All practice questions

Question 1941Question →

Let the functions $f$ and $g$ be defined for all real numbers by $f(x) = x^2 - 6x + 7$ and $g(x) = |2x - 5|$ . What is the sum of all real values of $x$ for which $f(g(x)) = 14$ ?

3

5

6

8

10

Show answer & explanation

Answer: 5

Answer

The sum of all possible real values of

x

is

5

.

The correct answer is

5

. Setting

f(g(x)) = 14

yields

(g(x))^2 - 6g(x) + 7 = 14

, which simplifies to

(g(x))^2 - 6g(x) - 7 = 0

. Factoring this quadratic gives

g(x) = 7

or

g(x) = -1

. Because

g(x) = |2x - 5|

cannot be negative, we discard the negative case. Solving

|2x - 5| = 7

gives

2x - 5 = 7 \Rightarrow x = 6

and

2x - 5 = -7 \Rightarrow x = -1

. The sum of these values is

6 + (-1) = 5

.

Step-by-Step Solution

1

Set up the composite equation

f(g(x)) = 14

by substituting

g(x)

into

f(x)

.

(g(x))^2 - 6g(x) + 7 = 14

This defines the relation between

g(x)

and the target output value.

2

Rearrange the equation into a standard quadratic form and solve for

g(x)

.

(g(x))^2 - 6g(x) - 7 = 0 \Rightarrow (g(x) - 7)(g(x) + 1) = 0

, so

g(x) = 7

or

g(x) = -1

.

Factoring the quadratic equation gives the possible values for the inner function

g(x)

.

3

Apply the definition of

g(x)

to solve for

x

and discard invalid cases.

Since

g(x) = |2x - 5|

must be non-negative, the case

g(x) = -1

has no real solutions. For the case

g(x) = 7

, we have

|2x - 5| = 7

.

The range of an absolute value function is

[0, \infty)

, making negative outputs impossible.

4

Solve the absolute value equation

|2x - 5| = 7

by setting up both positive and negative cases.

Either

2x - 5 = 7 \Rightarrow 2x = 12 \Rightarrow x = 6

, or

2x - 5 = -7 \Rightarrow 2x = -2 \Rightarrow x = -1

.

An absolute value equation

|u| = c

splits into

u = c

and

u = -c

.

5

Sum the valid solutions for

x

.

6 + (-1) = 5

The question asks for the sum of all real values of

x

that satisfy the equation.

Key Concept

Solving equations involving composite functions and absolute values

Estimated Time:2m 0s

Question 1942Question →

If $a = -27$ , $b = -\frac{1}{4}$ , and $c = -2$ , what is the value of the algebraic expression $a^{-4/3} - b^{-2} \cdot c^{-5}$ ?

-\frac{79}{162}

\frac{79}{162}

\frac{83}{162}

\frac{163}{2}

\frac{1295}{2592}

Show answer & explanation

Answer: $\frac{83}{162}$

Answer

The correct value of the algebraic expression is

\frac{83}{162}

.

Evaluating each term of the expression with the given values yields:

a^{-4/3} = \frac{1}{81}

,

b^{-2} = 16

, and

c^{-5} = -\frac{1}{32}

. Applying the order of operations, we multiply

16

by

-\frac{1}{32}

first to obtain

-\frac{1}{2}

. We then subtract this product from

\frac{1}{81}

, which simplifies to

\frac{1}{81} + \frac{1}{2} = \frac{83}{162}

.

Step-by-Step Solution

1

Substitute the given values of

a

,

b

, and

c

into the algebraic expression.

The expression is written as

(-27)^{-4/3} - (-\frac{1}{4})^{-2} \cdot (-2)^{-5}

.

This establishes the numerical expression to be evaluated.

2

Evaluate the first term,

(-27)^{-4/3}

.

(-27)^{-4/3} = \frac{1}{(-27)^{4/3}} = \frac{1}{((-27)^{1/3})^4} = \frac{1}{(-3)^4} = \frac{1}{81}

.

A negative exponent indicates a reciprocal, and a fractional exponent of

4/3

indicates taking the cube root and then raising to the fourth power.

3

Evaluate the second term,

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

.

(-\frac{1}{4})^{-2} = (-4)^2 = 16

.

Raising a fraction to a negative integer power is equivalent to raising its reciprocal to the corresponding positive integer power.

4

Evaluate the third term,

(-2)^{-5}

.

(-2)^{-5} = \frac{1}{(-2)^5} = -\frac{1}{32}

.

Evaluating a negative base raised to an odd negative power results in a negative unit fraction.

5

Substitute the evaluated terms back into the original expression and apply the order of operations.

E = \frac{1}{81} - 16 \cdot \left(-\frac{1}{32}\right) = \frac{1}{81} - \left(-\frac{16}{32}\right) = \frac{1}{81} + \frac{1}{2} = \frac{2 + 81}{162} = \frac{83}{162}

.

Multiplication must be performed before subtraction according to standard mathematical order of operations.

Key Concept

Evaluating expressions containing multiple variables with fractional exponents, negative bases, and standard order of operations.

