All practice questions

Which of the following expressions is equivalent to $\frac{12x^8}{4x^2}$ for all $x \neq 0$ ?

8x^4

8x^6

3x^4

3x^6

3x^{10}

Show answer & explanation

Answer: $3x^6$

Answer

The expression

3x^6

To simplify the expression, divide the coefficients first:

\frac{12}{4} = 3

. Then, apply the quotient rule of exponents to the variable terms, which states that for any non-zero base,

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

. Subtracting the exponents gives

x^{8-2} = x^6

. Combining these results yields the correct expression

3x^6

.

Step-by-Step Solution

1

Divide the numerical coefficients of the terms.

\frac{12}{4} = 3

When simplifying a fraction with algebraic terms, the coefficients are divided normally.

2

Apply the quotient rule of exponents to simplify the variable terms.

\frac{x^8}{x^2} = x^{8-2} = x^6

According to the quotient rule of exponents, when dividing expressions with the same base, subtract the exponent in the denominator from the exponent in the numerator.

3

Multiply the simplified coefficient and variable results together.

3x^6

Combining the divided coefficient and the simplified variable expression gives the final simplified result.

Key Concept

Quotient Rule of Exponents

Alternative Method

Alternatively, you can expand the exponent terms in the numerator and denominator:

\frac{12 \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x}{4 \cdot x \cdot x}

. Simplifying the coefficients gives

3

, and canceling two pairs of

x

from both the numerator and denominator leaves six factors of

x

in the numerator, which simplifies to

3x^6

.

Estimated Time:45s

Question 1842Question →

Which of the following is the solution set for the inequality $|5 - 2x| < 9$ ?

-2 < x < 7

x < -2

or

x > 7

-7 < x < 2

x > -2

x < 7

Show answer & explanation

Answer: $-2 < x < 7$

Answer

-2 < x < 7

The correct solution is found by setting up the compound inequality

-9 < 5 - 2x < 9

. Subtracting

5

from all parts gives

-14 < -2x < 4

. Finally, dividing by

-2

and reversing the inequality signs yields the interval

-2 < x < 7

.

Step-by-Step Solution

1

Rewrite the absolute value inequality as a compound inequality.

-9 < 5 - 2x < 9

An absolute value inequality of the form

|u| < c

represents all points within distance

c

from

0

, which translates to

-c < u < c

.

2

Subtract

5

from all three parts of the inequality.

-14 < -2x < 4

To isolate the variable term, we perform the inverse operation of adding

5

, which is subtracting

5

.

3

Divide all three parts by

-2

and reverse the inequality signs.

-2 < x < 7

Dividing by a negative number reverses the inequality direction. Doing so gives

7 > x > -2

, which is conventionally written as

-2 < x < 7

.

Key Concept

Solving absolute value inequalities of the form

|ax + b| < c

Question 1843Question →

A company's daily profit is modeled by the function $P(x) = 3x - 50$ , where $x$ represents the number of items sold. The number of items sold depends on the number of hours the store is open, $h$ , according to the function $x(h) = 5h$ . What is the daily profit, in dollars, when the store is open for $8$ hours?

-26

40

70

120

170

Show answer & explanation

Answer: 70

Answer

70

Evaluating the composite function in the correct order,

P(x(8))

, yields

70

. First, find the number of items sold when the store is open for 8 hours:

x(8) = 5(8) = 40

. Then, calculate the daily profit by substituting this output into the profit function:

P(40) = 3(40) - 50 = 120 - 50 = 70

.

Step-by-Step Solution

1

Determine the number of items sold,

x

, when the store is open for

8

hours by evaluating the function

x(h) = 5h

at

h = 8

.

x(8) = 5 \times 8 = 40

items.

The number of items sold is the input for the profit function.

2

Substitute the number of items sold (

x = 40

) into the profit function

P(x) = 3x - 50

.

P(40) = 3(40) - 50 = 120 - 50 = 70

dollars.

Evaluating the profit function at the given number of items sold determines the final daily profit.

Key Concept

Evaluating a composite function by finding the output of the inner function and using it as the input for the outer function.

Estimated Time:1m 0s

Question 1844Question →

What is the sum of the solutions to the equation $3(x - 2)^2 = -6x + 36$ ?

2

-2

\frac{10}{3}

6

-6

Show answer & explanation

Answer: 2

Answer

The sum of the solutions is 2.

The correct answer is 2. Expanding the equation correctly and rearranging it into standard form yields

x^2 - 2x - 8 = 0

. Factoring this expression gives

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

, which results in the solutions

x = 4

and

x = -2

. The sum of these two solutions is

4 + (-2) = 2

.

Step-by-Step Solution

1

Divide both sides of the equation by 3 to simplify.

(x - 2)^2 = -2x + 12

Dividing both sides by the common factor of 3 simplifies the coefficients, making the algebraic manipulation and factoring steps easier.

2

Expand the squared binomial on the left side.

x^2 - 4x + 4 = -2x + 12

Expanding the binomial

(x - 2)^2

using the identity

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

is necessary to write the equation in standard form.

