All practice questions

Question 361Question →

For the quadratic function $f$ , the table below shows three points that lie on its graph in the $xy$ -plane, where $k$ is a constant.

$x$	$f(x)$
$2$	$0$
$6$	$0$
$1$	$-15$

If the vertex of the graph of $y = f(x)$ is $(4, k)$ , what is the value of $k$ ?

Show answer & explanation

Answer: 12

Answer

The value of

k

is

12

.

The table indicates that the function has

x

-intercepts at

x = 2

and

x = 6

because

f(2) = 0

and

f(6) = 0

. Consequently, the quadratic function can be written in factored form as

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

for some constant

a

. Using the point

(1, -15)

to find

a

, we substitute

x = 1

and

f(1) = -15

, which yields

-15 = a(1 - 2)(1 - 6)

, or

-15 = 5a

, so

a = -3

. Since the vertex of the graph is

(4, k)

, the value of

k

is the function value at the vertex's

x

-coordinate, which is

f(4)

. Evaluating the function gives

k = -3(4 - 2)(4 - 6) = -3(2)(-2) = 12

.

Step-by-Step Solution

1

Write the quadratic function in factored form using the given

x

-intercepts.

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

The table shows that

f(2) = 0

and

f(6) = 0

, meaning the graph has

x

-intercepts at

x = 2

and

x = 6

.

2

Substitute the point

(1, -15)

into the factored equation to find the value of the constant

a

.

a = -3

Substituting

x = 1

and

f(x) = -15

gives

-15 = a(1 - 2)(1 - 6) \implies -15 = 5a \implies a = -3

.

3

Calculate the value of

k

by finding the function value at the vertex

x = 4

.

k = 12

Since the vertex is

(4, k)

, the value of

k

is

f(4)

. Substituting

x = 4

into

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

yields

k = -3(4 - 2)(4 - 6) = -3(2)(-2) = 12

.

Key Concept

Using intercepts and an additional point to determine the equation of a quadratic function, and evaluating it at the vertex.

Question 362Question →

A specialized cooling system is used to lower the temperature of a chemical solution in a laboratory. The temperature of the solution, $T$ , in degrees Celsius, can be modeled by the linear equation $T = 85.4 - 1.25m$ , where $m$ is the number of minutes since the cooling process began. According to the model, how many minutes does it take for the temperature of the solution to decrease by $15$ degrees Celsius?

Show answer & explanation

Answer: 12

Answer

12 minutes

The correct answer is 12. In the equation

T = 85.4 - 1.25m

, the coefficient of

m

is

-1.25

, which indicates that the temperature decreases by

1.25

degrees Celsius for each minute that passes. To find how many minutes it takes for the temperature to decrease by

15

degrees Celsius, divide the total decrease by the rate of decrease:

\frac{15}{1.25} = 12

.

Step-by-Step Solution

1

Identify the rate of change from the linear equation.

The rate of decrease is

1.25

degrees Celsius per minute.

The slope of the linear equation

T = 85.4 - 1.25m

is

-1.25

, representing the change in temperature per minute.

2

Divide the target temperature change by the rate of change to find the time.

12 minutes

To find the number of minutes for a

15

-degree decrease at a rate of

1.25

degrees per minute, calculate

\frac{15}{1.25}

.

Key Concept

Interpreting the slope of a linear relationship in context

Question 363Question →

A projectile is launched from the ground. Its height, in feet, $t$ seconds after launch is modeled by the function $h(t) = -16t^2 + v_0 t$ , where $v_0$ is the initial upward velocity in feet per second. If the projectile reaches its maximum height of $144$ feet, what is the value of $v_0$ ?

Show answer & explanation

Answer: 96

Answer

The correct answer is 96. The initial velocity of the projectile must be 96 feet per second to reach a maximum height of 144 feet.

The maximum height of a projectile modeled by a quadratic function is the

y

-value of its vertex. For

h(t) = -16t^2 + v_0 t

, the time

t

at the vertex is given by

t = -\frac{b}{2a} = -\frac{v_0}{2(-16)} = \frac{v_0}{32}

. Substituting this value of

t

back into the height equation yields the maximum height:

h\left(\frac{v_0}{32}\right) = -16\left(\frac{v_0}{32}\right)^2 + v_0\left(\frac{v_0}{32}\right) = -16\left(\frac{v_0^2}{1024}\right) + \frac{v_0^2}{32} = -\frac{v_0^2}{64} + \frac{v_0^2}{32} = \frac{v_0^2}{64}

. Given that the maximum height is

144

feet, we set

\frac{v_0^2}{64} = 144

, which simplifies to

v_0^2 = 144 \times 64

. Taking the square root of both sides gives

v_0 = 12 \times 8 = 96

.

