac{2}{5}x - 3

.

2

Substitute the point

(15, p)

into the linear equation.

p = 3

Since the line passes through the point

(15, p)

, the coordinates must satisfy the equation of the line. Substituting

x = 15

and

y = p

into

y = \frac{2}{5}x - 3

gives

p = \frac{2}{5}(15) - 3

, which simplifies to

p = 6 - 3 = 3

.

Key Concept

Using the slope and a point on a line to find another coordinate along the same line.

Question 1702Question →

In the quadratic equation $3x^2 + kx + 12 = 0$ , where $k$ is a constant, the difference between the two real solutions is $4$ . What is the value of $k^2$ ?

160

192

288

0

Show answer & explanation

Answer: 288

Answer

The correct value of

k^2

is 288.

The correct value is 288. The difference between the two solutions

r_1

and

r_2

of the quadratic equation

ax^2 + bx + c = 0

is given by

|r_1 - r_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}

. Substituting

a = 3

,

b = k

, and

c = 12

, we get

|r_1 - r_2| = \frac{\sqrt{k^2 - 4(3)(12)}}{3} = \frac{\sqrt{k^2 - 144}}{3}

. Setting this equal to the given difference of

4

yields

\frac{\sqrt{k^2 - 144}}{3} = 4

. Multiplying by 3 gives

\sqrt{k^2 - 144} = 12

, and squaring both sides gives

k^2 - 144 = 144

, which simplifies to

k^2 = 288

. Alternatively, using Viete's formulas,

r_1 + r_2 = -\frac{k}{3}

and

r_1 r_2 = 4

. Using the identity

(r_1 - r_2)^2 = (r_1 + r_2)^2 - 4r_1 r_2

, we have

4^2 = (-\frac{k}{3})^2 - 4(4)

, which simplifies to

16 = \frac{k^2}{9} - 16

, leading to

\frac{k^2}{9} = 32

, or

k^2 = 288

.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation

3x^2 + kx + 12 = 0

.

The coefficients are

a = 3

,

b = k

, and

c = 12

.

These values are needed to apply the difference of roots formula or Viete's relations.

2

State the formula for the difference between the two solutions

r_1

and

r_2

of a quadratic equation.

|r_1 - r_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}

. Substituting the coefficients gives

|r_1 - r_2| = \frac{\sqrt{k^2 - 4(3)(12)}}{3} = \frac{\sqrt{k^2 - 144}}{3}

.

This formula relates the difference of the roots directly to the coefficients of the equation.

3

Set the expression for the difference of the roots equal to the given difference of

4

and solve for

k^2

.

\frac{\sqrt{k^2 - 144}}{3} = 4 \Rightarrow \sqrt{k^2 - 144} = 12 \Rightarrow k^2 - 144 = 144 \Rightarrow k^2 = 288

.

Squaring both sides and isolating

k^2

yields its value.

Key Concept

Difference of roots and discriminant of a quadratic equation

Alternative Method

Use Viete's relations: The sum of the roots is

r_1 + r_2 = -\frac{k}{3}

and the product of the roots is

r_1 r_2 = \frac{12}{3} = 4

. The difference between the roots is given as

|r_1 - r_2| = 4

. Square this relation to get

(r_1 - r_2)^2 = 16

. Expand and rewrite the identity as

(r_1 + r_2)^2 - 4r_1 r_2 = 16

. Substituting the sum and product, we get

(-\frac{k}{3})^2 - 4(4) = 16 \Rightarrow \frac{k^2}{9} - 16 = 16 \Rightarrow \frac{k^2}{9} = 32 \Rightarrow k^2 = 288

.

Estimated Time:2m 0s

Question 1703Question →

A polynomial $p$ is defined by $p(x) = (x - 4)(x^2 + ax + 3)$ , where $a$ is a constant. If $p(1) = -18$ , what is the value of $a$ ?

Show answer & explanation

Answer: 2

Answer

2

Substituting

x = 1

into

p(x) = (x - 4)(x^2 + ax + 3)

gives

p(1) = (1 - 4)(1^2 + a(1) + 3) = -3(4 + a)

. Since we are given that

p(1) = -18

, we can set

-3(4 + a) = -18

. Dividing both sides of the equation by

-3

gives

4 + a = 6

. Subtracting

4

from both sides gives

a = 2

.

Step-by-Step Solution

1

Substitute

x = 1

into the expression for

p(x)

.

p(1) = (1 - 4)(1^2 + a(1) + 3) = -3(4 + a)

We are given that

p(1) = -18

, so substituting

x = 1

will allow us to set up an equation in terms of the constant

a

.

2

Set the simplified expression equal to

-18

and solve for

a

.

a = 2

Dividing both sides of

-3(4 + a) = -18

by

-3

yields

4 + a = 6

, and subtracting

4

from both sides gives the value of

a

.

Key Concept

Using given function values to solve for unknown coefficients in a factored polynomial.

Question 1704Question →

In the $xy$ -plane, the circle with equation $x^2 + y^2 = 13$ intersects the line with equation $y - x = 1$ at two points, $(x_1, y_1)$ and $(x_2, y_2)$ . What is the value of $y_1 + y_2$ ?

-1

1

3

-5

Show answer & explanation

Answer: 1

Answer

The sum of the

y

-coordinates of the intersection points is

1

.