3

Move all terms to the left side to set the equation equal to zero.

x^2 - 2x - 8 = 0

A quadratic equation must be in the standard form

ax^2 + bx + c = 0

to solve it by factoring.

4

Factor the quadratic trinomial.

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

We find two numbers that multiply to

-8

and add to

-2

. These numbers are

-4

and

2

.

5

Set each factor to zero to solve for the roots.

x = 4

or

x = -2

By the zero product property, if the product of two factors is zero, at least one factor must equal zero.

6

Calculate the sum of the solutions.

4 + (-2) = 2

The question asks for the sum of the solutions, so we add the two calculated values of

x

.

Key Concept

Solving quadratic equations by expanding, rearranging into standard form, and factoring over the integers.

Question 1845Question →

A set of $10$ distinct positive integers has a median of $25$ and a range of $30$ . What is the greatest possible value of the mean of these $10$ integers?

25

29

31

33

37

Show answer & explanation

Answer: 33

Answer

The greatest possible value of the mean of the 10 integers is 33.

The correct answer is 33. To find the greatest possible mean of the 10 distinct positive integers, we must maximize their sum. Let the sorted integers be

x_1 < x_2 < x_3 < x_4 < x_5 < x_6 < x_7 < x_8 < x_9 < x_{10}

. The median is the average of the 5th and 6th terms, so

\frac{x_5 + x_6}{2} = 25

, or

x_5 + x_6 = 50

. Since the integers are distinct, the maximum value for

x_5

is 24, which forces

x_6 = 26

. To maximize the sum of the first five terms, they should be consecutive integers ending at 24:

x_1 = 20, x_2 = 21, x_3 = 22, x_4 = 23, x_5 = 24

. Since the range is 30, the maximum value is

x_{10} = x_1 + 30 = 20 + 30 = 50

. To maximize the remaining terms in the upper half, we choose the largest possible distinct integers less than 50:

x_7 = 47, x_8 = 48, x_9 = 49

. The maximum sum is

20 + 21 + 22 + 23 + 24 + 26 + 47 + 48 + 49 + 50 = 330

, yielding a maximum mean of

330 / 10 = 33

.

Step-by-Step Solution

1

Define the variables and apply the median constraint.

Let the 10 sorted distinct positive integers be

x_1 < x_2 < x_3 < x_4 < x_5 < x_6 < x_7 < x_8 < x_9 < x_{10}

. The median is the average of the 5th and 6th terms:

\frac{x_5 + x_6}{2} = 25

, which means

x_5 + x_6 = 50

.

Since the number of terms is even, the median is the average of the two middle terms.

2

Maximize the first five integers to find the maximum value of the first term.

Since the integers are distinct, we must have

x_5 < x_6

. With

x_5 + x_6 = 50

, the maximum possible integer value for

x_5

is 24 (which makes

x_6 = 26

). To maximize the sum, we make the preceding terms as large as possible:

x_4 = 23

,

x_3 = 22

,

x_2 = 21

, and

x_1 = 20

.

To maximize the mean, we must maximize the sum of all terms, which requires making each term as large as possible within the distinct integer constraints.

3

Apply the range constraint to find the maximum value of the last term.

The range is 30, so

x_{10} - x_1 = 30

. Since the maximum value of

x_1

is 20, the maximum possible value for

x_{10}

is

20 + 30 = 50

.

The range of a dataset is the difference between the maximum and minimum values.

4

Maximize the remaining terms in the upper half of the dataset.

We have

x_6 = 26

. The remaining terms must satisfy

26 < x_7 < x_8 < x_9 < x_{10} = 50

. To maximize the sum, we choose the largest possible distinct integers for these slots:

x_9 = 49

,

x_8 = 48

, and

x_7 = 47

.

This maximizes the sum of the upper half of the dataset under the constraint that the maximum value is 50.

5

Calculate the maximum sum and the resulting maximum mean.

Sum =

20 + 21 + 22 + 23 + 24 + 26 + 47 + 48 + 49 + 50 = 330

. Mean =

\frac{330}{10} = 33

.

The mean is calculated by dividing the sum of the elements by the number of elements.

Key Concept

Maximizing the mean of a bounded dataset using median, range, and distinctness constraints.

Question 1846Question →

For a real number $x$ , the equation $2(x + 5) = 16$ is true. What is the value of the expression $3x - 1$ ?

Show answer & explanation

Answer: 8

Answer

The value of the expression

3x - 1

is 8.

Solving the equation

2(x + 5) = 16

gives

x = 3

. Evaluating the expression

3x - 1

for

x = 3

yields

3(3) - 1 = 8

.

Step-by-Step Solution

1

Distribute the 2 on the left side of the equation.

2x + 10 = 16

To eliminate the parentheses using the distributive property.

2

Subtract 10 from both sides of the equation.

2x = 6

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

3

Divide both sides of the equation by 2.

x = 3

To find the value of

x

.

4

Substitute the value of

x

into the expression

3x - 1

.