Step-by-Step Solution

1

Find the time

t

at which the maximum height occurs using the vertex formula.

t = \frac{v_0}{32}

For a quadratic function of the form

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

, the vertex occurs at

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

. Here,

a = -16

and

b = v_0

, so the vertex is at

t = -\frac{v_0}{2(-16)} = \frac{v_0}{32}

.

2

Substitute the time

t = \frac{v_0}{32}

into the height function to express the maximum height in terms of

v_0

.

h\left(\frac{v_0}{32}\right) = \frac{v_0^2}{64}

Evaluating

h\left(\frac{v_0}{32}\right)

gives

-16\left(\frac{v_0}{32}\right)^2 + v_0\left(\frac{v_0}{32}\right) = -16\left(\frac{v_0^2}{1024}\right) + \frac{v_0^2}{32} = -\frac{v_0^2}{64} + \frac{v_0^2}{32} = \frac{v_0^2}{64}

.

3

Set the maximum height expression equal to the given maximum height of

144

feet and solve for

v_0

.

v_0 = 96

Setting

\frac{v_0^2}{64} = 144

yields

v_0^2 = 144 \times 64 = 9216

. Taking the positive square root (since

v_0

represents a positive initial upward velocity) gives

v_0 = \sqrt{144} \times \sqrt{64} = 12 \times 8 = 96

.

Key Concept

Determining the vertex coordinates of a quadratic function to find maximum value in context.

Question 364Question →

If $27^{2x - 2} = \left(\frac{1}{3}\right)^{x - 8}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 2

Answer

The value of

x

is

2

.

To solve the equation

27^{2x - 2} = \left(\frac{1}{3}\right)^{x - 8}

, express both sides with the base

3

. Since

27 = 3^3

and

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

, the equation becomes

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

. Applying the exponent rule

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

yields

3^{6x - 6} = 3^{8 - x}

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

6x - 6 = 8 - x

. Adding

x

to both sides gives

7x - 6 = 8

, and adding

6

to both sides gives

7x = 14

. Dividing by

7

results in

x = 2

.

Step-by-Step Solution

1

Rewrite

27

and

\frac{1}{3}

as powers of

3

.

27 = 3^3

and

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

This allows both sides of the equation to be expressed with a common base of

3

so their exponents can be equated.

2

Substitute these bases back into the equation and apply exponent rules.

(3^3)^{2x - 2} = (3^{-1})^{x - 8} \implies 3^{3(2x - 2)} = 3^{-(x - 8)} \implies 3^{6x - 6} = 3^{8 - x}

Using the power rule

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

, multiply the outer exponent by the power of the base.

3

Equate the exponents and solve for

x

.

6x - 6 = 8 - x \implies 7x = 14 \implies x = 2

Since the bases are equal, their exponents must be equal for the equation to hold true.

Key Concept

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

Question 365Question →

A bakery recipe requires $12$ ounces of butter to make one batch of bread. How many pounds of butter are needed to make $8$ batches of bread? (Given that $1\text{ pound} = 16\text{ ounces}$ )

Show answer & explanation

Answer: 6

Answer

The bakery needs

6

pounds of butter.

The total amount of butter needed for

8

batches is calculated by multiplying the butter required for one batch by the number of batches:

8 \times 12 = 96

ounces. Given that

1\text{ pound} = 16\text{ ounces}

, we convert ounces to pounds by dividing the total ounces by

16

. Therefore,

96 \div 16 = 6

pounds of butter are needed.

Step-by-Step Solution

1

Calculate the total ounces of butter needed.

96

ounces

Since each batch requires

12

ounces,

8

batches require

8 \times 12 = 96

ounces.

2

Convert the total ounces to pounds.

6

pounds

Divide the total number of ounces by the conversion factor (

16

ounces per pound):

\frac{96}{16} = 6

.

Key Concept

To convert a quantity from a smaller unit to a larger unit, multiply the initial quantity by the number of groups to find the total in the smaller unit, and then divide by the conversion factor that relates the two units.