Estimated Time:2m 0s

Question 1943Question →

A rectangular photograph has a length of $x$ inches and a width that is $1.5$ inches shorter than its length. If the area of the photograph is $10$ square inches, what is the length, in inches, of the photograph?

2.5

3.25

4

5

5.75

Show answer & explanation

Answer: 4

Answer

4

The correct length is 4. When the length is 4 inches, the width is 1.5 inches shorter, which is 2.5 inches. The area is the product of these dimensions: 4 * 2.5 = 10 square inches, which matches the given condition.

Step-by-Step Solution

1

Set up the equation for the area of the photograph.

x(x - 1.5) = 10

The area of a rectangle is found by multiplying its length (

x

) by its width (

x - 1.5

).

2

Distribute the

x

and subtract

10

from both sides to write the equation in standard form.

x^2 - 1.5x - 10 = 0

Standard form for a quadratic equation is

ax^2 + bx + c = 0

.

3

Multiply the entire equation by

2

to clear the decimal coefficient.

2x^2 - 3x - 20 = 0

Working with integer coefficients makes factoring or applying the quadratic formula easier.

4

Factor the quadratic expression.

(2x + 5)(x - 4) = 0

Factoring helps find the roots of the equation directly.

5

Solve for

x

by setting each factor to zero.

x = -2.5

or

x = 4

If the product of two numbers is zero, at least one of the numbers must be zero.

6

Exclude the negative solution.

x = 4

The length of a photograph must be a positive value.

Key Concept

Solving a quadratic equation by translating a word problem and factoring.

Question 1944Question →

A large rectangle has a length of $4x + 3$ inches and a width of $3x - 2$ inches. A smaller rectangular region is cut out from the center. The cut-out region has a length of $2x - 1$ inches and a width of $x - 5$ inches. The area of the remaining region can be written as the polynomial $Ax^2 + Bx + C$ , where $A$ , $B$ , and $C$ are integers. What is the value of the coefficient $B$ ?

Show answer & explanation

Answer: 12

Answer

The correct answer is 12, which is the coefficient of the linear term in the simplified remaining area polynomial.

The remaining area is found by subtracting the area of the smaller rectangle from the area of the larger rectangle. The area of the larger rectangle is

(4x + 3)(3x - 2) = 12x^2 + x - 6

. The area of the smaller rectangle is

(2x - 1)(x - 5) = 2x^2 - 11x + 5

. Subtracting the two yields

(12x^2 + x - 6) - (2x^2 - 11x + 5) = 10x^2 + 12x - 11

. Matching this to

Ax^2 + Bx + C

, the coefficient

B

of the linear term is

12

.

Step-by-Step Solution

1

Find the area of the outer rectangle.

Area =

12x^2 + x - 6

Multiply the length

(4x + 3)

and the width

(3x - 2)

using the distributive property.

2

Find the area of the inner rectangle.

Area =

2x^2 - 11x + 5

Multiply the length

(2x - 1)

and the width

(x - 5)

using the distributive property.

3

Subtract the inner area from the outer area to find the remaining area.

Remaining Area =

10x^2 + 12x - 11

Subtract the polynomial

(2x^2 - 11x + 5)

from

(12x^2 + x - 6)

by distributing the negative sign and combining like terms.

4

Identify the coefficient

B

of the

x

term.

B = 12

Match the simplified polynomial

10x^2 + 12x - 11

with the form

Ax^2 + Bx + C

.

Key Concept

Operations on Polynomials (multiplication and subtraction of polynomials)

Question 1945Question →

A sequence is defined by the formula $a_n = 2 + 3 \cdot \frac{2^n \cdot 2^{n-1}}{6}$ for all integers $n \geq 1$ . What is the value of the second term, $a_2$ ?

4

6

\frac{20}{3}

\frac{26}{7}

10

Show answer & explanation

Answer: $6$

Answer

The value of the second term is 6.

The correct answer is 6. When substituting

n = 2

into the formula, the numerator simplifies to

2^2 \cdot 2^1 = 8

. The fraction becomes

\frac{8}{6} = \frac{4}{3}

. Multiplying by 3 yields 4, and adding 2 yields 6.

Step-by-Step Solution

1

Substitute

n = 2

into the sequence formula.

a_2 = 2 + 3 \cdot \frac{2^2 \cdot 2^{2-1}}{6}

To find the second term, we replace

n

with 2 throughout the expression.

2

Simplify the exponents in the numerator.

2^2 \cdot 2^{2-1} = 4 \cdot 2^1 = 4 \cdot 2 = 8

Evaluate the exponential terms in the numerator before performing other operations.

3

Substitute the simplified numerator back and evaluate the fraction.

\frac{8}{6} = \frac{4}{3}

Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, 2.

4

Multiply by 3 and add 2 to get the final answer.

2 + 3 \cdot \frac{4}{3} = 2 + 4 = 6

Perform multiplication before addition according to the order of operations.

Key Concept

Evaluating terms of a sequence given by an explicit formula involving exponent rules.