The correct answer is

1

. Substituting

x = y - 1

from the linear equation into the circle equation yields

(y-1)^2 + y^2 = 13

. Expanding and simplifying gives the quadratic equation

2y^2 - 2y - 12 = 0

. Dividing by 2 yields

y^2 - y - 6 = 0

, which factors into

(y - 3)(y + 2) = 0

. Therefore, the

y

-coordinates of the intersection points are

3

and

-2

. The sum of these coordinates is

3 + (-2) = 1

.

Step-by-Step Solution

1

Rearrange the linear equation to solve for

x

in terms of

y

.

x = y - 1

Expressing

x

in terms of

y

allows us to substitute it into the circle equation to find the

y

-coordinates directly.

2

Substitute

x = y - 1

into the circle equation

x^2 + y^2 = 13

.

(y - 1)^2 + y^2 = 13

This substitution reduces the system of equations to a single quadratic equation in terms of

y

.

3

Expand and simplify the quadratic equation into standard form

ay^2 + by + c = 0

.

y^2 - 2y + 1 + y^2 = 13 \implies 2y^2 - 2y - 12 = 0

Expanding the squared term and combining like terms puts the equation in a form that can be solved.

4

Divide the equation by 2 and factor the simplified quadratic equation.

y^2 - y - 6 = 0 \implies (y - 3)(y + 2) = 0

Simplifying and factoring helps find the individual

y

-coordinates of the two intersection points.

5

Solve for

y

and calculate the sum of the coordinates.

y = 3

or

y = -2

, and

3 + (-2) = 1

Solving the factored equation gives the

y

-coordinates of the two intersection points, which are then added to find the final value.

Key Concept

Solving a nonlinear system of equations representing a circle and a line via substitution.

Alternative Method

Instead of solving for

x

first, we can substitute

y = x + 1

into the circle equation to find the

x

-coordinates:

x^2 + (x + 1)^2 = 13 \implies 2x^2 + 2x - 12 = 0 \implies x^2 + x - 6 = 0 \implies (x + 3)(x - 2) = 0

. This gives

x = -3

or

x = 2

. We then find the corresponding

y

-values using

y = x + 1

: for

x = -3

,

y = -2

; for

x = 2

,

y = 3

. The sum of the

y

-coordinates is

-2 + 3 = 1

.

Estimated Time:1m 30s

Question 1705Question →

A linear relationship between $x$ and $y$ is represented by the values in the table below.

$x$	$y$
$-2$	$a$
$1$	$5$
$4$	$17$
$7$	$b$

What is the value of $b - a$ ?

2.25

9

36

22

Show answer & explanation

Answer: 36

Answer

36

The constant rate of change (slope) of the relationship is 4, which is found by dividing the difference in

y

-values by the difference in

x

-values for the given points:

\frac{17 - 5}{4 - 1} = 4

. The value of

b - a

represents the change in

y

over the interval from

x = -2

to

x = 7

. The length of this interval is

7 - (-2) = 9

. Multiplying the rate of change by this interval length gives the total change in

y

:

4 \times 9 = 36

. Alternatively, solving for the equation yields

y = 4x + 1

, where substituting

x = -2

gives

a = -7

and substituting

x = 7

gives

b = 29

, and their difference is

29 - (-7) = 36

.

Step-by-Step Solution

1

Find the constant rate of change (slope) of the linear relationship using the points

(1, 5)

and

(4, 17)

.

The slope

m

is

\frac{17 - 5}{4 - 1} = \frac{12}{3} = 4

.

Since the relationship is linear, the rate of change is constant between any two points.

2

Determine the value of

a

when

x = -2

.

Using the point

(1, 5)

and moving to

x = -2

, the change in

x

is

-3

. The corresponding change in

y

is

4 \times (-3) = -12

. Thus,

a = 5 - 12 = -7

.

This establishes the value of the first variable in the expression.

3

Determine the value of

b

when

x = 7

.

Using the point

(4, 17)

and moving to

x = 7

, the change in

x

is

+3

. The corresponding change in

y

is

4 \times 3 = 12

. Thus,

b = 17 + 12 = 29

.

This establishes the value of the second variable in the expression.

4

Calculate the value of

b - a

.

b - a = 29 - (-7) = 29 + 7 = 36

.

Subtracting a negative number is equivalent to adding its positive counterpart.

Key Concept

Linear rate of change and evaluation of linear relationships from tables of values

Alternative Method

Instead of calculating the individual values of

a

and

b

, recognize that

b - a

is the total change in

y

over the interval from

x = -2

to

x = 7

. The change in

x

is

7 - (-2) = 9

. Since the constant rate of change (slope) is

4

, the change in

y

is simply

4 \times 9 = 36

.

Estimated Time:1m 30s

Question 1706Question →

If $x$ is the real solution to the equation $\sqrt{5x + 6} - \sqrt{x + 3} = 3$ , what is the value of $x$ ?

Show answer & explanation

Answer: 6

Answer

6

The correct answer is

6

. Isolating the first radical gives

\sqrt{5x+6} = \sqrt{x+3} + 3

. Squaring both sides yields

5x+6 = x+12+6\sqrt{x+3}

, which simplifies to

4x-6 = 6\sqrt{x+3}

. Dividing by

2

and squaring both sides again gives

(2x-3)^2 = 9(x+3)

, leading to the quadratic equation

4x^2 - 21x - 18 = 0

. Factoring this equation yields

(4x+3)(x-6)=0

, which gives the potential solutions

x = 6

and

x = -\frac{3}{4}

. Checking both solutions in the original equation shows that

x = 6

is a valid solution because

\sqrt{36} - \sqrt{9} = 6 - 3 = 3

. The solution

x = -\frac{3}{4}

is extraneous because

\sqrt{2.25} - \sqrt{2.25} = 0 \neq 3

.