Question 366Question →

A rideshare driver has a daily goal of earning at least $150. So far today, the driver has earned$ 45. The driver earns $12 per ride plus an average tip of$ 3 per ride. If the driver must pay a daily vehicle fee of $15, what is the minimum number of additional rides the driver must complete today to meet or exceed the daily earnings goal?

Show answer & explanation

Answer: 8

Answer

The minimum number of additional rides the driver must complete today is 8.

The driver earns

12 plus a

3 tip per ride, which is

15 per ride. Starting with

45 and subtracting the

15 fee leaves the driver with

30 before completing any new rides. To reach at least

150, the driver needs to earn at least

120 more. Dividing

120 by the

15 rate per ride gives a minimum of 8 rides.

Step-by-Step Solution

1

Define the variable

x

for the number of additional rides and write an inequality representing the total net earnings.

45 + 12x + 3x - 15 \geq 150

To represent the condition that the driver's total earnings, including initial earnings and new rides, minus the fee, must be at least $150.

2

Simplify the left side of the inequality by combining the constants and the

x

terms.

30 + 15x \geq 150

To group like terms and simplify the expression before solving.

3

Subtract 30 from both sides of the inequality to isolate the variable term.

15x \geq 120

To isolate the term with the variable on one side of the inequality.

4

Divide both sides of the inequality by 15 to solve for

x

.

x \geq 8

To find the minimum value of

x

that satisfies the inequality.

Key Concept

Solving linear inequalities in one variable to find a minimum threshold value in a real-world scenario.

Question 367Question →

A map has a scale where $2$ centimeters represents $15$ kilometers. If the distance between two cities on the map is $8$ centimeters, what is the actual distance between the two cities, in kilometers?

Show answer & explanation

Answer: 60

Answer

The actual distance between the two cities is

60

kilometers.

The correct answer is

60

. The map scale is

2

centimeters to

15

kilometers. The map distance between the cities is

8

centimeters, which is

4

times the scale distance of

2

centimeters (since

8 \div 2 = 4

). Therefore, the actual distance between the cities is

4

times the scale distance of

15

kilometers, which is

15 \times 4 = 60

kilometers.

Step-by-Step Solution

1

Set up a proportion using the map scale ratio of centimeters to kilometers.

\frac{2}{15} = \frac{8}{x}

This establishes that the ratio of map distance to actual distance remains constant.

2

Solve for the unknown actual distance,

x

, by cross-multiplying.

2x = 120

Multiplying the numerator of each fraction by the denominator of the other solves the proportion.

3

Divide by the coefficient of

x

to find the final actual distance.

x = 60

Dividing

120

by

2

isolates

x

and gives the actual distance in kilometers.

Key Concept

Setting up and solving proportions using scale factors.

Question 368Question →

At the beginning of the year, a library had $800$ history books. In the first half of the year, the number of history books increased by $15\%$ . In the second half of the year, the library acquired more history books, representing a percent increase of $x\%$ over the number of history books at the midyear point. If the library had a total of $1{}012$ history books at the end of the year, what is the value of $x$ ?

Show answer & explanation

Answer: 10

Answer

The value of

x

is

10

.

To find the second percent increase, we must first calculate the intermediate midyear value. A

15\%

increase on

800

is

800 \times 1.15 = 920

. The subsequent increase of

x\%

is based on this midyear value of

920

. Setting up the equation

920 \times (1 + \frac{x}{100}) = 1{}012

and solving for

x

yields

1 + \frac{x}{100} = 1.10

, which gives

x = 10

.

Step-by-Step Solution

1

Calculate the number of history books at the midyear point after the first increase of

15\%

920

books

An increase of

15\%

on the initial

800

books is calculated as

800 \times (1 + 0.15) = 920

.

2

Set up an equation for the second percent increase of

x\%

from the midyear value to the final value of

1{}012

920 \times (1 + \frac{x}{100}) = 1{}012

The second increase is

x\%

of the midyear value of

920

, resulting in the final value of

1{}012

.

3

Solve the equation for

x

x = 10

Divide both sides of the equation by

920

to get

1 + \frac{x}{100} = 1.10

, subtract

1

to get

\frac{x}{100} = 0.10

, and multiply by

100

to find

x = 10

.

Key Concept

Calculating successive percent increases by applying the percent change to the intermediate base value.