Estimated Time:1m 0s

Question 1946Question →

A local bakery sells two types of muffins: blueberry and bran. Store 1 sells 30 blueberry muffins and 20 bran muffins. Store 2 sells 40 blueberry muffins and 35 bran muffins. Blueberry muffins cost $3 each, and bran muffins cost$ 2 each. This information is represented by the inventory matrix $M$ and the price matrix $P$ :

M = \begin{pmatrix} 30 & 20 \\ 40 & 35 \end{pmatrix}

P = \begin{pmatrix} 3 \\ 2 \end{pmatrix}

Which of the following matrices represents the total revenue from muffin sales for Store 1 and Store 2, calculated using the matrix product $MP$ ?

\begin{pmatrix} 90 & 40 \\ 120 & 70 \end{pmatrix}

\begin{pmatrix} 120 \\ 185 \end{pmatrix}

\begin{pmatrix} 130 \\ 190 \end{pmatrix}

\begin{pmatrix} 130 & 190 \end{pmatrix}

\begin{pmatrix} 150 \\ 210 \end{pmatrix}

Show answer & explanation

Answer: $\begin{pmatrix} 130 \\ 190 \end{pmatrix}$

Answer

The column matrix with elements 130 and 190

The correct answer is the column matrix with elements 130 and 190. Multiplying the first row of the inventory matrix by the price column matrix yields the revenue for Store 1 (

30 \times 3 + 20 \times 2 = 130

). Multiplying the second row by the price column matrix yields the revenue for Store 2 (

40 \times 3 + 35 \times 2 = 190

). Since the product of a 2 by 2 matrix and a 2 by 1 matrix is a 2 by 1 column matrix, the final result is correctly formatted.

Step-by-Step Solution

1

Set up the matrix multiplication of the 2 by 2 matrix and the 2 by 1 matrix.

MP = \begin{pmatrix} 30 & 20 \\ 40 & 35 \end{pmatrix} \begin{pmatrix} 3 \\ 2 \end{pmatrix}

To calculate total revenue, we multiply the quantity matrix by the price matrix.

2

Calculate the first element of the resulting matrix (Store 1 total revenue) by taking the dot product of the first row of the inventory matrix and the price matrix.

30 \times 3 + 20 \times 2 = 90 + 40 = 130

This pairs the quantity of each muffin type sold at Store 1 with its respective price.

3

Calculate the second element of the resulting matrix (Store 2 total revenue) by taking the dot product of the second row of the inventory matrix and the price matrix.

40 \times 3 + 35 \times 2 = 120 + 70 = 190

This pairs the quantity of each muffin type sold at Store 2 with its respective price.

4

Assemble the resulting products into a 2 by 1 column matrix.

\begin{pmatrix} 130 \\ 190 \end{pmatrix}

The product of a 2 by 2 matrix and a 2 by 1 matrix is a 2 by 1 matrix.

Key Concept

Matrix multiplication of a 2 by 2 matrix and a 2 by 1 matrix in a real-world context.

Question 1947Question →

For all real values of $t$ that satisfy the inequality $12 - 4|3 - 2t| \leq -8$ , which of the following inequalities represents the complete set of possible values of $t$ ?

t \leq -1

-1 \leq t \leq 4

t \geq -1

or

t \leq 4

t \leq -1

or

t \geq 4

t \leq 1

or

t \geq 2

Show answer & explanation

Answer: $t \leq -1$ or $t \geq 4$

Answer

t \leq -1

or

t \geq 4

The correct answer is

t \leq -1

or

t \geq 4

. To solve the inequality, we first subtract 12 from both sides to get

-4|3 - 2t| \leq -20

. Next, dividing both sides by

-4

and reversing the inequality sign gives

|3 - 2t| \geq 5

. This absolute value inequality splits into two cases:

3 - 2t \geq 5

or

3 - 2t \leq -5

. Solving the first case, we subtract 3 to get

-2t \geq 2

, and dividing by

-2

while reversing the inequality sign yields

t \leq -1

. Solving the second case, we subtract 3 to get

-2t \leq -8

, and dividing by

-2

while reversing the inequality sign yields

t \geq 4

. Combining these, we obtain the solution set

t \leq -1

or

t \geq 4

.

Step-by-Step Solution

1

Isolate the absolute value term by subtracting 12 from both sides of the inequality.

-4|3 - 2t| \leq -20

Before splitting an absolute value inequality, the absolute value expression must be isolated on one side.

2

Divide both sides of the inequality by

-4

and reverse the inequality sign.

|3 - 2t| \geq 5

Dividing or multiplying an inequality by a negative number requires reversing the direction of the inequality sign.

3

Split the absolute value inequality

|u| \geq c

(where

c > 0

) into two separate inequalities:

u \geq c

or

u \leq -c

.

3 - 2t \geq 5

or

3 - 2t \leq -5

An absolute value inequality of the form

|u| \geq c

represents values that are at least

c

units away from zero, which lie in two disjoint intervals.

4

Solve the first inequality:

3 - 2t \geq 5

.

-2t \geq 2 \implies t \leq -1

Subtracting 3 from both sides gives

-2t \geq 2

. Dividing by

-2

and reversing the inequality sign yields

t \leq -1

.

5

Solve the second inequality:

3 - 2t \leq -5

.