3(3) - 1 = 8

To evaluate the final expression as requested by the question.

Key Concept

Solving multi-step linear equations and evaluating algebraic expressions.

Estimated Time:45s

Question 1847Question →

What is the positive real solution to the equation $\frac{3}{x-2} - \frac{2}{x} = 1$ ?

Show answer & explanation

Answer: 4

Answer

The positive real solution to the equation is 4.

By multiplying the entire equation by the least common denominator,

x(x-2)

, the rational equation is cleared of fractions, resulting in

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

. Simplifying this yields the quadratic equation

x^2 - 3x - 4 = 0

. Factoring the quadratic equation gives

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

, which has roots of

x = 4

and

x = -1

. The positive solution is 4.

Step-by-Step Solution

1

Multiply both sides of the equation by the least common denominator,

x(x-2)

, to eliminate the fractions.

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

Eliminating the denominators simplifies the rational equation into a polynomial equation.

2

Distribute and combine like terms to simplify both sides of the equation.

x + 4 = x^2 - 2x

Simplification is necessary before rearranging the equation into a solvable form.

3

Rearrange the terms into standard quadratic form:

ax^2 + bx + c = 0

.

x^2 - 3x - 4 = 0

Setting the quadratic expression equal to zero allows for factoring and solving.

4

Factor the quadratic equation.

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

Finding the factors reveals the potential values for

x

.

5

Solve for the roots and select the positive real solution.

x = 4

The equation yields two solutions,

x = 4

and

x = -1

, and the question explicitly requests the positive solution.

Key Concept

Solving rational equations by converting them to quadratic equations using the least common denominator.

Question 1848Question →

For the imaginary unit $i$ , where $i^2 = -1$ , what is the value of the expression $(3 + 2i)^2 - (3 - 2i)^2$ ?

0

10

24

24i

10 + 24i

Show answer & explanation

Answer: $24i$

Answer

The correct answer is

24i

.

The expression can be simplified by expanding each binomial term first. The first term,

(3 + 2i)^2

, expands to

9 + 12i + 4i^2

. Since

i^2 = -1

, this simplifies to

9 + 12i - 4 = 5 + 12i

. The second term,

(3 - 2i)^2

, expands to

9 - 12i + 4i^2

, which simplifies to

9 - 12i - 4 = 5 - 12i

. Subtracting the second simplified term from the first gives

(5 + 12i) - (5 - 12i) = 5 - 5 + 12i - (-12i) = 24i

.

Step-by-Step Solution

1

Expand the first squared binomial expression,

(3 + 2i)^2

.

(3 + 2i)^2 = 9 + 12i + 4i^2 = 9 + 12i - 4 = 5 + 12i

Apply the binomial squaring formula

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

and use the property of the imaginary unit where

i^2 = -1

.

2

Expand the second squared binomial expression,

(3 - 2i)^2

.

(3 - 2i)^2 = 9 - 12i + 4i^2 = 9 - 12i - 4 = 5 - 12i

Apply the binomial squaring formula

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

and use the property of the imaginary unit where

i^2 = -1

.

3

Subtract the second expanded expression from the first.

(5 + 12i) - (5 - 12i) = 5 + 12i - 5 + 12i = 24i

Distribute the negative sign to both terms of the second complex number and combine the real and imaginary parts.

Key Concept

Complex Numbers and Operations

Question 1849Question →

The cubic polynomial $2x^3 + 5x^2 - 8x - 20$ can be factored completely over the integers into the form $(x - a)(x + b)(cx + d)$ , where $a$ , $b$ , $c$ , and $d$ are positive integers. What is the value of $a + b + c + d$ ?

Show answer & explanation

Answer: 11

Answer

The value of

a + b + c + d

is 11.

The polynomial

2x^3 + 5x^2 - 8x - 20

can be factored completely by first grouping the terms as

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

. Factoring the difference of squares

x^2 - 4

yields

(x - 2)(x + 2)(2x + 5)

. Matching this to the given form

(x - a)(x + b)(cx + d)

gives the positive integers

a = 2

,

b = 2

,

c = 2

, and

d = 5

. The sum of these values is

2 + 2 + 2 + 5 = 11

.

Step-by-Step Solution

1

Group the terms of the polynomial

2x^3 + 5x^2 - 8x - 20

.

(2x^3 + 5x^2) - (8x + 20)

Grouping the terms allows us to look for common factors within each pair of terms.

2

Factor out the greatest common factor (GCF) from each group.

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

The GCF of the first group

2x^3 + 5x^2

is

x^2

, and the GCF of the second group

8x + 20

is

4

.

3

Factor out the common binomial factor

(2x + 5)

.

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

Both terms share the binomial factor

(2x + 5)

.

4

Factor the difference of squares

x^2 - 4

.

(x - 2)(x + 2)(2x + 5)

The term

x^2 - 4

is a difference of squares, which factors as

(x - 2)(x + 2)

.