Alternative Method

Instead of calculating the intermediate number of books, we can express the final number of books as

800 \times 1.15 \times (1 + \frac{x}{100}) = 1{}012

. Simplifying

800 \times 1.15

gives

920

, so

920 \times (1 + \frac{x}{100}) = 1{}012

, leading to the same result of

x = 10

.

Estimated Time:1m 30s

Question 369Question →

A survey of 80 commuters asked about their primary method of transportation to work and their commute distance. The results are summarized in the table below.

Commute Distance	Bicycle	Walk	Total
Under 1 mile	15	25	40
1 mile or more	35	5	40
Total	50	30	80

Based on the table, what fraction of the commuters who commute a distance of under 1 mile choose walking as their primary transportation method?

Show answer & explanation

Answer: 0.625

Answer

5/8 or 0.625

To find the fraction of commuters who commute a distance of under 1 mile that choose walking, limit the sample space to the row representing 'Under 1 mile', which has a subtotal of 40 commuters. Within this row, 25 commuters walk. Thus, the fraction is 25/40, which simplifies to 5/8 (or 0.625).

Step-by-Step Solution

1

Identify the conditional group (denominator).

The row 'Under 1 mile' has a total of 40 commuters.

The phrase 'of the commuters who commute a distance of under 1 mile' restricts the group of interest to only those in this row.

2

Identify the target group within the restricted group (numerator).

25 commuters choose walking.

Within the 'Under 1 mile' row, we look at the 'Walk' column.

3

Calculate the conditional fraction.

25/40 = 5/8 (or 0.625).

Divide the target count by the conditional group total to find the fraction.

Key Concept

Conditional Probability from a Two-Way Table

Question 370Question →

In the $xy$ -plane, a line is represented by the equation $ay - bx = 24$ , where $a$ and $b$ are constants. If the line has a $y$ -intercept of $(0, -3)$ and passes through the point $(4, 5)$ , what is the slope of the line?

Show answer & explanation

Answer: 2

Answer

The slope of the line is 2.

Substituting the y-intercept

(0, -3)

into the equation

ay - bx = 24

yields

-3a = 24

, which gives

a = -8

. Substituting the point

(4, 5)

and

a = -8

into the equation yields

-8(5) - 4b = 24

, which simplifies to

-40 - 4b = 24

, giving

b = -16

. Re-assembling the equation gives

-8y + 16x = 24

. Solving for

y

in terms of

x

yields

y = 2x - 3

, where the coefficient of

x

is the slope, 2.

Step-by-Step Solution

1

Substitute the y-intercept

(0, -3)

into the given equation

ay - bx = 24

to find the value of

a

.

a = -8

Since the y-intercept lies on the line, its coordinates satisfy the line's equation.

2

Substitute the point

(4, 5)

and the value of

a = -8

into the equation to find the value of

b

.

b = -16

Since the point

(4, 5)

lies on the line, its coordinates must satisfy the equation.

3

Write the resulting equation

-8y + 16x = 24

in slope-intercept form (

y = mx + c

) to identify the slope.

y = 2x - 3

, which gives a slope of

2

.

The slope of a line in the form

y = mx + c

is represented by the coefficient

m

of

x

.

Key Concept

Finding the slope of a line from a given equation by determining its constant coefficients using known points.

Estimated Time:1m 30s

Question 371Question →

For the quadratic function $f(x) = -x^2 + bx + c$ , where $b$ and $c$ are positive constants, the maximum value of $f(x)$ is $k$ . The distance between the two $x$ -intercepts of the graph of $y = f(x)$ in the $xy$ -plane is equal to $\frac{2}{3}k$ . If the graph of $y = f(x)$ passes through the point $(1, 8)$ , what is the value of $b$ ?

Show answer & explanation

Answer: 4

Answer

The correct answer is 4.

The correct answer is 4. By expressing the maximum value

k

and the distance between the

x

-intercepts

d

in terms of the constants

b

and

c

, we set up the equation

d = \frac{2}{3}k

. This simplifies to

b^2 + 4c = 36

. Using the point

(1, 8)

, we establish

c = 9 - b

. Substituting this into the first equation yields

b^2 - 4b = 0

, giving

b = 4

as the only positive solution.

Step-by-Step Solution

1

Find the maximum value

k

of the function

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

in terms of

b

and

c

.

k = \frac{b^2}{4} + c

The maximum value of a quadratic function with a negative leading coefficient occurs at its vertex, where

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

.

