All practice questions

Question 301Question →

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 302Question →

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 303Question →

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 304Question →

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 305Question →

\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 306Question →

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

Question 307Question →

A researcher uses the linear equation $T = 18.5 + 4.2d$ to estimate the temperature, $T$ , in degrees Celsius, of the Earth's crust at a depth of $d$ kilometers below the surface in a certain region. What is the estimated increase in temperature, in degrees Celsius, for each increase of 5 kilometers in depth?

Show answer & explanation

Answer: 21

Answer

21

The coefficient of

d

in the equation is

4.2

, representing a temperature increase of

4.2

degrees Celsius for every 1 kilometer increase in depth. To find the temperature increase for an increase of 5 kilometers in depth, multiply this rate of change by 5:

4.2 \times 5 = 21

.

Step-by-Step Solution

1

Identify the slope of the linear equation.

The slope is

4.2

.

In the equation

T = 18.5 + 4.2d

, the term

4.2d

indicates that for every 1 kilometer increase in depth (

d

), the temperature (

T

) increases by

4.2

degrees Celsius.

2

Calculate the total temperature increase for a 5-kilometer depth increase.

The increase is

21

degrees Celsius.

Since the rate of temperature increase is

4.2

degrees Celsius per kilometer, a depth increase of

5

kilometers results in a temperature increase of

4.2 \times 5 = 21

degrees Celsius.

Key Concept

Interpreting the slope of a linear equation in context as a constant rate of change.

Estimated Time:1m 30s

Question 308Question →

What is the sum of all real solutions to the equation $\sqrt{3x + 7} - \frac{10}{\sqrt{3x + 7}} = 3$ ?

Show answer & explanation

Answer: 6

Answer

The only real solution is 6, so the sum of all real solutions is 6.

The correct answer is 6. By substituting

u = \sqrt{3x + 7}

, the equation becomes

u - 10/u = 3

, which simplifies to the quadratic

u^2 - 3u - 10 = 0

. Factoring gives

(u - 5)(u + 2) = 0

. Since the principal square root must be non-negative,

u

must be positive, so we reject

u = -2

and keep

u = 5

. Solving

\sqrt{3x + 7} = 5

by squaring both sides yields

3x + 7 = 25

, which simplifies to

x = 6

. The extraneous solution

x = -1

(which comes from

u = -2

) must be discarded because substituting it back into the original equation results in

-3

instead of

3

. Thus, the only real solution is 6.

Step-by-Step Solution

1

Substitute

u = \sqrt{3x + 7}

into the equation.

u - \frac{10}{u} = 3

, where

u > 0

.

This substitution simplifies the radical equation into a rational equation of a single variable.

2

Multiply the entire equation by

u

and solve the resulting quadratic equation.

u^2 - 3u - 10 = 0 \implies (u - 5)(u + 2) = 0

, which yields

u = 5

or

u = -2

.

Clearing the denominator transforms the rational expression into a standard quadratic equation.

3

Apply the constraint that

u = \sqrt{3x + 7}

must be non-negative.

u = 5

is the only valid root;

u = -2

has no real solution and must be discarded.

The output of a principal square root is non-negative by definition.

4

Substitute back

\sqrt{3x + 7} = 5

and solve for

x

.

3x + 7 = 25 \implies 3x = 18 \implies x = 6

.

Squaring both sides eliminates the radical and allows us to isolate the variable

x

.

5

Verify if solving

\sqrt{3x+7} = -2

leads to an extraneous solution.

Squaring both sides of

\sqrt{3x+7} = -2

gives

3x+7 = 4 \implies x = -1

. Substituting

x = -1

into the original equation yields

\sqrt{4} - \frac{10}{\sqrt{4}} = 2 - 5 = -3 \neq 3

. Therefore,

x = -1

is extraneous.

Verifying solutions is necessary to identify and discard values that do not satisfy the original equation.

Key Concept

Solving equations using substitution and identifying extraneous solutions

Estimated Time:2m 30s

Question 309Question →

A commercial printing press uses a continuous roll of paper to print newspapers at a constant rate. After $10$ minutes of operation, the remaining length of the paper roll is $12,500$ feet. After $25$ minutes of operation, the remaining length of the paper roll is $8,000$ feet. If the relationship between the printing time, in minutes, and the remaining length of the paper roll, in feet, is linear, how many minutes after the printing press starts operating will the remaining length of the paper roll be $2,000$ feet?