Step-by-Step Solution

1

Isolate the first radical term and square both sides.

5x + 6 = x + 12 + 6\sqrt{x + 3}

Isolating one radical term simplifies the squaring process to eliminate that radical.

2

Isolate the remaining radical term and simplify the expression.

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

Simplifying and isolating the remaining radical term prepares the equation for a second squaring step.

3

Square both sides to eliminate the remaining radical and form a quadratic equation.

4x^2 - 21x - 18 = 0

Squaring both sides removes the last radical, turning the radical equation into a polynomial equation.

4

Factor the quadratic equation to find all candidate solutions.

x = 6

or

x = -\frac{3}{4}

Factoring

(4x + 3)(x - 6) = 0

gives the roots of the quadratic equation.

5

Substitute candidate solutions back into the original equation to identify extraneous roots.

The only valid solution is

x = 6

; the solution

x = -\frac{3}{4}

is extraneous.

Squaring equations can introduce extraneous solutions, so candidate solutions must be verified in the original equation.

Key Concept

Solving equations with multiple radicals by isolating terms, squaring both sides, and verifying candidate solutions for extraneous roots.

Question 1707Question →

In the $xy$ -plane, the graph of the quadratic function $f(x) = -(x - d)^2 + d^2$ , where $d$ is a positive constant, has vertex $V$ and intersects the $x$ -axis at points $P$ and $Q$ . If the area of triangle $PVQ$ is $64$ , what is the value of $d$ ?

2

4

8

16

Show answer & explanation

Answer: 4

Answer

The value of

d

is

4

.

The function

f(x) = -(x - d)^2 + d^2

is in vertex form,

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

, so its vertex is

V(d, d^2)

. Since

d > 0

, the vertex is in the first quadrant, and the height of the triangle is

d^2

. Setting

f(x) = 0

gives the

x

-intercepts

P(0,0)

and

Q(2d,0)

, meaning the base of the triangle has a length of

2d

. Using the area of a triangle formula, the area is

\frac{1}{2} \times 2d \times d^2 = d^3

. Since the area is given as

64

, we set

d^3 = 64

, which yields

d = 4

.

Step-by-Step Solution

1

Identify the vertex

V

of the quadratic function

f(x) = -(x - d)^2 + d^2

.

The vertex is

V(d, d^2)

.

The function is written in vertex form,

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

, where the vertex is

(h, k)

.

2

Find the

x

-intercepts

P

and

Q

of the function by setting

f(x) = 0

.

x = 0

and

x = 2d

.

Setting

-(x - d)^2 + d^2 = 0

gives

(x - d)^2 = d^2

, which simplifies to

x - d = \pm d

.

3

Calculate the area of triangle

PVQ

in terms of

d

.

The area is

d^3

.

The base of the triangle along the

x

-axis is the distance between the intercepts,

2d - 0 = 2d

. The height is the

y

-coordinate of the vertex,

d^2

. The area is

\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} (2d)(d^2) = d^3

.

4

Solve for

d

using the given area of

64

.

d = 4

.

Setting the area expression

d^3

equal to

64

and taking the cube root of both sides gives

d = 4

.

Key Concept

Finding the vertex and intercepts of a quadratic function in vertex form and applying geometric formulas to analyze the graph.

Question 1708Question →

What is the sum of all real solutions to the equation $\frac{x - 5}{x - 1} = \frac{2}{x + 1} - \frac{8}{x^2 - 1}$ ?

1

5

6

-6

Show answer & explanation

Answer: 5

Answer

The sum of all real solutions to the equation is

5

.

Multiplying the equation by the least common denominator,

(x - 1)(x + 1)

, yields

(x - 5)(x + 1) = 2(x - 1) - 8

. Expanding and simplifying leads to the quadratic equation

x^2 - 6x + 5 = 0

, which factors as

(x - 1)(x - 5) = 0

. This gives potential solutions of

x = 1

and

x = 5

. Since

x = 1

makes the denominators of the original terms zero, it is extraneous and must be discarded. The only valid solution is

x = 5

, and therefore the sum of all real solutions is

5

.

Step-by-Step Solution

1

Find the least common denominator of the rational terms.

The denominators are

x - 1

,

x + 1

, and

x^2 - 1

. Since

x^2 - 1 = (x - 1)(x + 1)

, the least common denominator is

(x - 1)(x + 1)

.

Finding the least common denominator allows us to eliminate the fractions by multiplying both sides.

2

Multiply the entire equation by the least common denominator

(x - 1)(x + 1)

, assuming

x \neq 1

and

x \neq -1

.

(x - 5)(x + 1) = 2(x - 1) - 8

This clears all rational expressions, leaving a polynomial equation.

3

Expand both sides of the equation and combine like terms.

x^2 - 4x - 5 = 2x - 2 - 8 \Rightarrow x^2 - 4x - 5 = 2x - 10

Expanding allows the simplification of terms on each side of the equation.

4

Rearrange the equation to set it equal to zero.

x^2 - 6x + 5 = 0

Moving all terms to one side forms a standard quadratic equation which can then be solved.

5

Factor the quadratic equation.

(x - 1)(x - 5) = 0

, which gives potential solutions

x = 1

and

x = 5

.

Factoring allows us to find the roots of the quadratic equation.

6

Check the potential solutions in the original equation to identify any extraneous solutions.

Substituting

x = 1

results in division by zero in the terms

\frac{x - 5}{x - 1}

and

\frac{8}{x^2 - 1}

, so

x = 1

is extraneous. Substituting

x = 5

yields

0 = 0

, meaning

x = 5

is a valid real solution.