2

Find the distance

d

between the

x

-intercepts of the graph in terms of

b

and

c

.

d = \sqrt{b^2 + 4c}

The

x

-intercepts are the roots of

-x^2 + bx + c = 0

, which are

x = \frac{b \pm \sqrt{b^2 + 4c}}{2}

. The distance between them is the difference of these roots.

3

Use the relation

d = \frac{2}{3}k

to find the value of the expression

b^2 + 4c

.

b^2 + 4c = 36

Substituting the expressions for

d

and

k

gives

\sqrt{b^2 + 4c} = \frac{1}{6}(b^2 + 4c)

. Solving this radical equation yields

b^2 + 4c = 36

.

4

Use the point

(1, 8)

to express

c

in terms of

b

.

c = 9 - b

Since the graph passes through

(1, 8)

, substituting

x = 1

and

y = 8

into

y = -x^2 + bx + c

yields

8 = -1 + b + c

, which simplifies to

c = 9 - b

.

5

Substitute

c = 9 - b

into

b^2 + 4c = 36

and solve for

b

.

b = 4

Substituting yields

b^2 + 4(9 - b) = 36 \implies b^2 - 4b = 0

. Solving for

b

gives

b = 0

or

b = 4

. Since

b

is a positive constant, we have

b = 4

.

Key Concept

Quadratic Functions and Graphs

Question 372Question →

A solar panel generates $120$ watt-hours of electricity every $3$ hours. At this rate, how many watt-hours of electricity will the solar panel generate in $8$ hours?

Show answer & explanation

Answer: 320

Answer

The solar panel will generate

320

watt-hours of electricity in

8

hours.

To find the total electricity generated, we determine the unit rate of energy generation:

\frac{120 \text{ watt-hours}}{3 \text{ hours}} = 40

watt-hours per hour. Multiplying this rate by the new duration of

8

hours gives

40 \times 8 = 320

watt-hours. Proportional reasoning also works:

\frac{120}{3} = \frac{x}{8}

, which yields

x = 320

.

Step-by-Step Solution

1

Calculate the hourly rate of electricity generation.

The rate is

40

watt-hours per hour.

Dividing the energy produced (

120

watt-hours) by the elapsed time (

3

hours) gives the unit rate.

2

Calculate the total energy generated over the new duration of

8

hours.

The total energy is

320

watt-hours.

Multiplying the unit rate (

40

watt-hours per hour) by the target duration (

8

hours) gives the total energy generated.

Key Concept

Calculating and applying unit rates to solve proportional relationships.

Alternative Method

Set up a proportion where the ratio of energy to time remains constant:

\frac{120}{3} = \frac{x}{8}

. Cross-multiply to solve for

x

:

3x = 120 \times 8

, which simplifies to

3x = 960

. Divide by

3

to find

x = 320

.

Estimated Time:45s

Question 373Question →

A technician at a chemistry lab prepares a solution by mixing $3$ milliliters of acid with every $7$ milliliters of distilled water. If the technician uses $35$ milliliters of distilled water to prepare the solution, how many milliliters of acid are needed?

Show answer & explanation

Answer: 15

Answer

The correct answer is 15.

To maintain the ratio of

3

milliliters of acid for every

7

milliliters of distilled water, the proportion

\frac{\text{acid}}{\text{water}} = \frac{3}{7} = \frac{x}{35}

is established. Multiplying both sides by

35

gives

x = 35 \times \frac{3}{7} = 15

milliliters of acid.

Step-by-Step Solution

1

Set up the proportion relating acid to distilled water.

\frac{3}{7} = \frac{x}{35}

, where

x

is the required amount of acid in milliliters.

The ratio of acid to distilled water must remain constant.

2

Solve the proportion for

x

.

x = 15

Multiply both sides of the equation by 35 to isolate

x

.

Key Concept

Solving proportions to find an unknown quantity when the ratio is constant.

Alternative Method

Find the scale factor:

35

divided by

7

is

5

. Multiply the acid amount (

3

) by this scale factor of

5

to get

15

.

Estimated Time:45s

Question 374Question →

A catering company charges a flat setup fee of $500 plus$ 35 per guest. A company has budgeted at most $2,400 for a banquet. If the company also wants to purchase a celebration cake for$ 150, what is the maximum number of guests they can invite to the banquet without exceeding their budget?

Show answer & explanation

Answer: 50

Answer

The maximum number of guests the company can invite without exceeding their budget is 50.