Show answer & explanation

Answer: 45

Answer

The remaining length of the paper roll will be

2,000

feet after

45

minutes of operation.

We are given that the remaining length of the paper roll is a linear function of time,

t

. Let

L(t)

be the remaining length of the paper roll, in feet, after

t

minutes of operation. We can represent the given information as two coordinate points:

(10, 12500)

and

(25, 8000)

. First, find the slope, which represents the constant rate at which the paper is consumed:

m = \frac{8000 - 12500}{25 - 10} = \frac{-4500}{15} = -300

feet per minute. Next, write the linear equation using the point-slope form:

L(t) - 12500 = -300(t - 10)

, which simplifies to

L(t) = 15500 - 300t

. To find the time when the remaining length is

2,000

feet, set

L(t) = 2000

and solve for

t

:

2000 = 15500 - 300t

, which simplifies to

300t = 13500

, giving

t = 45

.

Step-by-Step Solution

1

Calculate the rate of paper consumption (the slope of the linear equation) using the two given points,

(10, 12500)

and

(25, 8000)

.

The rate of paper consumption is

-300

feet per minute.

To establish the linear relationship, we first need the constant rate of change (slope) from the two known coordinate points.

2

Use the point-slope equation of a line,

y - y_1 = m(x - x_1)

, with the point

(10, 12500)

and slope

m = -300

, to find the equation relating the remaining length,

L

, to the time,

t

.

L(t) = 15500 - 300t

We need the full linear model to calculate the remaining length at any specific time.

3

Substitute

L(t) = 2000

into the linear equation and solve for

t

.

t = 45

This gives the specific operating time in minutes when the remaining paper roll length is

2,000

feet.

Key Concept

Writing and solving linear equations in two variables from two coordinate points.

Question 310Question →

If $x$ and $y$ satisfy the system of equations below, what is the value of $x + y$ ?

\begin{aligned} 8^{x-y} &= 2^{y+3} \\ 9^{x} &= 27^{y-1} \end{aligned}

Show answer & explanation

Answer: 36

Answer

36

The correct answer is 36. By rewriting

8^{x-y} = 2^{y+3}

with base 2, we obtain the relation

3x - 4y = 3

. Similarly, rewriting

9^x = 27^{y-1}

with base 3 gives

2x - 3y = -3

. Solving this system of equations simultaneously yields

x = 21

and

y = 15

. Adding these values together gives the final result of 36.

Step-by-Step Solution

1

Rewrite the first equation using the common base of 2.

2^{3(x-y)} = 2^{y+3}

, which simplifies to

3x - 4y = 3

.

Since

8 = 2^3

, we can apply the power of a power rule

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

to rewrite

8^{x-y}

as

2^{3(x-y)}

. Setting the exponents equal gives the linear equation

3(x-y) = y+3

, which simplifies to

3x - 4y = 3

.

2

Rewrite the second equation using the common base of 3.

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

, which simplifies to

2x - 3y = -3

.

Since

9 = 3^2

and

27 = 3^3

, we rewrite the terms using base 3. Equating the exponents gives

2x = 3(y-1)

, which simplifies to

2x - 3y = -3

.

3

Solve the system of two linear equations.

x = 21

and

y = 15

.

Using the elimination method, we can multiply the first equation by 3 and the second equation by 4 to align the coefficients of

y

, obtaining

9x - 12y = 9

and

8x - 12y = -12

. Subtracting the equations gives

x = 21

. Substituting this back yields

y = 15

.

4

Find the sum of

x

and

y

.

36

The question asks for the value of

x + y

. Substituting

x = 21

and

y = 15

gives

21 + 15 = 36

.

Key Concept

Solving systems of exponential equations by expressing bases as powers of common bases to establish a system of linear equations.

Question 311Question →

If $\sqrt{4x + 9} = 5$ , what is the value of $x$ ?

Show answer & explanation

Answer: 4

Answer

4

Squaring both sides of the equation

\sqrt{4x + 9} = 5

yields

4x + 9 = 25

. Subtracting 9 from both sides gives

4x = 16

, and dividing by 4 yields

x = 4

. Checking this solution in the original equation yields

\sqrt{4(4) + 9} = \sqrt{25} = 5

, which confirms it is correct.