Multiplying by variables can introduce extraneous solutions that make the original denominators zero.

7

Calculate the sum of all valid solutions.

Since

x = 5

is the only valid solution, the sum of all real solutions is

5

.

The question asks for the sum of all real solutions.

Key Concept

Solving rational equations by clearing denominators and checking for extraneous solutions.

Question 1709Question →

\begin{aligned} 3x - 2y &= 14 \\ 5x + 6y &= 42 \end{aligned}

If $(x, y)$ is the solution to the system of equations above, what is the value of $x - y$ ?

2

4

5

8

Show answer & explanation

Answer: 4

Answer

The value of

x - y

is

4

.

To find the value of

x - y

, we first solve the system of linear equations. Multiplying the first equation,

3x - 2y = 14

, by

3

gives

9x - 6y = 42

. Adding this equation to the second equation,

5x + 6y = 42

, eliminates

y

and yields

14x = 84

, which simplifies to

x = 6

. Substituting

x = 6

back into the first equation gives

3(6) - 2y = 14

, or

18 - 2y = 14

, which simplifies to

2y = 4

, so

y = 2

. Therefore, the value of

x - y

is

6 - 2 = 4

.

Step-by-Step Solution

1

Multiply the first equation by

3

to align the coefficients of the

y

terms for elimination.

9x - 6y = 42

This makes the coefficients of

y

opposite in sign and equal in magnitude to the second equation.

2

Add the modified first equation to the second equation to eliminate

y

and solve for

x

.

(9x - 6y) + (5x + 6y) = 42 + 42 \implies 14x = 84 \implies x = 6

Adding the equations eliminates

y

, resulting in a single-variable linear equation.

3

Substitute

x = 6

back into the first equation to solve for

y

.

3(6) - 2y = 14 \implies 18 - 2y = 14 \implies -2y = -4 \implies y = 2

Substituting the known value of

x

yields the value of the other coordinate.

4

Evaluate the expression

x - y

using the values

x = 6

and

y = 2

.

x - y = 6 - 2 = 4

This computes the requested quantity.

Key Concept

Solving systems of linear equations using the elimination method and evaluating a linear combination of the variables.

Question 1710Question →

If $4^x - 3^{x - 0.5} = 3^{x + 0.5} - 2^{2x-1}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 1.5

Answer

The correct answer is

1.5

(or

\frac{3}{2}

)

By rewriting

4^x

as

2^{2x}

and grouping the base 2 and base 3 terms, the equation becomes

2^{2x} + 2^{2x-1} = 3^{x+0.5} + 3^{x-0.5}

. Factoring out the variable terms gives

2^{2x}(1 + 2^{-1}) = 3^x(3^{0.5} + 3^{-0.5})

. Simplifying the coefficients results in

\frac{3}{2} \cdot 2^{2x} = \frac{4}{\sqrt{3}} \cdot 3^x

. Dividing both sides to separate the variables and constants yields

\frac{2^{2x}}{3^x} = \frac{8}{3\sqrt{3}}

. Representing both sides with the common base

\frac{4}{3}

gives

\left(\frac{4}{3}\right)^x = \left(\frac{4}{3}\right)^{1.5}

. Equating the exponents leads to

x = 1.5

.

Step-by-Step Solution

1

Rewrite the base 4 term as a power of 2

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

Expressing exponential terms with prime bases simplifies finding relationships between them.

2

Group terms with like bases on opposite sides of the equation

2^{2x} + 2^{2x-1} = 3^{x+0.5} + 3^{x-0.5}

Separating different base systems allows us to factor out common exponential functions.

3

Factor out the common exponential factors from each side

2^{2x}(1 + 2^{-1}) = 3^x(3^{0.5} + 3^{-0.5})

This isolates the constant coefficients from the exponential functions of

x

.

4

Simplify the constant terms on both sides of the equation

2^{2x}\left(\frac{3}{2}\right) = 3^x\left(\sqrt{3} + \frac{1}{\sqrt{3}}\right) = 3^x\left(\frac{4}{\sqrt{3}}\right)

Reducing the coefficients makes it easier to combine the constant terms.

5

Divide both sides to group the exponential terms together

\frac{2^{2x}}{3^x} = \frac{4}{\sqrt{3}} \cdot \frac{2}{3} = \frac{8}{3\sqrt{3}}

This sets up the equation in the form of

f(x) = C

where

f(x)

has a combined base.

6

Express both sides of the equation with a common base of

\frac{4}{3}

\left(\frac{4}{3}\right)^x = \left(\frac{4}{3}\right)^{1.5}

Since

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

, the left side simplifies to

\left(\frac{4}{3}\right)^x

. The right side can be rewritten as

\frac{8}{3\sqrt{3}} = \frac{2^3}{3^{1.5}} = \frac{(4^{0.5})^3}{3^{1.5}} = \frac{4^{1.5}}{3^{1.5}} = \left(\frac{4}{3}\right)^{1.5}

.

7

Equate the exponents

x = 1.5

Since the bases are identical and positive, the exponents must be equal.

Key Concept

Solving exponential equations using base conversion, exponent rules, and factoring.

Question 1711Question →

A municipal recycling facility processes plastic waste. The total mass of unprocessed plastic, $M$ , in tons, remaining at the facility $h$ hours after the facility opens on a given day can be modeled by the equation $M = 142 - 8.5h$ . Which of the following is the best interpretation of the number $8.5$ in this context?