The total cost of the banquet consists of a flat setup fee (

500), a celebration cake (

150), and a variable cost of

35 per guest (

35g

, where

g

represents the number of guests). Since the company has budgeted at most

2,400, the sum of these expenses must be less than or equal to

2,400. This is represented by the inequality

35g + 500 + 150 \le 2400

. Combining the constants yields

35g + 650 \le 2400

. Subtracting 650 from both sides gives

35g \le 1750

. Dividing both sides by 35 results in

g \le 50$. Therefore, the maximum number of guests they can invite without exceeding their budget is 50.

Step-by-Step Solution

1

Set up the inequality representing the total expenses.

35g + 500 + 150 \le 2400

The sum of the fixed setup fee, the cake cost, and the variable per-guest cost must not exceed the maximum budget of $2,400.

2

Simplify the inequality by combining the constants.

35g + 650 \le 2400

Combining the fixed expenses (

500 setup fee and

150 cake) simplifies the calculation.

3

Isolate the variable term by subtracting the fixed cost from both sides.

35g \le 1750

This determines the portion of the budget that can be allocated specifically to guests.

4

Solve for the variable by dividing both sides by the cost per guest.

g \le 50

Dividing the remaining guest budget by the rate per guest gives the maximum number of guests allowed.

Key Concept

Formulating and solving multi-step linear inequalities in one variable to determine a maximum value in context.

Question 375Question →

A polynomial function $P(x)$ of degree 4 with real coefficients is symmetric about the line $x = 2$ in the $xy$ -plane. If $P(x)$ is divisible by $x^2 - 4x + 3$ , the remainder when $P(x)$ is divided by $x - 4$ is $36$ , and $P(2) = -8$ , what is the value of $P(5)$ ?

Show answer & explanation

Answer: 136

Answer

The value of

P(5)

is 136.

The correct answer of 136 is found by representing the symmetric fourth-degree polynomial as

P(x) = a(x - 2)^4 + b(x - 2)^2 + c

, solving for the coefficients using the roots at

x=3

, the point

(2, -8)

, and the remainder point

(4, 36)

, and then evaluating the resulting expression

(x-2)^4 + 7(x-2)^2 - 8

at

x=5

.

Step-by-Step Solution

1

Express the fourth-degree polynomial in a form that reflects its symmetry.

P(x) = a(x - 2)^4 + b(x - 2)^2 + c

Since the graph of

P(x)

is symmetric about the line

x = 2

, the polynomial expression in terms of

(x - 2)

must contain only even powers.

2

Use the divisibility condition to establish an equation for the coefficients.

P(3) = a(3-2)^4 + b(3-2)^2 + c = a + b + c = 0

The divisor

x^2 - 4x + 3

factors into

(x-1)(x-3)

. By the Factor Theorem,

P(3) = 0

and

P(1) = 0

.

3

Use the given value

P(2) = -8

to find the constant term

c

.

c = -8

Substituting

x = 2

into

P(x) = a(x - 2)^4 + b(x - 2)^2 + c

makes the terms with

(x-2)

equal to zero, leaving

P(2) = c

.

4

Formulate a system of linear equations for

a

and

b

.

a + b = 8

and

4a + b = 11

Substituting

c = -8

into

a + b + c = 0

gives

a + b = 8

. By the Remainder Theorem,

P(4) = 36

, which gives

a(4-2)^4 + b(4-2)^2 - 8 = 36 \implies 16a + 4b = 44 \implies 4a + b = 11

.

5

Solve the system of equations for

a

and

b

.

a = 1

and

b = 7

Subtracting

a + b = 8

from

4a + b = 11

yields

3a = 3 \implies a = 1

, which then gives

b = 7

.

6

Evaluate the polynomial at

x = 5

.

P(5) = 136

Substitute

a = 1

,

b = 7

,

c = -8

, and

x = 5

into the symmetric polynomial form:

P(5) = (5-2)^4 + 7(5-2)^2 - 8 = 3^4 + 7(3^2) - 8 = 81 + 63 - 8 = 136

.

Key Concept

Polynomial Factors and Graphs

Question 376Question →

At a certain car dealership, the number of hybrid vehicles sold in 2024 was $40\%$ of the number of gasoline vehicles sold in 2024. In 2025, the number of hybrid vehicles sold increased by $35\%$ compared to 2024, and the number of gasoline vehicles sold decreased by $10\%$ compared to 2024. If the dealership only sold hybrid and gasoline vehicles in both years, what percent of the total vehicles sold in 2025 were hybrid vehicles?