5

Compare the factored expression with the template

(x - a)(x + b)(cx + d)

to determine the values of

a

,

b

,

c

, and

d

.

a = 2

,

b = 2

,

c = 2

, and

d = 5

Comparing the terms yields

x - a = x - 2 \implies a = 2

,

x + b = x + 2 \implies b = 2

, and

cx + d = 2x + 5 \implies c = 2, d = 5

. All values are positive integers as required.

6

Calculate the sum

a + b + c + d

.

11

Substituting the values of the variables into the expression gives

2 + 2 + 2 + 5 = 11

.

Key Concept

Factoring a cubic polynomial by grouping and then factoring the resulting difference of squares.

Alternative Method

Instead of factoring by grouping, we can use the Rational Root Theorem to find rational roots of the polynomial. The possible rational roots of

2x^3 + 5x^2 - 8x - 20 = 0

are of the form

\pm \frac{p}{q}

, where

p

is a factor of

20

and

q

is a factor of

2

. Testing values shows that

x = 2

and

x = -2

are roots, which corresponds to the linear factors

(x - 2)

and

(x + 2)

. Dividing the original cubic by their product,

(x^2 - 4)

, yields the remaining linear factor

(2x + 5)

.

Estimated Time:1m 30s

Question 1850Question →

For the functions $f(x) = x^2 - 3x - 1$ and $g(x) = |x - 1|$ , what is the product of all real values of $x$ for which $f(g(x)) = 3$ ?

5

0

-15

-5

-6

Show answer & explanation

Answer: -15

Answer

The product of all real values of

x

is

-15

.

The correct answer is

-15

. By substituting

g(x) = |x - 1|

into

f(x) = x^2 - 3x - 1

, we get

|x - 1|^2 - 3|x - 1| - 1 = 3

. Letting

u = |x - 1|

yields the quadratic

u^2 - 3u - 4 = 0

, which factors into

(u - 4)(u + 1) = 0

. Since

u \geq 0

, we have

u = 4

. Solving

|x - 1| = 4

yields

x = 5

and

x = -3

. The product of these solutions is

5 \times (-3) = -15

.

Step-by-Step Solution

1

Substitute the function

g(x) = |x - 1|

into

f(x) = x^2 - 3x - 1

to form the composite function

f(g(x))

.

f(g(x)) = |x - 1|^2 - 3|x - 1| - 1

To evaluate the composite function before setting it equal to

3

.

2

Set the composite function equal to

3

and substitute

u = |x - 1|

(where

u \geq 0

) to rewrite the equation as a quadratic in terms of

u

.

u^2 - 3u - 1 = 3 \Rightarrow u^2 - 3u - 4 = 0

Using substitution simplifies the absolute value equation into a standard quadratic equation.

3

Factor the quadratic equation

(u - 4)(u + 1) = 0

to solve for

u

, and discard the extraneous solution.

u = 4

or

u = -1

. Since

u = |x - 1| \geq 0

, we discard

u = -1

, leaving

u = 4

.

The absolute value of a real expression must be non-negative, so

u

cannot be

-1

.

4

Solve the absolute value equation

|x - 1| = 4

for all possible real values of

x

.

x - 1 = 4 \Rightarrow x = 5

, and

x - 1 = -4 \Rightarrow x = -3

.

An absolute value equation

|A| = B

(where

B > 0

) yields two scenarios:

A = B

and

A = -B

.

5

Calculate the product of the two real solutions found.

5 \times (-3) = -15

To find the final product of all real values of

x

that satisfy the equation.

Key Concept

Function composition and solving absolute value equations using quadratic substitution

Estimated Time:2m 0s

Question 1851Question →

For all real numbers $x$ such that $x \neq 1$ and $x \neq -1$ , which of the following is equivalent to the expression $\left(\frac{1}{x-1} + \frac{1}{x+1}\right)^2$ ?

\frac{1}{x^2}

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

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

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

\frac{4x^2}{(x^2-1)^2}

Show answer & explanation

Answer: $\frac{4x^2}{(x^2-1)^2}$

Answer

The expression is equivalent to the fraction with numerator four times x squared and denominator the square of the difference x squared minus one.

The correct answer is found by first rewriting the terms inside the parentheses with the common denominator

(x^2-1)

, which simplifies the sum to

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

. Squaring the numerator

(2x)^2

yields

4x^2

, and squaring the denominator yields

(x^2-1)^2

.

Step-by-Step Solution

1

Find a common denominator for the two fractions inside the parentheses.

The common denominator is

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

. The expression inside the parentheses becomes

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

.

To add rational expressions, they must have a common denominator.

2

Combine the numerators over the common denominator.

The sum is

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

.

Combine the like terms in the numerator to simplify the expression before squaring.

3

Square the simplified rational expression by squaring the numerator and the denominator separately.

\left(\frac{2x}{x^2-1}\right)^2 = \frac{(2x)^2}{(x^2-1)^2} = \frac{4x^2}{(x^2-1)^2}.

Apply the power of a quotient property to obtain the final simplified expression.