The facility has

142

tons of unprocessed plastic when it opens.

The mass of unprocessed plastic at the facility decreases by

8.5

tons each hour.

The facility takes

8.5

hours to process each ton of plastic.

The mass of unprocessed plastic at the facility decreases by

133.5

tons each hour.

Show answer & explanation

Answer: The mass of unprocessed plastic at the facility decreases by $8.5$ tons each hour.

Answer

The mass of unprocessed plastic at the facility decreases by

8.5

tons each hour.

In the equation

M = 142 - 8.5h

, the total mass of unprocessed plastic

M

decreases by

8.5

tons for each hour

h

that passes. The number

8.5

is the magnitude of the slope of the linear relationship, which represents the rate of change. Since the coefficient of

h

is negative, the amount of unprocessed plastic decreases by

8.5

tons per hour.

Step-by-Step Solution

1

Identify the structure of the linear equation

M = 142 - 8.5h

.

The equation is in the form

y = b + mx

, where

b

is the y-intercept (

142

) and

m

is the slope (

-8.5

).

This helps separate the initial value from the rate of change.

2

Analyze the coefficient of the independent variable

h

.

The coefficient is

-8.5

.

The coefficient of

h

represents the rate of change of the mass

M

per hour.

3

Interpret the meaning of the rate of change in context.

A rate of change of

-8.5

means that the mass of unprocessed plastic decreases by

8.5

tons for each hour that passes.

The negative sign indicates a decrease, and the unit is tons per hour.

Key Concept

Interpreting the slope of a linear relationship in a real-world context.

Question 1712Question →

If $\frac{2}{x} + \frac{3}{x - 2} = 2$ and $x > 1$ , what is the value of $x - 1$ ?

Show answer & explanation

Answer: 3

Answer

The correct answer is 3.

By multiplying both sides of the equation by the common denominator

x(x - 2)

, we obtain the quadratic equation

2x^2 - 9x + 4 = 0

. Factoring this equation yields the solutions

x = 0.5

and

x = 4

. Since the problem specifies that

x > 1

, we select

x = 4

. Substituting this value into the expression

x - 1

gives the final answer of 3.

Step-by-Step Solution

1

Multiply the equation by the common denominator

x(x - 2)

to eliminate the denominators.

2(x - 2) + 3x = 2x(x - 2)

This clears the fractions and allows us to work with a polynomial equation.

2

Expand and simplify both sides of the equation.

5x - 4 = 2x^2 - 4x

Combining like terms simplifies the expression before solving.

3

Rearrange the terms to set the quadratic equation equal to zero.

2x^2 - 9x + 4 = 0

Putting the equation in standard form

ax^2 + bx + c = 0

is necessary for factoring.

4

Factor the quadratic expression.

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

Factoring allows us to find the roots of the quadratic equation.

5

Solve for

x

and apply the condition

x > 1

.

x = 4

The equation yields two solutions:

x = 0.5

and

x = 4

. The condition

x > 1

excludes

x = 0.5

, leaving

x = 4

as the only valid solution.

6

Calculate the value of the requested expression

x - 1

.

3

Substituting

x = 4

into

x - 1

gives

4 - 1 = 3

.

Key Concept

Solving rational equations by clearing denominators and solving the resulting quadratic equation while considering domain constraints.

Alternative Method

Instead of factoring, the quadratic formula can be used to solve

2x^2 - 9x + 4 = 0

, where

x = \frac{9 \pm \sqrt{(-9)^2 - 4(2)(4)}}{2(2)} = \frac{9 \pm 5}{4}

, giving

x = 0.5

and

x = 4

. Applying

x > 1

leaves

x = 4

, so

x - 1 = 3

.

Estimated Time:1m 30s

Question 1713Question →

For all $x > 0$ , which of the following is equivalent to the expression $\frac{4x^2 - 25}{2x^{\frac{3}{2}} + 5x^{\frac{1}{2}}}$ ?

2x - 5

2x^{\frac{3}{2}} - 5x^{\frac{1}{2}}

2x^{\frac{1}{2}} - 5x^{-\frac{1}{2}}

2x^{\frac{1}{2}} + 5x^{-\frac{1}{2}}

Show answer & explanation

Answer: $2x^{\frac{1}{2}} - 5x^{-\frac{1}{2}}$

Answer

The correct equivalent expression is

2x^{\frac{1}{2}} - 5x^{-\frac{1}{2}}

.

To simplify the expression

\frac{4x^2 - 25}{2x^{\frac{3}{2}} + 5x^{\frac{1}{2}}}

, we first factor the numerator and the denominator. The numerator is a difference of squares:

4x^2 - 25 = (2x - 5)(2x + 5)

. In the denominator, we can factor out

x^{\frac{1}{2}}

to get

x^{\frac{1}{2}}(2x + 5)

. Substituting these factored forms gives

\frac{(2x - 5)(2x + 5)}{x^{\frac{1}{2}}(2x + 5)}

. Canceling the common factor

(2x + 5)

yields

\frac{2x - 5}{x^{\frac{1}{2}}}

. Dividing each term in the numerator by the denominator gives

\frac{2x}{x^{\frac{1}{2}}} - \frac{5}{x^{\frac{1}{2}}}

. Applying exponent rules, this simplifies to

2x^{1 - \frac{1}{2}} - 5x^{-\frac{1}{2}} = 2x^{\frac{1}{2}} - 5x^{-\frac{1}{2}}

, which is the correct expression.

Step-by-Step Solution

1

Factor the numerator of the expression as a difference of squares.