Show answer & explanation

Answer: 37.5

Answer

37.5

To find the percentage of hybrid vehicles sold in 2025, we first express the number of vehicles sold in 2024 in terms of a single variable. Let the number of gasoline vehicles sold in 2024 be

g

. Since the hybrid sales were

40\%

of the gasoline sales, the number of hybrid vehicles sold in 2024 is

0.40g

. In 2025, hybrid sales increased by

35\%

, resulting in

0.40g \times 1.35 = 0.54g

hybrid vehicles sold. Gasoline sales decreased by

10\%

, resulting in

g \times 0.90 = 0.90g

gasoline vehicles sold. The total number of vehicles sold in 2025 is the sum of these two quantities:

0.54g + 0.90g = 1.44g

. The percentage of hybrid vehicles sold in 2025 is the ratio of hybrid sales to total sales:

\frac{0.54g}{1.44g} \times 100 = 37.5\%

.

Step-by-Step Solution

1

Define variables for the number of vehicles sold in 2024.

Let the number of gasoline vehicles sold in 2024 be

g

. The number of hybrid vehicles sold in 2024 is

0.40g

.

Establishing initial values in terms of a single variable allows for comparison after percentage changes are applied.

2

Determine the number of hybrid and gasoline vehicles sold in 2025 after their respective percentage changes.

Hybrid vehicles sold in 2025:

0.40g \times (1 + 0.35) = 0.54g

. Gasoline vehicles sold in 2025:

g \times (1 - 0.10) = 0.90g

.

Applying the percent increase of

35\%

to the 2024 hybrid vehicles and the percent decrease of

10\%

to the 2024 gasoline vehicles.

3

Calculate the total number of vehicles sold in 2025.

Total vehicles sold in 2025:

0.54g + 0.90g = 1.44g

.

Summing the two types of vehicles sold in 2025 to find the new total.

4

Compute the percentage of hybrid vehicles out of the total vehicles sold in 2025.

\frac{0.54g}{1.44g} \times 100 = 37.5\%

.

Dividing the 2025 hybrid sales by the total 2025 sales and multiplying by 100 to convert to a percentage.

Key Concept

Calculating percent change and conditional percentages from initial relative ratios.

Question 377Question →

If

x

is a real number that satisfies the equation below, what is the value of

x

?

\frac{x + 2}{x - 4} - \frac{3}{x} = \frac{12}{x^2 - 4x}

Show answer & explanation

Answer: 1

Answer

The only valid real solution to the equation is

1

.

Multiplying the equation by the common denominator

x(x - 4)

simplifies the equation to

x^2 - x = 0

. Solving this quadratic equation yields

x = 0

and

x = 1

. Because

x = 0

leads to a division by zero in the original equation, it is extraneous. Therefore, the only valid real solution is

1

.

Step-by-Step Solution

1

Determine the common denominator and restrictions for the rational equation.

The common denominator is

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

. The restrictions are

x \neq 0

and

x \neq 4

.

Finding the common denominator allows us to eliminate fractions, while the restrictions help us identify potential extraneous solutions.

2

Multiply the entire equation by the common denominator

x(x - 4)

.

x(x + 2) - 3(x - 4) = 12

This step clears the rational expressions, leaving a polynomial equation.

3

Expand and simplify the polynomial equation.

x^2 + 2x - 3x + 12 = 12

, which simplifies to

x^2 - x = 0

.

Expanding the terms allows us to combine like terms and set the quadratic equation to zero.

4

Factor the quadratic equation to solve for

x

.

x(x - 1) = 0

, giving candidate solutions

x = 0

or

x = 1

.

Applying the zero-product property identifies the roots of the quadratic equation.

5

Verify the candidate solutions against the initial restrictions.

Since

x = 0

makes the denominators in the original equation equal to zero, it is extraneous. The candidate solution

x = 1

is valid.

We must verify solutions because multiplying by a variable expression can introduce extraneous roots that make the original expressions undefined.

Key Concept

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

Estimated Time:1m 30s

Question 378Question →

During a science experiment, a liquid is cooled at a constant rate of $0.5$ degrees Celsius per minute. What is the cooling rate of the liquid, in degrees Celsius per hour?