Step-by-Step Solution

1

Square both sides of the equation

4x + 9 = 25

To eliminate the square root on the left side of the equation.

2

Subtract 9 from both sides of the equation

4x = 16

To isolate the variable term on the left side of the equation.

3

Divide both sides of the equation by 4

x = 4

To solve for

x

.

4

Substitute the solved value back into the original equation

\sqrt{4(4) + 9} = 5 \implies 5 = 5

To ensure the solution is valid and not extraneous.

Key Concept

Solving radical equations of the form

\sqrt{ax + b} = c

by squaring both sides.

Question 312Question →

The polynomial function $p$ is defined by $p(x) = (x - 3)(x - 5)(x - r)$ , where $r$ is a constant. If the graph of $y = p(x)$ in the $xy$ -plane intersects the $y$ -axis at $(0, -60)$ , what is the value of $r$ ?

Show answer & explanation

Answer: 4

Answer

The value of

r

is

4

.

Evaluating the polynomial at

x = 0

yields the

y

-intercept of the graph, which is given as

-60

. Substituting

0

for

x

in the equation

p(x) = (x - 3)(x - 5)(x - r)

gives

p(0) = (0 - 3)(0 - 5)(0 - r) = (-3)(-5)(-r) = -15r

. Setting this expression equal to

-60

yields

-15r = -60

. Dividing both sides of the equation by

-15

yields

r = 4

.

Step-by-Step Solution

1

Identify the meaning of the

y

-intercept.

Evaluating

p(x)

at

x = 0

must equal

-60

, so

p(0) = -60

.

The

y

-intercept of a graph occurs where the

x

-coordinate is

0

.

2

Substitute

x = 0

into the polynomial expression.

p(0) = (0 - 3)(0 - 5)(0 - r) = (-3)(-5)(-r) = -15r

.

This simplifies the polynomial expression in terms of the unknown constant

r

.

3

Set the expression equal to the given

y

-intercept value and solve for

r

.

-15r = -60 \implies r = 4

.

Dividing both sides by

-15

isolates the variable

r

.

Key Concept

Using the y-intercept of a polynomial function to find an unknown root/constant in its factored form.

Question 313Question →

In the $xy$ -plane, the graph of the quadratic function $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, has a vertex $(h, k)$ that lies on the line $y = 4x + 8$ . If the graph of $y = f(x)$ has $x$ -intercepts at $x = -2$ and $x = 6$ , what is the value of $c$ ?

Show answer & explanation

Answer: 12

Answer

The value of c is 12.

The correct answer is 12. The axis of symmetry of the quadratic function lies midway between its x-intercepts at

x = -2

and

x = 6

, giving an x-coordinate of

h = 2

for the vertex. Since the vertex lies on the line

y = 4x + 8

, its y-coordinate is

k = 4(2) + 8 = 16

. Substituting the vertex

(2, 16)

into the vertex form of a quadratic function gives

f(x) = a(x - 2)^2 + 16

. Using the x-intercept at

(6, 0)

to solve for

a

gives

0 = a(6 - 2)^2 + 16

, which yields

a = -1

. Evaluating the function at

x = 0

to find the constant term

c

gives

c = f(0) = -(0 - 2)^2 + 16 = 12

.

Step-by-Step Solution

1

Determine the x-coordinate of the vertex of the quadratic function.

h = 2

The axis of symmetry of a parabola lies midway between its x-intercepts. The midpoint of the intercepts

x = -2

and

x = 6

is calculated as

\frac{-2 + 6}{2} = 2

.

2

Determine the y-coordinate of the vertex using the given linear equation.

k = 16

Since the vertex

(h, k)

lies on the line

y = 4x + 8

, substituting

x = h = 2

into the line equation gives

k = 4(2) + 8 = 16

.

3

Write the quadratic function in vertex form and solve for the leading coefficient

a

.

a = -1

The vertex form is

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

. Substituting the vertex

(2, 16)

yields

f(x) = a(x - 2)^2 + 16

. Using the x-intercept

(6, 0)

, substitute

x = 6

and

f(x) = 0

into the equation:

0 = a(6 - 2)^2 + 16

, which simplifies to

16a = -16

, resulting in

a = -1

.