-2t \leq -8 \implies t \geq 4

Subtracting 3 from both sides gives

-2t \leq -8

. Dividing by

-2

and reversing the inequality sign yields

t \geq 4

.

6

Combine the solutions from both cases to express the complete solution set.

t \leq -1

or

t \geq 4

The complete solution set is the union of the solutions to both cases.

Key Concept

Solving absolute value inequalities of the form

|ax + b| \geq c

by translating them into compound inequalities and carefully reversing the inequality sign when multiplying or dividing by negative numbers.

Question 1948Question →

Match each of the unsimplified algebraic expressions on the left with its equivalent simplified form on the right. (Assume all variables represent real numbers.)

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct matches pair the first expression with

6r^2 - 5rs + 6s^2

, the second expression with

2r^2 - 2rs + s^2

, the third expression with

3s^2 - 4rs

, and the fourth expression with

6r^2 - 2r^2s + 5s^2

.

Each expression is correctly simplified by performing distribution first (carefully tracking negative signs and binomial expansion rules) and then combining terms that share the exact same variable powers.

Step-by-Step Solution

1

Simplify the expression

3r(2r - s) - 2s(r - 3s)

.

6r^2 - 5rs + 6s^2

Distribute

3r

to get

6r^2 - 3rs

. Then distribute

-2s

to get

-2rs + 6s^2

(noting that a negative times a negative is positive). Combine the like terms

-3rs

and

-2rs

to get

-5rs

.

2

Simplify the expression

(2r - s)^2 - 2(r^2 - rs)

.

2r^2 - 2rs + s^2

Square the binomial

(2r - s)^2

using the formula

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

to get

4r^2 - 4rs + s^2

. Distribute

-2

to get

-2r^2 + 2rs

. Combine

4r^2 - 2r^2

to get

2r^2

, and

-4rs + 2rs

to get

-2rs

.

3

Simplify the expression

\frac{1}{2}(4r^2 - 6rs) - (r - s)(2r + 3s)

.

3s^2 - 4rs

Distribute

\frac{1}{2}

to get

2r^2 - 3rs

. Expand the binomial product to get

2r^2 + 3rs - 2rs - 3s^2 = 2r^2 + rs - 3s^2

. Subtract this entire expression:

(2r^2 - 3rs) - (2r^2 + rs - 3s^2) = 2r^2 - 3rs - 2r^2 - rs + 3s^2

. Combine like terms to get

-4rs + 3s^2

.

4

Simplify the expression

r^2(6 - s) - s(r^2 - 5s)

.

6r^2 - 2r^2s + 5s^2

Distribute

r^2

to get

6r^2 - r^2s

. Distribute

-s

to get

-sr^2 + 5s^2

. Combine the like terms

-r^2s

and

-sr^2

(since multiplication is commutative) to get

-2r^2s

.

Key Concept

Simplifying expressions by applying the distributive property, expanding products of binomials, and combining like terms.

Question 1949Question →

Which of the following represents the completely factored form of the expression $4x^4 - 16x^2$ ?

4x^2(x - 2)^2

4x^3(x - 4)

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

(2x^2 - 4x)^2

16x^2(x^2 - 1)

Show answer & explanation

Answer: $4x^2(x - 2)(x + 2)$

Answer

The completely factored form of the expression is

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

.

The correct answer is found by first identifying the greatest common factor of the terms in the polynomial

4x^4 - 16x^2

, which is

4x^2

. Factoring this out results in

4x^2(x^2 - 4)

. Next, the binomial

x^2 - 4

is recognized as a difference of squares and is factored into

(x - 2)(x + 2)

. Putting these together gives the completely factored form

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

.

Step-by-Step Solution

1

Identify and factor out the greatest common factor (GCF) of the terms in the expression

4x^4 - 16x^2

.

The GCF of

4x^4

and

16x^2

is

4x^2

. Factoring it out yields

4x^2(x^2 - 4)

.

Factoring out the greatest common factor simplifies the polynomial and reveals remaining factorable patterns.

2

Factor the remaining binomial expression inside the parentheses,

x^2 - 4

.

Since

x^2 - 4

is a difference of squares (

x^2 - 2^2

), it factors into

(x - 2)(x + 2)

.

A difference of squares of the form

a^2 - b^2

always factors into

(a - b)(a + b)

.

3

Combine the factored parts to write the completely factored expression.

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

This combines the GCF and the factored difference of squares to represent the original expression in its simplest factored parts.

Key Concept

Factoring a polynomial completely by first extracting the greatest common factor (GCF) and then applying the difference of squares formula.

Estimated Time:1m 0s

Question 1950Question →

If $(x - 4)^2 = -4x + 28$ , what is the sum of the solutions to the equation?

-8

-4

4

8

28

Show answer & explanation

Answer: 4

Answer

The sum of the solutions to the equation is

4

.

The correct answer of

4

is found by first expanding

(x - 4)^2

to obtain

x^2 - 8x + 16 = -4x + 28

. Setting the equation to zero gives the quadratic equation

x^2 - 4x - 12 = 0

. Factoring the trinomial yields

(x - 6)(x + 2) = 0

, which results in the solutions

x = 6

and

x = -2

. Adding these solutions together gives

6 + (-2) = 4

.