Key Concept

Simplifying rational expressions and applying the power of a quotient rule.

Estimated Time:1m 30s

Question 1852Question →

Let the function $f$ be defined by $f(x) = \frac{x - 1}{x + 1}$ for all real numbers $x \neq -1$ . Let $f^1(x) = f(x)$ , and let $f^n(x) = f(f^{n-1}(x))$ for all integers $n \geq 2$ . What is the value of $30 \cdot f^{2026}(3)$ ?

Show answer & explanation

Answer: -10

Answer

The value of the expression is -10.

Evaluating successive iterations of

f(3)

reveals a repeating sequence:

f^1(3) = 1/2

,

f^2(3) = -1/3

,

f^3(3) = -2

,

f^4(3) = 3

, and then

f^5(3) = 1/2

. This indicates a cycle of period 4. Dividing the power 2026 by the period 4 gives a remainder of 2, meaning

f^{2026}(3)

is equal to

f^2(3) = -1/3

. Multiplying this by 30 yields the final answer of -10.

Step-by-Step Solution

1

Calculate the first few compositions of the function evaluated at the given input

x = 3

.

f^1(3) = 1/2

,

f^2(3) = -1/3

,

f^3(3) = -2

, and

f^4(3) = 3

.

To look for a repeating pattern or periodic behavior in the iterated function composition.

2

Identify the period of the repeating cycle.

The cycle has a length of 4, repeating the values

[1/2, -1/3, -2, 3]

.

Since

f^4(3) = 3

, evaluating further iterations will yield the same sequence of values.

3

Use modular arithmetic to find the value of the 2026th composition.

2026 \equiv 2 \pmod 4

, meaning

f^{2026}(3) = f^2(3) = -1/3

.

Since the cycle repeats every 4 iterations, dividing 2026 by 4 yields a remainder of 2, indicating the second value in the sequence.

4

Multiply the computed composition value by the given coefficient.

30 \cdot (-1/3) = -10

.

To find the final value of the expression

30 \cdot f^{2026}(3)

.

Key Concept

Evaluating repeated function compositions using periodicity and modular arithmetic.

Question 1853Question →

A gardener starts with $15$ flowers already planted in a garden. She plans to plant additional flowers at a constant rate of $8$ flowers per hour. How many hours will it take the gardener to have a total of $79$ flowers planted?

Show answer & explanation

Answer: 8

Answer

It will take the gardener

8

hours to have a total of

79

flowers planted.

The total number of flowers planted can be modeled by the linear equation

15 + 8h = 79

, where

h

is the number of hours. Subtracting

15

from both sides of the equation gives

8h = 64

. Dividing both sides by

8

reveals that

h = 8

hours.

Step-by-Step Solution

1

Set up the linear equation representing the total flowers planted over time.

15 + 8h = 79

, where

h

is the number of hours.

The gardener begins with

15

flowers and adds

8

flowers for each hour

h

, with the final goal of

79

total flowers.

2

Isolate the variable term by subtracting the initial number of flowers from the total.

8h = 64

Subtracting

15

from both sides of the equation isolates the term containing the variable

h

.

3

Divide by the rate to solve for the number of hours.

h = 8

Dividing the remaining flowers to be planted (

64

) by the planting rate (

8

flowers per hour) yields the total number of hours required.

Key Concept

Translating a real-world scenario into a linear equation and solving for the unknown variable.

Question 1854Question →

If $x$ and $y$ are real numbers, the expression $3x(x - 2y)^2 - x^2(3x - 10y) - y^2(2x - y)$ can be simplified to which of the following?

-22x^2y + 10xy^2 - y^3

-2x^2y + 10xy^2 - y^3

-2x^2y + 10xy^2 + y^3

10x^2y + 10xy^2 + y^3

-2x^2y + 10xy^2 + y^2

Show answer & explanation

Answer: $-2x^2y + 10xy^2 + y^3$

Answer

The correct expression is

-2x^2y + 10xy^2 + y^3

.

The correct expression

-2x^2y + 10xy^2 + y^3

is obtained by correctly expanding the binomial

(x-2y)^2

into

x^2 - 4xy + 4y^2

, distributing the outer terms

3x

,

-x^2

, and

-y^2

to all terms inside their respective parentheses, and then combining the coefficients of the matching variable terms: the

x^3

terms cancel out, the

x^2y

terms combine to

-2x^2y

, the

xy^2

terms combine to

10xy^2

, and the

y^3

term remains.

Step-by-Step Solution

1

Expand the binomial expression

(x - 2y)^2

.

x^2 - 4xy + 4y^2

Before distributing

3x

, the squared binomial must be expanded using the identity

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

.

2

Multiply the expanded binomial by the coefficient

3x

.

3x^3 - 12x^2y + 12xy^2

Distribute

3x

to each of the three terms in the expanded expression:

3x(x^2) = 3x^3

,

3x(-4xy) = -12x^2y

, and

3x(4y^2) = 12xy^2

.

3

Distribute

-x^2

to

(3x - 10y)

.