4

Expand the quadratic function or evaluate it at

x = 0

to find the value of

c

.

c = 12

In standard form

f(x) = ax^2 + bx + c

, the constant term

c

is equal to

f(0)

. Substituting

x = 0

into

f(x) = -(x - 2)^2 + 16

yields

f(0) = -(0 - 2)^2 + 16 = -4 + 16 = 12

.

Key Concept

Using symmetry and the vertex form of a quadratic function to determine its standard form coefficients.

Alternative Method

Alternatively, since the x-intercepts are

-2

and

6

, the quadratic function can be written in factored form as

f(x) = a(x + 2)(x - 6)

. Expanding this gives

f(x) = a(x^2 - 4x - 12) = ax^2 - 4ax - 12a

. Comparing this to the standard form

f(x) = ax^2 + bx + c

, we see that the x-coordinate of the vertex is

h = -\frac{b}{2a} = -\frac{-4a}{2a} = 2

. Using the line equation

y = 4x + 8

at

x = 2

, we find the vertex y-coordinate is

k = 16

. Since the vertex is

(2, 16)

, we evaluate the factored form at

x = 2

:

16 = a(2 + 2)(2 - 6) \Rightarrow 16 = -16a \Rightarrow a = -1

. The constant term

c

is

-12a

, so

c = -12(-1) = 12

.

Estimated Time:2m 30s

Question 314Question →

If $(x, y)$ is a solution to the system of equations below and $x > 1$ , what is the value of $x$ ?

\begin{aligned} y - 2x &= 3 \\ y^2 - 5x^2 &= 20 \end{aligned}

Show answer & explanation

Answer: 11

Answer

The value of

x

is 11.

The system can be solved by substituting the first equation into the second. Rearranging the first equation gives

y = 2x + 3

. Substituting this expression for

y

in the second equation gives

(2x + 3)^2 - 5x^2 = 20

. Expanding the binomial yields

4x^2 + 12x + 9 - 5x^2 = 20

, which simplifies to

-x^2 + 12x + 9 = 20

. Rearranging into standard quadratic form gives

x^2 - 12x + 11 = 0

. Factoring the quadratic equation results in

(x - 11)(x - 1) = 0

. Thus,

x = 11

or

x = 1

. Since the problem specifies

x > 1

, the value of

x

must be 11.

Step-by-Step Solution

1

Rearrange the first equation to express

y

in terms of

x

.

y = 2x + 3

This allows for substitution into the second equation.

2

Substitute

y = 2x + 3

into the second equation.

(2x + 3)^2 - 5x^2 = 20

To eliminate

y

and obtain an equation in terms of

x

only.

3

Expand and simplify the equation.

x^2 - 12x + 11 = 0

Expanding

(2x + 3)^2

gives

4x^2 + 12x + 9

. Subtracting

5x^2

yields

-x^2 + 12x + 9 = 20

. Subtracting 20 from both sides gives

-x^2 + 12x - 11 = 0

. Multiplying the entire equation by

-1

results in standard form.

4

Factor the quadratic equation.

(x - 11)(x - 1) = 0

To find the possible values of

x

.

5

Apply the given constraint

x > 1

.

x = 11

The solutions are

x = 11

and

x = 1

. The constraint

x > 1

excludes

x = 1

, leaving

x = 11

as the only valid solution.

Key Concept

Solving a system of nonlinear equations by substitution and factoring a quadratic equation.

Question 315Question →

If $x$ is a real number that satisfies the equation $\frac{x}{x-3} - \frac{6}{x^2-3x} = \frac{2}{x}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 2

Answer

2

The correct answer is 2. Multiplying the entire equation by the lowest common denominator,

x(x-3)

, yields

x^2 - 6 = 2(x - 3)

. Distributing the right side gives

x^2 - 6 = 2x - 6

. Adding 6 to both sides and subtracting

2x

results in

x^2 - 2x = 0

. Factoring this equation gives

x(x-2) = 0

, which has solutions

x = 0

and

x = 2

. However,

x = 0

is extraneous because it results in division by zero in the original equation. Thus, the only valid real solution is 2.

Step-by-Step Solution

1

Identify the common denominator and multiply the equation by it.

x^2 - 6 = 2(x - 3)

To clear the fractions and simplify the rational equation.

2

Distribute and simplify the equation.

x^2 - 2x = 0

To set up the quadratic equation in standard form.

3

Factor and solve for

x

.

x = 0

or

x = 2

To find potential solutions to the equation.