4x^2 - 25 = (2x - 5)(2x + 5)

To simplify the rational expression, we need to factor both the numerator and the denominator to identify common factors.

2

Factor out the common term

x^{\frac{1}{2}}

from the denominator.

2x^{\frac{3}{2}} + 5x^{\frac{1}{2}} = x^{\frac{1}{2}}(2x + 5)

Factoring out

x^{\frac{1}{2}}

reveals the common binomial factor

(2x + 5)

in the denominator.

3

Substitute the factored forms back into the original expression and cancel the common factor

(2x + 5)

.

\frac{(2x - 5)(2x + 5)}{x^{\frac{1}{2}}(2x + 5)} = \frac{2x - 5}{x^{\frac{1}{2}}}

Since

x > 0

, the term

2x + 5

is non-zero, allowing us to cancel it from both the numerator and the denominator.

4

Divide each term in the numerator by

x^{\frac{1}{2}}

and apply the rules of exponents.

\frac{2x}{x^{\frac{1}{2}}} - \frac{5}{x^{\frac{1}{2}}} = 2x^{1 - \frac{1}{2}} - 5x^{-\frac{1}{2}} = 2x^{\frac{1}{2}} - 5x^{-\frac{1}{2}}

This simplifies the rational expression to its final equivalent form.

Key Concept

Equivalent Algebraic Expressions

Question 1714Question →

\begin{aligned} 5x + 3y &= 22 \\ 3x + 5y &= 18 \end{aligned}

If $(x, y)$ is the solution to the system of equations above, what is the value of $x - y$ ?

Show answer & explanation

Answer: 2

Answer

The value of the expression

x - y

is 2.

Subtracting the second equation,

3x + 5y = 18

, from the first equation,

5x + 3y = 22

, yields

(5x - 3x) + (3y - 5y) = 22 - 18

, which simplifies to

2x - 2y = 4

. Dividing both sides of this equation by 2 isolates the expression

x - y

and yields a final value of 2.

Step-by-Step Solution

1

Subtract the second equation from the first equation.

2x - 2y = 4

Subtracting the equations aligns the coefficients of

x

and

y

to form a multiple of the target expression

x - y

.

2

Divide both sides of the equation by 2.

x - y = 2

Dividing the expression

2x - 2y

by 2 isolates the target expression

x - y

.

Key Concept

Solving systems of linear equations by linear combination and algebraic manipulation.

Question 1715Question →

If $\frac{8^x}{2^y} = 64$ , which of the following equations correctly expresses $y$ in terms of $x$ ?

y = 3x - 64

y = 3x - 6

y = 3x + 6

y = 6 - 3x

Show answer & explanation

Answer: $y = 3x - 6$

Answer

y = 3x - 6

To solve the equation, express all terms using the common base of 2. Substituting

8 = 2^3

and

64 = 2^6

into the equation gives

\frac{(2^3)^x}{2^y} = 2^6

. Applying the power rule of exponents transforms the numerator into

2^{3x}

, which leads to the equation

\frac{2^{3x}}{2^y} = 2^6

. Using the quotient rule of exponents, the division simplifies to

2^{3x - y} = 2^6

. Since the bases are now identical, their exponents must be equal:

3x - y = 6

. Isolating the variable yields the correct formulation.

Step-by-Step Solution

1

Rewrite all bases in the equation as powers of the common base 2.

Since

8 = 2^3

and

64 = 2^6

, the equation becomes

\frac{(2^3)^x}{2^y} = 2^6

.

Expressing all exponential terms using a single base allows for the application of exponent rules.

2

Apply the power of a power property to simplify the numerator on the left side.

The term

(2^3)^x

simplifies to

2^{3x}

, yielding the equation

\frac{2^{3x}}{2^y} = 2^6

.

The exponent rule

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

states that a base raised to a power and then another power requires multiplying the exponents.

3

Apply the quotient property of exponents to combine the left side of the equation.

The left side simplifies to

2^{3x - y}

, resulting in

2^{3x - y} = 2^6

.

The quotient rule

\frac{a^m}{a^n} = a^{m-n}

states that dividing powers with the same base requires subtracting the exponent of the denominator from the exponent of the numerator.

4

Set the exponents equal to each other and solve for the desired variable.

3x - y = 6

, which simplifies to

y = 3x - 6

when solved for

y

.

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

Key Concept

Solving exponential equations by base conversion and rules of exponents

Alternative Method

Instead of converting both sides to base 2, you can also substitute a test value for

x

. For example, if

x = 2

, then the equation becomes

\frac{8^2}{2^y} = 64

, which simplifies to

\frac{64}{2^y} = 64 \implies 2^y = 1 \implies y = 0

. Substituting

x = 2

into the correct option yields

y = 3(2) - 6 = 0

, confirming the answer.

Estimated Time:1m 30s

Question 1716Question →

The concentration of a medication in a patient's bloodstream, in milligrams per liter ( $\text{mg/L}$ ), $h$ hours after injection is modeled by an exponential decay function. If the concentration of the medication decreases by $48.8\%$ every $3$ hours, which of the following is closest to the percent decrease of the medication concentration each hour?

16.3%

20.0%

80.0%

86.6%

Show answer & explanation

Answer: 20.0%

Answer

20.0%

The correct answer is 20.0%. Since the concentration decreases by 48.8% every 3 hours, the remaining concentration after 3 hours is 51.2%, which corresponds to a factor of 0.512 of the initial concentration. The hourly decay factor

b

satisfies the equation

b^3 = 0.512

. Taking the cube root of both sides yields

b = 0.8

. A decay factor of 0.8 means that 80% of the medication remains each hour, representing a percent decrease of 20% each hour.