-3x^3 + 10x^2y

Distribute

-x^2

to both terms, noting that multiplying two negative signs yields a positive term:

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

and

-x^2(-10y) = +10x^2y

.

4

Distribute

-y^2

to

(2x - y)

.

-2xy^2 + y^3

Distribute

-y^2

to both terms, adding the exponents of

y

where appropriate:

-y^2(2x) = -2xy^2

and

-y^2(-y^1) = +y^3

.

5

Combine all parts and group the like terms together.

-2x^2y + 10xy^2 + y^3

Group and add the coefficients of identical variable terms:

(3x^3 - 3x^3) + (-12x^2y + 10x^2y) + (12xy^2 - 2xy^2) + y^3 = 0x^3 - 2x^2y + 10xy^2 + y^3

.

Key Concept

Simplifying polynomial expressions with multiple variables by expanding binomials, distributing coefficients (including negative signs), and combining like terms.

Question 1855Question →

A company produces solar-powered chargers. The daily production cost, in dollars, is a linear function of the number of chargers produced. The setup cost is the cost when $0$ chargers are produced. If the company produces $15$ chargers, the total daily cost is $400$ dollars. If the company produces $25$ chargers, the total daily cost is $580$ dollars. The company updates its production process, which reduces the cost per charger by $20\%$ but increases the setup cost by $50$ dollars. Under the updated process, what is the total daily cost, in dollars, to produce $30$ chargers?

562

dollars

586

dollars

612

dollars

714

dollars

828

dollars

Show answer & explanation

Answer: $612$ dollars

Answer

The correct daily cost under the updated process is

612

dollars.

The correct answer is

612

dollars. First, the relationship between unit cost and setup cost is modeled as a linear equation. Subtracting the cost of producing

15

chargers (

400

dollars) from the cost of producing

25

chargers (

580

dollars) gives the cost of producing the

10

additional units, which is

180

dollars. This determines the unit cost is

18

dollars per charger. Substituting this value back shows the original setup cost is

130

dollars. The updated rate decreases the unit cost to

14.40

dollars (

80\%

of

18

) and increases the setup cost to

180

dollars (

130 + 50

). For

30

chargers, the total daily cost is

14.40(30) + 180 = 612

dollars.

Step-by-Step Solution

1

Set up a system of linear equations using the cost function

C(c) = mc + S

, where

m

is the cost per charger,

S

is the setup cost, and

c

is the number of chargers.

Equation 1:

15m + S = 400

Equation 2:

25m + S = 580

This establishes the relationship between production volume and total cost under the initial process.

2

Solve the system of equations for the unit cost

m

and the setup cost

S

.

Subtracting Equation 1 from Equation 2 yields

10m = 180 \implies m = 18

.
Substituting

m = 18

into Equation 1 yields

15(18) + S = 400 \implies 270 + S = 400 \implies S = 130

.

This determines the original pricing parameters.

3

Apply the updates to the cost parameters.

New unit cost:

18 \times (1 - 0.20) = 14.40

dollars.
New setup cost:

130 + 50 = 180

dollars.

This accounts for the

20\%

decrease in the variable cost and the

50

dollar increase in the fixed setup cost.

4

Evaluate the new linear cost function for

30

chargers.

C_{\text{new}}(30) = 14.40(30) + 180 = 432 + 180 = 612

dollars.

This calculates the total daily cost under the updated process.

Key Concept

Translating and Solving Algebraic Word Problems

Alternative Method

Instead of solving for the setup cost first, note that under the original process, the cost of

30

chargers would be the cost of

15

chargers plus the cost of

15

more units:

400 + 15(18) = 670

dollars. The updated process reduces the rate of each of the

30

units by

20\%

of

18

dollars (saving

3.60 \times 30 = 108

dollars) and increases the setup cost by

50

dollars. Thus, the new cost is

670 - 108 + 50 = 612

dollars.

Estimated Time:2m 0s

Question 1856Question →

When the expression $2x(3x - y)^2 - 3y^2(x - 2y) - x^2(18x - 15y)$ is simplified by combining like terms, what is the coefficient of $x^2y$ ?

-27

-12

3

15

-15

Show answer & explanation

Answer: 3

Answer

The coefficient of

x^2y

is

3

.

The correct coefficient of

3

is obtained by expanding the expression step-by-step. First, the square of the binomial

(3x - y)^2

is

9x^2 - 6xy + y^2

. Distributing

2x

gives

18x^3 - 12x^2y + 2xy^2

. Next, distributing

-x^2

to

(18x - 15y)

gives

-18x^3 + 15x^2y

. Combining the

x^2y

terms gives

-12x^2y + 15x^2y = 3x^2y

, meaning the coefficient is

3

.

Step-by-Step Solution

1

Expand the squared binomial term

(3x - y)^2

using the algebraic identity

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

.

(3x - y)^2 = 9x^2 - 6xy + y^2

This is necessary to remove the parentheses before distributing the outer variable.

2

Multiply the term

2x

by each term inside the expanded binomial expression:

2x(9x^2 - 6xy + y^2)

.