Step-by-Step Solution

1

Expand the squared binomial on the left side of the equation.

x^2 - 8x + 16 = -4x + 28

Applying the binomial expansion formula

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

to

(x - 4)^2

yields

x^2 - 8x + 16

.

2

Rearrange the equation into standard quadratic form by adding

4x

and subtracting

28

from both sides.

x^2 - 4x - 12 = 0

To solve a quadratic equation by factoring, all terms must be moved to one side so that the equation is set to zero.

3

Factor the quadratic expression

x^2 - 4x - 12

.

(x - 6)(x + 2) = 0

Find two integers that multiply to the constant term

-12

and add to the linear coefficient

-4

. These numbers are

-6

and

2

.

4

Set each binomial factor to zero and solve for

x

.

x = 6

or

x = -2

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

5

Find the sum of the solutions.

6 + (-2) = 4

Adding the two individual solutions gives the requested sum.

Key Concept

Solving quadratic equations by expanding binomials, rearranging terms into standard form, factoring the trinomial, and applying the zero-product property.

Alternative Method

Once the equation is rewritten in the standard form

x^2 - 4x - 12 = 0

, Vieta's formulas can be used to find the sum of the solutions directly. For any quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is given by

-\frac{b}{a}

. Substituting

a = 1

and

b = -4

into the formula gives

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

.

Estimated Time:1m 0s

Question 1951Question →

The first term of an arithmetic sequence is $8$ , and the second term is $5$ . What is the $6$ th term of this sequence?

-15

-10

-7

-4

23

Show answer & explanation

Answer: -7

Answer

-7

The correct answer is

-7

. Since the sequence is arithmetic, it changes by a constant common difference,

d

, with each step. Calculating

d

gives

5 - 8 = -3

. To find the

6

th term, we start at the first term,

8

, and add the common difference

5

times (representing the steps from the first to the sixth term):

8 + 5(-3) = 8 - 15 = -7

.

Step-by-Step Solution

1

Determine the common difference,

d

, of the arithmetic sequence.

d = -3

Subtract the first term from the second term:

5 - 8 = -3

.

2

Set up the equation for the

n

th term of an arithmetic sequence,

a_n = a_1 + (n-1)d

.

a_6 = 8 + (6-1)(-3)

We want to find the 6th term (

n = 6

) starting with

a_1 = 8

and

d = -3

.

3

Perform the operations to find the final value.

a_6 = -7

First multiply

5 \times (-3) = -15

, then add to

8

to get

8 - 15 = -7

.

Key Concept

Finding a specific term of an arithmetic sequence using its first term and common difference.

Estimated Time:45s

Question 1952Question →

A landscaping company charges a flat equipment fee plus a fixed hourly rate for lawn maintenance. For a job that took $4$ hours, the company charged a total of $\$190$ . For a different job that took $7$ hours, the company charged a total of $\$295$ . The company also offers a package that includes a $15\%$ discount on the hourly rate, but the flat equipment fee remains the same. Under this discounted package, what is the total charge, in dollars, for a job that takes $8$ hours?

Show answer & explanation

Answer: 288

Answer

The total charge for an

8

-hour job under the discounted package is

288

dollars.

By translating the problem into the equations

F + 4H = 190

and

F + 7H = 295

, we find that the hourly rate

H

is

35

dollars and the flat fee

F

is

50

dollars. Applying a

15\%

discount to the hourly rate yields a new rate of

29.75

dollars per hour. The total cost for

8

hours is then

50 + 8(29.75) = 288

dollars.

Step-by-Step Solution

1

Define variables for the flat fee and hourly rate, and translate the given information into a system of linear equations.

F + 4H = 190

and

F + 7H = 295

, where

F

is the flat fee and

H

is the hourly rate.

To represent the cost structure algebraically.

2

Solve the system of equations by elimination or substitution.

H = 35

and

F = 50

.

To determine the individual cost components (flat fee and hourly rate).

3

Apply the

15\%

discount to the hourly rate.

H_{\text{discounted}} = 35 \times (1 - 0.15) = 29.75

.

To find the new hourly rate under the discounted package.

4

Calculate the total cost for

8

hours of work with the flat fee and the discounted hourly rate.

\text{Total Cost} = 50 + 8(29.75) = 288

.

To answer the question asking for the total charge of an

8

-hour job under the discounted package.

Key Concept

Translating verbal statements into a system of linear equations and solving them to evaluate a modified expression.

Question 1953Question →

If $a$ and $b$ represent positive real numbers, which of the following is an equivalent form of the expression below?

\frac{(a^{-1} + b^{-1})^{-2}}{ab}

\frac{a^2 + b^2}{ab}

\frac{(a+b)^2}{4ab}

\frac{ab}{(a+b)^2}

\frac{(a+b)^2}{ab}

\frac{1}{a^2 b^2 (a+b)^2}

Show answer & explanation

Answer: $\frac{ab}{(a+b)^2}$

Answer

The correct answer is the fraction with

ab

in the numerator and the square of the sum

(a+b)

in the denominator, which is \frac{ab}{(a+b)^2}.