4

Check for extraneous solutions by substituting the values back into the original denominators.

x = 2

x = 0

makes the denominators equal to zero and is therefore extraneous.

Key Concept

Solving rational equations and identifying extraneous solutions.

Question 316Question →

If $\frac{x - 3}{4} = \frac{9}{x - 3}$ and $x > 3$ , what is the value of $x$ ?

Show answer & explanation

Answer: 9

Answer

9

Cross-multiplying the equation

\frac{x - 3}{4} = \frac{9}{x - 3}

gives

(x - 3)^2 = 36

. Taking the square root of both sides gives

x - 3 = 6

or

x - 3 = -6

. Solving for

x

yields

x = 9

or

x = -3

. Since the question specifies the constraint

x > 3

, the only valid solution is

9

.

Step-by-Step Solution

1

Cross-multiply the equation to eliminate the denominators.

(x - 3)^2 = 36

Multiplying both sides of the equation by

4(x - 3)

simplifies the rational equation into a quadratic form.

2

Take the square root of both sides of the equation.

x - 3 = 6

or

x - 3 = -6

Taking the square root of a squared term yields both positive and negative root options.

3

Solve each linear equation for

x

and apply the constraint

x > 3

.

x = 9

Adding

3

to both sides gives

x = 9

or

x = -3

. The constraint

x > 3

excludes the negative solution, leaving

9

as the only valid value.

Key Concept

Solving rational equations by cross-multiplication and factoring the resulting quadratic equation while adhering to domain constraints.

Question 317Question →

A linear function $f$ is defined such that its graph passes through the points $(2, 5)$ and $(4, 11)$ . The graph of another linear function, $g$ , is perpendicular to the graph of $f$ . If the graph of $g$ passes through the point $(3, 10)$ , what is the $x$ -coordinate of the $x$ -intercept of the graph of $g$ ?

Show answer & explanation

Answer: 33

Answer

The correct answer is

33

.

The slope of the linear function

f

is found using the two given points:

\frac{11 - 5}{4 - 2} = 3

. Because the graph of the linear function

g

is perpendicular to the graph of

f

, its slope is the negative reciprocal of

3

, which is

-\frac{1}{3}

. Using the point-slope form with the point

(3, 10)

, the equation of

g

is

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

, which simplifies to

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

. The

x

-intercept is the value of

x

when

y = 0

. Setting

y

to

0

gives

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

, which yields

x = 33

.

Step-by-Step Solution

1

Calculate the slope of the linear function

f

The slope of

f

is

3

.

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

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

. Substituting

(2, 5)

and

(4, 11)

yields

\frac{11 - 5}{4 - 2} = \frac{6}{2} = 3

.

2

Determine the slope of the perpendicular linear function

g

The slope of

g

is

-\frac{1}{3}

.

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

3

is

-\frac{1}{3}

.

3

Write the equation representing function

g

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

Using the point-slope equation

y - y_1 = m(x - x_1)

with the slope

m = -\frac{1}{3}

and the point

(3, 10)

, we get

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

. Distributing and simplifying yields

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

.

4

Find the

x

-intercept of the graph of

g

33

The

x

-intercept is the point on the graph where

y = 0

. Substituting

0

for

y

in the equation of

g

gives

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

.

Key Concept

The relationship between the slopes of perpendicular lines and finding intercepts of linear functions.

Estimated Time:1m 30s

Question 318Question →

In the polynomial function $p(x) = x^3 - 5x^2 + 2x + k$ , the constant $k$ is chosen such that $p(x)$ is divisible by $x - 4$ . What is the value of $k$ ?

Show answer & explanation

Answer: 8

Answer

The value of

k

is 8.

According to the Factor Theorem, a polynomial

p(x)

is divisible by

x - c

if and only if

p(c) = 0

. Since

p(x)

is divisible by

x - 4

, we must have

p(4) = 0

. Substituting

4

for

x

in the equation for

p(x)

yields

4^3 - 5(4)^2 + 2(4) + k = 0

. Simplifying the numerical terms gives

64 - 80 + 8 + k = 0

, which simplifies further to

-8 + k = 0

. Therefore, the value of the constant

k

is

8

.

Step-by-Step Solution

1

Apply the Factor Theorem to relate the factor

x - 4

to the value of the polynomial.

Since

x - 4

is a factor of

p(x)

,

p(4) = 0

.