18x^3 - 12x^2y + 2xy^2

Applying the distributive property expands the first part of the expression.

3

Distribute

-x^2

to the terms inside the parentheses

(18x - 15y)

, paying close attention to the signs.

-18x^3 + 15x^2y

This expands the third part of the expression and correctly distributes the negative sign.

4

Identify and combine the like terms of the form

x^2y

from the expanded parts of the expression.

-12x^2y + 15x^2y = 3x^2y

To find the coefficient of

x^2y

, we only need to sum the coefficients of the terms that contain exactly

x^2y

.

Key Concept

Simplifying Expressions and Combining Like Terms

Estimated Time:1m 30s

Question 1857Question →

An electronics retailer sells tablet computers. The retailer's weekly revenue from tablets, in thousands of dollars, is modeled by the expression $\frac{2}{5}(3x - 4)$ , where $x$ represents the average number of tablets sold per day. The weekly operating costs, in thousands of dollars, are modeled by the expression $\frac{1}{3}\left(2x - \frac{1}{2}\right)$ . The retailer's weekly profit is equal to the profit of a competitor, which is modeled by $\frac{1}{4}(x + 5) - \frac{11}{6}$ thousand dollars. If the retailer and the competitor earn the same weekly profit, what is the value of $12x - 5$ ?

31

3

\frac{767}{17}

51

\frac{367}{97}

Show answer & explanation

Answer: 31

Answer

31

The correct answer is 31. By equating the profit models, we get

\frac{2}{5}(3x - 4) - \frac{1}{3}(2x - 0.5) = \frac{1}{4}(x + 5) - \frac{11}{6}

. Multiplying the entire equation by the least common multiple of the denominators (60) yields

24(3x - 4) - 20(2x - 0.5) = 15(x + 5) - 110

. Expanding the terms gives

72x - 96 - 40x + 10 = 15x + 75 - 110

, which simplifies to

32x - 86 = 15x - 35

. Solving for

x

gives

17x = 51

, or

x = 3

. Evaluating the required expression

12x - 5

for

x = 3

gives

12(3) - 5 = 31

.

Step-by-Step Solution

1

Set up the equation equating the retailer's profit (revenue minus cost) to the competitor's profit.

\frac{2}{5}(3x - 4) - \frac{1}{3}\left(2x - \frac{1}{2}\right) = \frac{1}{4}(x + 5) - \frac{11}{6}

Profit is calculated as revenue minus operating costs. Since the retailer and the competitor earn the same profit, their profit models are equal.

2

Multiply the entire equation by 60 to eliminate all fractional denominators.

24(3x - 4) - 20\left(2x - \frac{1}{2}\right) = 15(x + 5) - 10(11)

The least common multiple (LCM) of 5, 3, 4, and 6 is 60. Multiplying both sides by 60 simplifies the equation into integer terms.

3

Distribute the coefficients to remove parentheses, taking care with the negative signs.

72x - 96 - 40x + 10 = 15x + 75 - 110

Distributing

-20

across

\left(2x - \frac{1}{2}\right)

yields

-40x + 10

, and distributing

24

across

(3x - 4)

yields

72x - 96

.

4

Combine like terms on both sides of the equation.

32x - 86 = 15x - 35

On the left side,

72x - 40x = 32x

and

-96 + 10 = -86

. On the right side,

75 - 110 = -35

.

5

Isolate the variable term

x

on one side and the constants on the other.

17x = 51 \implies x = 3

Subtracting

15x

from both sides gives

17x - 86 = -35

. Adding

86

to both sides gives

17x = 51

. Dividing by 17 yields

x = 3

.

6

Substitute

x = 3

into the requested expression

12x - 5

to find the final value.

12(3) - 5 = 31

The question asks for the value of the expression

12x - 5

rather than just the variable

x

.

Key Concept

Solving multi-step linear equations containing fractions by clearing denominators and distributing terms correctly.

Question 1858Question →

For all non-zero real numbers $x$ and $y$ , which of the following is equivalent to the expression $\frac{\left( 2x^{-1} y^2 + \frac{1}{2} x^{-1} y^2 \right)^{-3}}{(x^2 y^{-3})^{-2}}$ ?

\frac{5x^7}{2y^{12}}

\frac{8x^7}{125y^{12}}

\frac{65x^7}{8y^{12}}

\frac{8}{125x^{12}y^{36}}

\frac{x^7}{y^{12}}

Show answer & explanation

Answer: $\frac{8x^7}{125y^{12}}$

Answer

\frac{8x^7}{125y^{12}}

To find the equivalent expression, we first combine the like terms inside the parentheses in the numerator to get

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

. Raising this product to the power of

-3

yields

\left(\frac{5}{2}\right)^{-3} (x^{-1})^{-3} (y^2)^{-3} = \frac{8}{125} x^3 y^{-6}

. Next, the denominator simplifies to

(x^2 y^{-3})^{-2} = x^{-4} y^6

. Dividing the numerator by the denominator requires subtracting the exponents of like bases: for

x

, we have

3 - (-4) = 7

, and for

y

, we have

-6 - 6 = -12

. This results in

\frac{8}{125} x^7 y^{-12}

, which is equivalent to the correct expression

\frac{8x^7}{125y^{12}}

.