The correct answer is found by first rewriting

a^{-1} + b^{-1}

as

\frac{1}{a} + \frac{1}{b}

, which simplifies to

\frac{a+b}{ab}

using a common denominator. Raising this to the power of

-2

gives

\frac{a^2 b^2}{(a+b)^2}

. Finally, dividing this expression by

ab

reduces the exponents of

a

and

b

by 1, resulting in

\frac{ab}{(a+b)^2}

.

Step-by-Step Solution

1

Rewrite the negative exponents in the numerator as reciprocals.

The expression inside the parentheses becomes

\frac{1}{a} + \frac{1}{b}

.

By the definition of negative exponents,

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

.

2

Find a common denominator to add the fractions inside the parentheses.

\frac{1}{a} + \frac{1}{b} = \frac{b}{ab} + \frac{a}{ab} = \frac{a+b}{ab}

.

To add fractions, they must share a common denominator, which is the product of

a

and

b

.

3

Apply the negative power of

-2

to the simplified fraction.

\left(\frac{a+b}{ab}\right)^{-2} = \left(\frac{ab}{a+b}\right)^2 = \frac{a^2 b^2}{(a+b)^2}

.

An expression raised to a negative exponent is equal to the reciprocal of the expression raised to the positive exponent.

4

Divide the result by the denominator

ab

.

\frac{a^2 b^2}{(a+b)^2} \cdot \frac{1}{ab} = \frac{ab}{(a+b)^2}

.

Dividing by a term is equivalent to multiplying by its reciprocal, and applying the exponent quotient rule simplifies

\frac{a^2 b^2}{ab}

to

ab

.

Key Concept

Properties of exponents including negative power rules, quotient rules, and algebraic fraction addition.

Alternative Method

An alternative method is to substitute small positive integer values for

a

and

b

. Let

a = 1

and

b = 2

. The expression evaluates to

\frac{(1^{-1} + 2^{-1})^{-2}}{1 \cdot 2} = \frac{(1 + 0.5)^{-2}}{2} = \frac{1.5^{-2}}{2} = \frac{(3/2)^{-2}}{2} = \frac{4/9}{2} = \frac{2}{9}

. Evaluating the correct expression with these values yields

\frac{1 \cdot 2}{(1+2)^2} = \frac{2}{9}

, which matches.

Estimated Time:2m 0s

Question 1954Question →

What is the real solution to the equation $\sqrt{3x + 1} = x - 3$ ?

1

1 and 8

8

-2 and 5

-2

Show answer & explanation

Answer: 8

Answer

The only real solution is 8.

The correct answer is the single value 8. Squaring both sides of the equation

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

leads to the quadratic equation

x^2-9x+8=0

, which factors as

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

. This yields potential solutions of 8 and 1. Checking 8 in the original equation gives

\sqrt{25} = 5

, which is correct. Checking 1 in the original equation gives

\sqrt{4} = -2

, which is incorrect because the principal square root must be non-negative. Therefore, 1 is an extraneous solution, and 8 is the only valid solution.

Step-by-Step Solution

1

Square both sides of the equation to eliminate the radical.

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

To clear the square root and obtain a polynomial equation.

2

Expand the right side and move all terms to the right side to set the quadratic equation to zero.

x^2 - 9x + 8 = 0

Expanding

(x - 3)^2

gives

x^2 - 6x + 9

. Subtracting

3x

and

1

from both sides gives the standard form of the quadratic equation.

3

Factor the quadratic equation.

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

Finding two numbers that multiply to 8 and add to -9, which are -8 and -1, allows us to find potential solutions

x = 8

and

x = 1

.

4

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

Checking

x = 8

:

\sqrt{3(8) + 1} = \sqrt{25} = 5

and

8 - 3 = 5

(True). Checking

x = 1

:

\sqrt{3(1) + 1} = \sqrt{4} = 2

and

1 - 3 = -2

(False).

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

Key Concept

Solving radical equations and identifying extraneous solutions

Estimated Time:1m 30s

Question 1955Question →

The first three terms of a geometric sequence of positive numbers are $x$ , $y$ , and $z$ , in that order. An arithmetic sequence has first three terms $x$ , $y$ , and $z - 4$ , in that order. If $x = 4$ , what is the value of $y$ ?

2

4

6

8

16

Show answer & explanation

Answer: 8

Answer

8

The correct answer is 8. By defining the geometric sequence terms as

4, y, z

, we have the property

y^2 = 4z

. For the arithmetic sequence

4, y, z-4

, the common difference property gives

y - 4 = z - 4 - y

, which simplifies to

2y = z

. Substituting

z = 2y

into the first equation yields

y^2 = 8y

. Since all terms must be positive, dividing by

y

gives

y = 8

.

Step-by-Step Solution

1

Set up the equation for the geometric sequence.

y^2 = 4z

Since

4, y, z

is a geometric sequence, the ratio of consecutive terms must be equal:

\frac{y}{4} = \frac{z}{y}

, which simplifies to

y^2 = 4z

.

2

Set up the equation for the arithmetic sequence.

2y = z

Since

4, y, z - 4

is an arithmetic sequence, the difference between consecutive terms must be equal:

y - 4 = (z - 4) - y

. Simplifying this gives

2y = z

.