Step-by-Step Solution

1

Determine the remaining fraction of the medication after 3 hours.

Remaining fraction = 1 - 0.488 = 0.512

Since the concentration decreases by 48.8% every 3 hours, the remaining concentration is 51.2% of its value at the beginning of the 3-hour interval.

2

Set up the equation for the hourly retention factor,

b

.

b^3 = 0.512

An exponential decay model of the form

C(h) = C_0 \cdot b^h

retains a factor of

b^3

over a 3-hour period.

3

Solve for the hourly retention factor,

b

.

b = \sqrt[3]{0.512} = 0.8

Taking the cube root of 0.512 gives the hourly factor, indicating that the concentration retains 80% of its value each hour.

4

Calculate the hourly percent decrease from the hourly retention factor.

Percent decrease =

(1 - 0.8) \times 100\% = 20.0\%

The hourly percent decrease is the difference between the initial 100% and the remaining 80% retained each hour.

Key Concept

Converting decay factors across different time intervals using exponent properties.

Estimated Time:2m 0s

Question 1717Question →

A quadratic function $f$ is defined by $f(x) = a(x - h)^2 + k$ , where $a$ , $h$ , and $k$ are constants, and its graph in the $xy$ -plane has vertex $(h, k)$ in the first quadrant. The $y$ -intercept of the graph of $f$ is $(0, 4)$ . The function $g$ is defined by $g(x) = f(x - 2) + 12$ . If the $y$ -intercept of the graph of $g$ is $(0, 8)$ , and the vertex of the graph of $f$ lies on the line $y = 5x$ , what is the value of $k$ ?

4

5

\frac{20}{3}

20

Show answer & explanation

Answer: 5

Answer

The value of

k

is 5.

By using the given vertex relation

k = 5h

, we can substitute this into the

y

-intercept equations for

f(x)

and

g(x)

. The

y

-intercept of

f

is at

(0,4)

, which translates to the equation

ah^2 + k = 4

. The

y

-intercept of

g

is at

(0,8)

, which leads to

a(h+2)^2 + k = -4

. Eliminating the parameter

a

by equating the two representations of

a

in terms of

h

yields the quadratic equation

3h^2 + h - 4 = 0

. Factoring this equation gives

(3h+4)(h-1)=0

. Since the vertex is in the first quadrant, both coordinates must be positive, which restricts

h

to the positive value

h = 1

. Substituting

h = 1

back into the vertex relation

k = 5h

yields

k = 5

.

Step-by-Step Solution

1

Use the vertex line relation to express

k

in terms of

h

.

k = 5h

Since the vertex

(h, k)

lies on the line

y = 5x

, substituting the coordinates into the line's equation gives

k = 5h

.

2

Set up an equation using the

y

-intercept of

f(x)

.

ah^2 + 5h = 4 \implies a = \frac{4 - 5h}{h^2}

The

y

-intercept of

f(x)

is

(0, 4)

, so

f(0) = a(0-h)^2 + k = 4

. Substituting

k = 5h

gives

ah^2 + 5h = 4

.

3

Set up an equation using the

y

-intercept of

g(x)

.

a(h + 2)^2 + 5h = -4 \implies a = \frac{-5h - 4}{(h+2)^2}

The function

g(x) = f(x - 2) + 12 = a(x - 2 - h)^2 + k + 12

. The

y

-intercept is

(0, 8)

, so

g(0) = a(-2 - h)^2 + k + 12 = 8

. Simplifying and substituting

k = 5h

gives

a(h+2)^2 + 5h = -4

.

4

Equate the expressions for

a

and solve the resulting quadratic equation for

h

.

3h^2 + h - 4 = 0 \implies (3h + 4)(h - 1) = 0

Equating the two expressions for

a

gives

\frac{4 - 5h}{h^2} = \frac{-(5h + 4)}{(h + 2)^2}

. Cross-multiplying yields

(4 - 5h)(h^2 + 4h + 4) = -h^2(5h + 4) \implies 16 - 4h - 16h^2 - 5h^3 = -5h^3 - 4h^2 \implies 12h^2 + 4h - 16 = 0 \implies 3h^2 + h - 4 = 0

.

5

Determine

h

and compute

k

based on the quadrant constraints.

h = 1 \implies k = 5

Since the vertex

(h, k)

is in the first quadrant, we must have

h > 0

. Thus,

h = 1

is the only valid solution. Using

k = 5h

, we find

k = 5(1) = 5

.

Key Concept

Analyzing quadratic vertex form and transformations using algebraic systems.

Estimated Time:3m 0s

Question 1718Question →

For centuries, painters relied on natural ultramarine, a blue pigment made from grinding lapis lazuli, to achieve deep blue hues in their works; because this material was imported from Afghanistan, it was prohibitively expensive. In the early eighteenth century, a chemist accidentally created Prussian ______ new pigment quickly became popular among artists because it was both affordable and chemically stable.

Which choice completes the text so that it conforms to the conventions of Standard English?

blue, but the

blue, the

blue; the

blue the

Show answer & explanation

Answer: blue; the

Answer

The correct choice is the option that reads 'blue; the', as it properly uses a semicolon to link two independent clauses.

The correct choice is the option that contains 'blue; the'. It correctly uses a semicolon to separate two independent clauses. Since both clauses can stand alone as complete sentences, a semicolon is an appropriate punctuation mark to link them without a coordinating conjunction.