Step-by-Step Solution

1

Combine the like terms inside the parentheses in the numerator.

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

Before applying the outer negative exponent, it is mathematically simpler to combine the like terms inside the grouping.

2

Apply the power of

-3

to the term in the numerator.

\left(\frac{5}{2} x^{-1} y^2\right)^{-3} = \left(\frac{5}{2}\right)^{-3} (x^{-1})^{-3} (y^2)^{-3} = \frac{8}{125} x^3 y^{-6}

The power of a product rule

(ab)^n = a^n b^n

and the power of a power rule

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

are applied to expand the term.

3

Simplify the denominator by applying the power of

-2

.

(x^2 y^{-3})^{-2} = (x^2)^{-2} (y^{-3})^{-2} = x^{-4} y^6

The power of a product rule is applied to the denominator to resolve the outer exponent.

4

Divide the simplified numerator by the simplified denominator using the quotient rule for exponents.

\frac{\frac{8}{125} x^3 y^{-6}}{x^{-4} y^6} = \frac{8}{125} x^{3 - (-4)} y^{-6 - 6} = \frac{8}{125} x^7 y^{-12} = \frac{8x^7}{125y^{12}}

The quotient rule

a^m / a^n = a^{m-n}

is used to subtract the exponents of the corresponding variables, and negative exponents are rewritten in the denominator.

Key Concept

Properties of Exponents in Algebraic Expressions

Alternative Method

Alternatively, you can write out all variables with positive exponents before simplifying. Rewrite the term in the numerator as

\frac{2y^2}{x} + \frac{y^2}{2x} = \frac{5y^2}{2x}

. Raising this to the

-3

power flips the fraction and cubes it, yielding

\left(\frac{2x}{5y^2}\right)^3 = \frac{8x^3}{125y^6}

. Simplifying the denominator yields

\frac{1}{(x^2 y^{-3})^2} = \frac{1}{x^4 y^{-6}} = \frac{y^6}{x^4}

. Dividing the numerator by the denominator yields

\frac{8x^3}{125y^6} \div \frac{y^6}{x^4} = \frac{8x^3}{125y^6} \cdot \frac{x^4}{y^6} = \frac{8x^7}{125y^{12}}

.

Estimated Time:2m 0s

Question 1859Question →

A dog-walking service charges a flat fee of $10$ dollars per visit plus $20$ dollars for each dog walked. If a client was charged a total of $90$ dollars for a single visit, how many dogs were walked during that visit?

9

5

4

7

60

Show answer & explanation

Answer: 4

Answer

4

The correct answer is

4

. The relation between the total cost and the number of dogs walked is

10 + 20d = 90

. Subtracting

10

from both sides yields

20d = 80

, and dividing by

20

yields

d = 4

.

Step-by-Step Solution

1

Set up the equation based on the scenario details.

Let

d

be the number of dogs walked. The total cost is represented by the equation

10 + 20d = 90

.

The total cost of

90

dollars consists of a flat fee of

10

dollars plus

20

dollars per dog walked.

2

Subtract the flat fee from both sides of the equation.

20d = 80

Subtracting

10

from both sides isolates the cost of walking the dogs.

3

Divide both sides by the per-dog rate to solve for

d

.

d = 4

Dividing by

20

yields the number of dogs walked.

Key Concept

Translating verbal descriptions of multi-step scenarios into linear equations and solving them.

Question 1860Question →

For all non-zero real numbers $x$ , the expression $x^4 \cdot (x^3)^k$ is equivalent to $x^{10}$ . What is the value of the integer $k$ ?

Show answer & explanation

Answer: 2

Answer

The correct answer is 2.

First, use the power of a power property to write

(x^3)^k

as

x^{3k}

. The expression then becomes

x^4 \cdot x^{3k}

. Next, use the product of powers property to combine the terms into

x^{4+3k}

. Since the expression is equivalent to

x^{10}

, set the exponents equal:

4 + 3k = 10

. Solving this equation gives

3k = 6

, which simplifies to

k = 2

.

Step-by-Step Solution

1

Apply the power of a power rule

(x^a)^b = x^{ab}

to simplify

(x^3)^k

.

x^{3k}

To raise a power to another power, multiply the exponents.

2

Apply the product of powers rule

x^a \cdot x^b = x^{a+b}

to combine the terms

x^4 \cdot x^{3k}

.

x^{4+3k}

When multiplying exponential terms with the same base, add their exponents.

3

Set the combined exponent

4+3k

equal to the target exponent

10

and solve for

k

.

k = 2

Since the bases are equal and non-zero, their exponents must be equal.

Key Concept

Properties of exponents in algebraic expressions (power of a power rule and product of powers rule)

Estimated Time:45s

Answer

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