3

Substitute the arithmetic equation into the geometric equation.

y^2 = 8y

Substituting

z = 2y

into

y^2 = 4z

yields

y^2 = 4(2y) = 8y

.

4

Solve for the variable

y

.

y = 8

Rearranging the equation gives

y^2 - 8y = 0

, which factors as

y(y - 8) = 0

. Since the sequence consists of positive numbers,

y

must be positive, so

y = 8

.

Key Concept

Relating arithmetic and geometric sequence properties to solve a system of non-linear equations

Question 1956Question →

A square has a side length of $x^3 - 3$ inches. Which of the following expressions represents the area, in square inches, of the square?

x^6 - 9

x^9 - 6x^3 + 9

x^6 - 6x^3 + 9

x^6 + 9

x^9 - 9

Show answer & explanation

Answer: $x^6 - 6x^3 + 9$

Answer

The expression

x^6 - 6x^3 + 9

To find the area of a square, square its side length:

(x^3 - 3)^2

. Expanding this gives

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

.

Step-by-Step Solution

1

Write the formula for the area of a square.

A = s^2

, where

s

is the side length.

The area of a square is the side length squared.

2

Substitute the given side length

s = x^3 - 3

into the area formula.

A = (x^3 - 3)^2

We replace

s

with the algebraic expression for the side length.

3

Expand the squared binomial

(x^3 - 3)^2

using the identity

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

.

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

Expanding the binomial yields the correct simplified polynomial in standard descending order.

Key Concept

Expanding a squared binomial and applying exponent rules for powers of powers.

Estimated Time:1m 0s

Question 1957Question →

The weight of Box A is $1.5$ pounds more than $\frac{2}{3}$ of the weight of Box B. If the weight of Box A is $13.5$ pounds, what is the weight of Box B, in pounds?

8

10.5

13.5

18

22.5

Show answer & explanation

Answer: 18

Answer

The weight of Box B is 18 pounds.

The weight of 18 pounds is correct. Translating the word problem yields the linear equation

A = \frac{2}{3}B + 1.5

. Substituting the weight of Box A (

13.5

) gives

13.5 = \frac{2}{3}B + 1.5

. Subtracting

1.5

from both sides results in

12 = \frac{2}{3}B

. Multiplying both sides by the reciprocal

\frac{3}{2}

isolates

B

, giving

B = 12 \times \frac{3}{2} = 18

.

Step-by-Step Solution

1

Translate the verbal description into an algebraic equation.

Let

A

represent the weight of Box A and

B

represent the weight of Box B. The description translates to the equation:

A = \frac{2}{3}B + 1.5

.

This establishes the mathematical relationship between the weights of the two boxes.

2

Substitute the given value for Box A into the equation and isolate the term containing the variable B.

13.5 = \frac{2}{3}B + 1.5 \implies 12 = \frac{2}{3}B

.

Substituting

13.5

for

A

and subtracting

1.5

from both sides simplifies the equation to isolate the fraction term.

3

Solve for B by multiplying both sides of the equation by the reciprocal of the coefficient.

B = 12 \times \frac{3}{2} = 18

.

Multiplying by the reciprocal

\frac{3}{2}

isolates

B

to find its value.

Key Concept

Translating real-world descriptions into multi-step linear equations and solving them using inverse operations.

Estimated Time:1m 0s

Question 1958Question →

A digital marketing firm allocates a total monthly advertising budget of $B$ dollars between search engine ads and social media ads. The amount allocated to social media ads is $\$800$ less than twice the amount allocated to search engine ads. The firm then increases its total monthly budget by $20\%$ and allocates this entire increase to search engine ads. As a result, the new amount allocated to search engine ads is equal to the amount originally allocated to social media ads. What was the firm's original total monthly advertising budget, $B$ ?

\$1,600

\$2,200

\$2,400

\$4,000

\$8,000

Show answer & explanation

Answer: $\$4,000$

Answer

The original total monthly advertising budget was

\$4,000

.

By defining the original search engine advertising budget as

S

, the original social media advertising budget as

M = 2S - 800

, and the original total budget as

B = 3S - 800

, we can set up the equation

S + 0.20B = M

using the condition that the new search engine budget equals the original social media budget. Solving this equation yields

S = 1600

. Substituting

S = 1600

into the expression for the total budget gives

B = 3(1600) - 800 = 4000

dollars.

Step-by-Step Solution

1

Define the variables representing each budget allocation.

Let

S

represent the original amount allocated to search engine ads, and let

M

represent the original amount allocated to social media ads. The original total budget is

B = S + M

.

Establishing clear variables is necessary to translate the verbal relationships into equations.

2

Translate the relationship between the two advertising categories into an equation.

M = 2S - 800

The problem states that the social media ads budget is

\$800

less than twice the search engine ads budget.

3

Express the original total budget

B

in terms of a single variable,

S

.

B = S + (2S - 800) = 3S - 800

Substituting the expression for

M

into the total budget equation simplifies the system to a single variable.

4

Set up the equation for the new budget allocation after the

20\%

increase.