By the Factor Theorem, if

x - c

is a factor of a polynomial

p(x)

, then

p(c) = 0

.

2

Substitute

x = 4

into the polynomial expression.

4^3 - 5(4)^2 + 2(4) + k = 0

We substitute

x = 4

into

p(x) = x^3 - 5x^2 + 2x + k

and set the expression to

0

.

3

Simplify the numerical expression to solve for

k

.

64 - 80 + 8 + k = 0 \implies -8 + k = 0 \implies k = 8

Evaluate powers, multiply, and solve the resulting linear equation for

k

.

Key Concept

Factor Theorem

Estimated Time:1m 0s

Question 319Question →

The price of a ticket to a museum is $25. If a$ 6\%$ sales tax is added to the price of the ticket, what is the total cost of the ticket, in dollars?

Show answer & explanation

Answer: 26.5

Answer

The correct answer is 26.5.

To find the total cost of the museum ticket, the

6\%

sales tax must be added to the original price. The tax amount is calculated by converting the percent to a decimal and multiplying it by the original price:

25 \times 0.06 = 1.5

dollars. Adding this tax to the original price of

25

gives the total cost:

25 + 1.5 = 26.5

dollars. Alternatively, this can be solved by multiplying the original price by

1.06

:

25 \times 1.06 = 26.5

dollars.

Step-by-Step Solution

1

Calculate the amount of sales tax.

1.5

dollars

To find

6\%

of

25

, convert the percentage to a decimal by dividing by 100 (

6\% = 0.06

) and multiply by the ticket price:

25 \times 0.06 = 1.5

.

2

Add the sales tax to the original ticket price to get the total cost.

26.5

dollars

The total cost is the sum of the original price and the tax:

25 + 1.5 = 26.5

.

Key Concept

Calculating a final value after a percentage increase

Alternative Method

The total cost of the ticket including a

6\%

tax can be found in one step by multiplying the original price by

1.06

:

25 \times 1.06 = 26.5

.

Estimated Time:45s

Question 320Question →

For all $x \neq 2$ , the expression $\frac{3x^2 + kx - 8}{x - 2}$ is equivalent to $3x + 10 + \frac{12}{x - 2}$ , where $k$ is a constant. What is the value of $k$ ?

Show answer & explanation

Answer: 4

Answer

The value of the constant

k

is

4

.

The correct answer is

4

. By rewriting the right-hand side of the equation with a common denominator of

x - 2

, the expression becomes

\frac{(3x + 10)(x - 2) + 12}{x - 2}

. Expanding and simplifying the numerator yields

3x^2 + 4x - 8

. Comparing this to the numerator of the left-hand side,

3x^2 + kx - 8

, shows that the coefficient of the

x

term,

k

, must equal

4

.

Step-by-Step Solution

1

Set up the equation representing the equivalence of the two expressions.

\frac{3x^2 + kx - 8}{x - 2} = 3x + 10 + \frac{12}{x - 2}

This establishes the relationship that must hold for all

x \neq 2

.

2

Find a common denominator for the terms on the right-hand side.

3x + 10 + \frac{12}{x - 2} = \frac{(3x + 10)(x - 2)}{x - 2} + \frac{12}{x - 2}

To combine the terms into a single rational expression, they must have the same denominator.

3

Expand and simplify the numerator on the right-hand side.

(3x + 10)(x - 2) + 12 = 3x^2 + 4x - 8

Expanding the product of binomials and combining like terms yields the simplified numerator.

4

Equate the numerators of the simplified expressions to solve for

k

.

3x^2 + kx - 8 = 3x^2 + 4x - 8

, which means

k = 4

.

Since the denominators are equal, the numerators must be identical for all values of

x \neq 2

, meaning their corresponding coefficients must match.

Key Concept

Equivalent Algebraic Expressions

Alternative Method

An alternative method is to substitute a convenient value for

x

that is not equal to

2

. Substituting

x = 1

into both sides of the equivalence gives:

\frac{3(1)^2 + k(1) - 8}{1 - 2} = 3(1) + 10 + \frac{12}{1 - 2}

. Simplifying this yields

\frac{k - 5}{-1} = 13 - 12

, which simplifies to

-k + 5 = 1

. Solving for

k

results in

k = 4

.

Estimated Time:1m 30s

Answer

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