Step-by-Step Solution

1

Analyze the grammatical structure of the clauses surrounding the blank.

The text contains two independent clauses: 'In the early eighteenth century, a chemist accidentally created Prussian blue' and 'the new pigment quickly became popular among artists because it was both affordable and chemically stable.'

Identifying that both clauses are independent determines that they must be joined by appropriate punctuation or a coordinating conjunction.

2

Determine the acceptable grammatical methods for linking independent clauses.

Two independent clauses can be joined by a period, a semicolon, or a comma followed by a coordinating conjunction.

This establishes the rules for selecting a grammatically correct option.

3

Evaluate the given choices against the grammatical rules and context.

The option containing 'blue; the' correctly uses a semicolon to link the clauses. The option with a comma splice ('blue, the') and the run-on option ('blue the') violate standard conventions, while the option with 'but' introduces an incorrect contrast relationship.

This selects the correct option and eliminates the incorrect ones.

Key Concept

Linking Independent Clauses

Estimated Time:1m 0s

Question 1719Question →

Consider the equation:

\frac{x + 3}{x - 1} + \frac{4}{x} = \frac{4}{x^2 - x}

What is the value of $x + 3$ for the real value of $x$ that satisfies this equation?

-5

4

11

2

Show answer & explanation

Answer: -5

Answer

-5

The correct answer is

-5

. Multiplying the rational equation by the common denominator

x(x - 1)

gives the simplified equation

x(x + 3) + 4(x - 1) = 4

. Expanding and rearranging the terms leads to the quadratic equation

x^2 + 7x - 8 = 0

. Factoring this quadratic yields

(x + 8)(x - 1) = 0

, giving the potential solutions

x = -8

and

x = 1

. Substituting these values back into the original equation shows that

x = 1

causes division by zero, making it extraneous. Thus, the only valid solution is

x = -8

. Substituting this valid solution into the expression

x + 3

yields

-8 + 3 = -5

.

Step-by-Step Solution

1

Find the common denominator of the rational terms.

The denominators are

x - 1

,

x

, and

x^2 - x

. The common denominator is

x(x - 1) = x^2 - x

.

Finding a common denominator allows us to eliminate the fractions by multiplying both sides.

2

Multiply the entire equation by the common denominator

x(x - 1)

to clear the denominators.

x(x + 3) + 4(x - 1) = 4

This simplifies the rational equation into a polynomial equation.

3

Expand and simplify the resulting equation.

x^2 + 3x + 4x - 4 = 4

, which simplifies to

x^2 + 7x - 4 = 4

.

Expanding the terms allows us to collect like terms and identify the type of polynomial equation.

4

Write the quadratic equation in standard form and factor it.

x^2 + 7x - 8 = 0

, which factors as

(x + 8)(x - 1) = 0

.

Setting the quadratic equation to zero allows us to find its roots by factoring.

5

Find the potential solutions and check for extraneous values in the original equation.

The potential solutions are

x = -8

and

x = 1

. The value

x = 1

is extraneous because it makes the denominators

x - 1

and

x^2 - x

equal to zero. The only valid solution is

x = -8

.

We must check if the roots make any denominator in the original equation zero, as division by zero is undefined.

6

Evaluate the expression

x + 3

using the valid solution.

-8 + 3 = -5

The question asks for the value of the expression

x + 3

, not the value of

x

itself.

Key Concept

Solving rational equations and checking for extraneous solutions

Question 1720Question →

A parabola in the $xy$ -plane has equation $y = a(x - h)^2 + k$ , where $a$ , $h$ , and $k$ are constants. The parabola passes through the points $(2, 13)$ and $(8, 13)$ in the $xy$ -plane. If the minimum value of the quadratic function defined by this equation is $-5$ , what is the value of $a$ ?

Show answer & explanation

Answer: 2

Answer

The correct answer is 2.

The correct answer is 2. The axis of symmetry of the parabola is halfway between the x-coordinates of the two symmetric points

(2, 13)

and

(8, 13)

, which is

x = 5

. Since the minimum value of the function is

-5

, the vertex of the parabola is

(5, -5)

. Writing the equation in vertex form,

y = a(x - 5)^2 - 5

, and substituting the point

(2, 13)

yields

13 = a(2 - 5)^2 - 5

, which simplifies to

18 = 9a

, so

a = 2

.

Step-by-Step Solution

1

Find the axis of symmetry

x = h

using the symmetric points

(2, 13)

and

(8, 13)

.

h = 5

Since the points

(2, 13)

and

(8, 13)

have the same

y

-coordinate, the axis of symmetry of the parabola must lie exactly halfway between their

x

-coordinates:

h = \frac{2 + 8}{2} = 5

.

2

Determine the vertex

(h, k)

of the parabola.

(5, -5)

The minimum value of a quadratic function occurs at its vertex. The given minimum value is

-5

, which represents the

y

-coordinate (

k

) of the vertex. Combining this with

h = 5

, the vertex is

(5, -5)

.

3

Write the equation of the parabola in vertex form.

y = a(x - 5)^2 - 5

The vertex form of a quadratic equation is

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

. Substituting the values

h = 5

and

k = -5

yields this equation.

4

Substitute one of the given points into the equation to solve for

a

.

a = 2

Substituting

(2, 13)

into the equation gives

13 = a(2 - 5)^2 - 5 \Rightarrow 18 = 9a \Rightarrow a = 2

.

Key Concept

Finding the equation of a parabola using symmetry and vertex form

Estimated Time:2m 0s

Answer

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