S + 0.20B = M

An increase of

20\%

of the total budget is

0.20B

, which is added entirely to the search engine ads budget, making it equal to the original social media ads budget.

5

Substitute the expressions for

B

and

M

in terms of

S

into the new budget equation.

S + 0.20(3S - 800) = 2S - 800

Substituting the relationships found in steps 2 and 3 allows us to solve for

S

directly.

6

Solve the equation for

S

.

S + 0.60S - 160 = 2S - 800 \Rightarrow 1.60S - 160 = 2S - 800 \Rightarrow 640 = 0.40S \Rightarrow S = 1,600

Distributing, combining like terms, and isolating the variable yields the original search engine ads allocation.

7

Calculate the original total budget,

B

.

B = 3(1,600) - 800 = 4,800 - 800 = 4,000

The question asks for the original total budget

B

, so we must evaluate the expression for

B

using the value of

S

.

Key Concept

Translating verbal descriptions of algebraic relationships into systems of linear equations and solving them.

Question 1959Question →

When the expression $(5x^2 - 8x + 2) - (2x^2 - 4x - 9)$ is written in the standard form $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are integers, what is the value of $b$ ?

Show answer & explanation

Answer: -4

Answer

The correct answer is

-4

, which is the coefficient of the

x

term after simplifying the expression.

To subtract polynomials, we distribute the negative sign to all terms of the polynomial being subtracted and then combine like terms. Simplifying

(5x^2 - 8x + 2) - (2x^2 - 4x - 9)

gives

5x^2 - 8x + 2 - 2x^2 + 4x + 9 = 3x^2 - 4x + 11

. Comparing this to

ax^2 + bx + c

shows that

b

, the coefficient of the

x

term, is

-4

.

Step-by-Step Solution

1

Distribute the subtraction sign to all terms inside the second set of parentheses.

-2x^2 + 4x + 9

Subtracting a polynomial is equivalent to adding its opposite, which means changing the sign of every term in that polynomial.

2

Group and combine like terms from both polynomials.

3x^2 - 4x + 11

Like terms (terms with the same variable raised to the same power) can be combined by adding or subtracting their coefficients:

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

,

(-8x + 4x) = -4x

, and

(2 + 9) = 11

.

3

Compare the simplified expression to the standard form

ax^2 + bx + c

to identify the value of

b

.

b = -4

In the standard form

ax^2 + bx + c

, the coefficient of the

x

term is represented by

b

. In the simplified expression

3x^2 - 4x + 11

, the coefficient of

x

is

-4

.

Key Concept

Polynomial Subtraction and Combining Like Terms

Question 1960Question →

If $i$ represents the imaginary unit, and the complex number $z$ satisfies the equation $z(2 + i) - 5i^{97} = \frac{25}{3 - 4i}$ , what is the value of $z^2$ ?

-18i

-2i

16

18i

50i

Show answer & explanation

Answer: 18i

Answer

18i

Simplifying the original equation yields the complex number

z = 3 + 3i

. Squaring this number gives

(3 + 3i)^2 = 9 + 18i + 9i^2 = 9 + 18i - 9 = 18i

.

Step-by-Step Solution

1

Simplify the fraction on the right side of the equation by multiplying the numerator and denominator by the complex conjugate of the denominator,

3 + 4i

.

The fraction simplifies to

3 + 4i

.

Multiplying by the conjugate rationalizes the denominator:

\frac{25}{3 - 4i} = \frac{25(3 + 4i)}{(3 - 4i)(3 + 4i)} = \frac{25(3 + 4i)}{9 + 16} = 3 + 4i

.

2

Simplify the power of the imaginary unit,

i^{97}

, by dividing the exponent by 4 to find the remainder.

5i^{97} = 5i

Since

97 = 4 \times 24 + 1

, the expression simplifies as

i^{97} = (i^4)^{24} \cdot i = 1^{24} \cdot i = i

.

3

Substitute the simplified expressions back into the original equation and isolate the term containing

z

.

z(2 + i) = 3 + 9i

Substituting gives

z(2 + i) - 5i = 3 + 4i

. Adding

5i

to both sides yields

z(2 + i) = 3 + 9i

.

4

Solve for

z

by dividing both sides by

2 + i

, then simplify by multiplying by the conjugate of the denominator,

2 - i

.

z = 3 + 3i

Performing the division:

z = \frac{3 + 9i}{2 + i} = \frac{(3 + 9i)(2 - i)}{(2 + i)(2 - i)} = \frac{6 - 3i + 18i - 9i^2}{4 - i^2} = \frac{15 + 15i}{5} = 3 + 3i

.

5

Calculate the value of

z^2

by squaring the complex number

3 + 3i

.

z^2 = 18i

Squaring the binomial gives

(3 + 3i)^2 = 9 + 18i + 9i^2 = 9 + 18i - 9 = 18i

.

Key Concept

Solving equations containing complex numbers by performing operations such as multiplication, division using complex conjugates, and simplifying powers of

i

.

Estimated Time:2m 30s

Answer

Step-by-Step Solution

Key Concept

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Common Mistakes

Alternative Method

Answer

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Common Mistakes

Alternative Method

Answer

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