Show answer & explanation

Answer: 30

Answer

The cooling rate of the liquid is

30

degrees Celsius per hour.

To find the cooling rate in degrees Celsius per hour, we multiply the rate per minute by the conversion factor. There are

60

minutes in

1

hour. Multiplying

0.5

degrees Celsius per minute by

60

minutes per hour yields

30

degrees Celsius per hour.

Step-by-Step Solution

1

Identify the initial cooling rate per minute.

0.5

degrees Celsius per minute

This is the rate given in the problem statement.

2

Convert minutes to hours by multiplying the rate by

60

.

30

degrees Celsius per hour

An hour contains

60

minutes, so the total temperature decrease in one hour will be

60

times the decrease in one minute.

Key Concept

Unit conversions involving rates

Question 379Question →

In the quadratic equation $2x^2 - 12x + k = 0$ , $k$ is a constant. If the sum of the squares of the solutions to the equation is $26$ , what is the value of $k$ ?

Show answer & explanation

Answer: 10

Answer

The value of

k

is

10

.

The correct answer is

10

. By expressing the sum of the squares of the solutions as

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

, we can substitute the sum of the solutions (

-\frac{-12}{2} = 6

) and the product of the solutions (

\frac{k}{2}

) directly into the expression. This gives

26 = 36 - k

. Solving for

k

yields

k = 10

. Alternatively, solving the quadratic equation using the quadratic formula yields solutions

3 + \frac{\sqrt{144-8k}}{4}

and

3 - \frac{\sqrt{144-8k}}{4}

. Squaring these solutions and setting their sum equal to

26

simplifies to

18 + 2\left(\frac{144-8k}{16}\right) = 26

, which also solves to

k = 10

.

Step-by-Step Solution

1

Find the sum and product of the solutions using the coefficients of the quadratic equation.

The sum of the solutions is

6

and the product of the solutions is

\frac{k}{2}

.

By Vieta's formulas, for any quadratic equation

ax^2 + bx + c = 0

with solutions

x_1

and

x_2

, the sum of the solutions is

x_1 + x_2 = -\frac{b}{a}

and the product of the solutions is

x_1 x_2 = \frac{c}{a}

.

2

Apply the algebraic identity to express the sum of the squares of the solutions in terms of their sum and product.

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

This identity allows us to substitute the sum and product of the solutions directly without solving for the individual solutions first.

3

Substitute the values and solve for

k

.

26 = 6^2 - 2\left(\frac{k}{2}\right) \implies 26 = 36 - k \implies k = 10

Substituting the given sum of squares (

26

), the sum of solutions (

6

), and the product of solutions (

\frac{k}{2}

) allows us to solve for the unknown constant

k

directly.

Key Concept

Using the relationship between the roots and coefficients of a quadratic equation (Vieta's formulas) in combination with algebraic identities to solve for unknown constants.

Question 380Question →

A commercial bakery uses an automated flour silo. The mass of the flour in the silo, $F$ , in kilograms, is modeled as a linear function of the time $t$ , in hours, after the bakery opens. The table below shows the mass of the flour remaining in the silo at two different times during the day:

Time (hours), $t$	Mass of flour (kilograms), $F$
$3$	$1,850$
$7$	$1,490$

If the mass of the flour in the silo decreases at a constant rate of $r$ kilograms per hour, what is the value of $r$ ?

Show answer & explanation

Answer: 90

Answer

90

The rate of decrease of the flour is represented by the magnitude of the slope of the linear relationship. Using the points

(3, 1850)

and

(7, 1490)

from the table, the slope is calculated as

\frac{1490 - 1850}{7 - 3} = \frac{-360}{4} = -90

. This indicates that the mass of the flour decreases by

90

kilograms per hour. Therefore, the value of

r

is

90

.

Step-by-Step Solution

1

Identify the data points representing time and mass from the table.

(t_1, F_1) = (3, 1850)

and

(t_2, F_2) = (7, 1490)

We need two coordinates to find the slope of the linear relationship.

2

Calculate the slope (rate of change) of the linear function.

Slope =

\frac{1490 - 1850}{7 - 3} = -90

The slope formula

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

gives the rate of change of the mass of the flour per hour.

3

Determine the value of

r

based on the rate of decrease.

r = 90

The rate of decrease is the positive magnitude of the rate of change.

Key Concept

Interpreting rate of change (slope) from tabular data in a linear context

Estimated Time:1m 30s

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