All practice questions

Question 1721Question →

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

x + 2

x

x^2 + 2x

x^2 + 2

x^3 + 2

Show answer & explanation

Answer: $x + 2$

Answer

The expression is equivalent to

x + 2

.

Combining the fractions gives the combined numerator over the denominator. Expanding the binomial in the numerator yields the sum of terms, which simplifies after subtracting one. Factoring out the variable in the numerator and dividing by the same variable simplifies the expression to a first-degree binomial.

Step-by-Step Solution

1

Combine the two rational terms since they already share a common denominator of

x

.

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

When subtracting fractions with the same denominator, subtract the numerators and keep the denominator.

2

Expand the squared binomial

(x+1)^2

.

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

The square of a binomial is given by

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

.

3

Simplify the numerator by combining like terms (

1 - 1 = 0

).

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

Combine the constant terms in the numerator.

4

Factor

x

out of each term in the numerator and divide by the denominator

x

.

x + 2

Since

x \neq 0

, we can cancel the common factor of

x

from both the numerator and the denominator.

Key Concept

Simplifying rational expressions by combining fractions and expanding binomials

Estimated Time:45s

Question 1722Question →

Which of the following represents the complete set of real solutions to the equation $\sqrt{4x + 28} - x = 4$ ?

-6

and

2

-2

and

6

2

only

6

only

-6

only

Show answer & explanation

Answer: $2$ only

Answer

The correct answer is

2

only.

The correct answer is

2

only. Isolating the radical in the original equation gives

\sqrt{4x+28} = x+4

. Squaring both sides yields

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

. Rearranging into standard quadratic form gives

x^2+4x-12 = 0

. Factoring the quadratic yields

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

, giving the candidate solutions

x = -6

and

x = 2

. Testing

x = 2

in the original equation gives

\sqrt{36} - 2 = 4

, which is valid. Testing

x = -6

gives

\sqrt{4} - (-6) = 8 \neq 4

, meaning

-6

is extraneous.

Step-by-Step Solution

1

Isolate the radical expression on one side of the equation.

\sqrt{4x + 28} = x + 4

Isolating the radical allows us to square both sides directly to eliminate the square root.

2

Square both sides of the equation.

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

Squaring a square root eliminates the radical, and squaring the binomial

(x+4)

yields

x^2 + 8x + 16

.

3

Move all terms to one side of the equation to form a standard quadratic equation.

x^2 + 4x - 12 = 0

Subtracting

4x

and

28

from both sides of the equation simplifies it to the form

ax^2 + bx + c = 0

.

4

Factor the quadratic equation to find potential solutions.

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

, which gives potential solutions

x = -6

and

x = 2

.

Factoring is a standard method to solve quadratic equations of this form.

5

Substitute the potential solutions back into the original equation to check for extraneous solutions.

For

x = 2

:

\sqrt{4(2)+28}-2 = 6-2 = 4

, which is true. For

x = -6

:

\sqrt{4(-6)+28}-(-6) = 2+6 = 8 \neq 4

, which is false.

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

Key Concept

Solving radical equations by isolating the radical, squaring both sides, and identifying extraneous solutions.

Question 1723Question →

The square of 3 less than twice a certain real number is equal to 3 less than 7 times that number. If $p$ and $q$ are the two distinct real solutions to this equation with $p > q$ , what is the value of $4p - 4q$ ?

9

11

13

14

1

Show answer & explanation

Answer: 13

Answer

13

Correctly expanding the squared expression leads to the quadratic equation

4x^2 - 19x + 12 = 0

. Factoring this equation yields the solutions

4

and

3/4

. Since the problem specifies

p > q

, we have

p = 4

and

q = 3/4

. Substituting these values into the expression

4p - 4q

gives

4(4) - 4(3/4) = 16 - 3 = 13

.

Step-by-Step Solution

1

Translate the verbal description into an algebraic equation.

(2x - 3)^2 = 7x - 3

'Twice a number' is represented as

2x

, '3 less than twice the number' is

2x - 3

, and 'the square of' that is

(2x - 3)^2

. This is set equal to '3 less than 7 times that number', which is

7x - 3

.

2

Expand the binomial and write the equation in standard quadratic form

ax^2 + bx + c = 0

.

4x^2 - 19x + 12 = 0

Expanding the left side yields

4x^2 - 12x + 9 = 7x - 3

. Subtracting

7x

and adding

3

to both sides results in the standard form.

3

Factor the quadratic equation by grouping.

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

We need two numbers that multiply to

4 \times 12 = 48

and add to

-19

. These numbers are

-16

and

-3

. Rewriting the equation as

4x^2 - 16x - 3x + 12 = 0

allows us to factor out

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

, which simplifies to

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

.

4

Find the roots of the equation and evaluate the target expression

4p - 4q

.

p = 4

,

q = 3/4

, and the final value is

13

.

Setting each factor to zero gives

x = 3/4

and

x = 4

. Since

p > q

, we assign

p = 4

and

q = 3/4

. Evaluating the expression yields

4(4) - 4(3/4) = 16 - 3 = 13

.

Key Concept

Solving Quadratic Equations by Factoring

Estimated Time:2m 30s

Question 1724Question →

A researcher records the following list of seven temperatures (in degrees Fahrenheit) during a week, where $x$ and $y$ are unknown values:

88, 92, 75, 85, x, y, 90

The mean temperature for the seven days is $85.0^\circ\text{F}$ , and the range of the temperatures is $22.0^\circ\text{F}$ . If the highest temperature of the week is $y$ and the lowest temperature of the week is $x$ , what is the median temperature of the week?

82.5^\circ\text{F}

85.0^\circ\text{F}

86.5^\circ\text{F}

88.0^\circ\text{F}

90.0^\circ\text{F}

Show answer & explanation

Answer: $88.0^\circ\text{F}$

Answer

88.0^\circ\text{F}

To find the median temperature, we must first find the values of

x

and

y

. Since the mean of the seven temperatures is

85.0^\circ\text{F}

, their sum is

7 \times 85.0 = 595.0

. The sum of the five known temperatures is

88 + 92 + 75 + 85 + 90 = 430.0

. Therefore, the sum of the two unknown temperatures is

x + y = 595.0 - 430.0 = 165.0

. We are also given that the range of the temperatures is

22.0^\circ\text{F}

, and that

y

is the highest and

x

is the lowest temperature, which means

y - x = 22.0

. Solving this system of equations (

y + x = 165.0

and

y - x = 22.0

) yields

x = 71.5

and

y = 93.5

. Placing the seven temperatures in ascending order gives

71.5, 75, 85, 88, 90, 92, 93.5

. The median is the middle value (the 4th value) in this sorted list, which is

88.0^\circ\text{F}

.

Step-by-Step Solution

1

Determine the sum of the temperatures using the given mean of

85.0^\circ\text{F}

.

Total sum =

595.0

Since the mean of 7 temperatures is

85.0

, the sum of all temperatures must be

7 \times 85.0 = 595.0

.

2

Set up an equation for the sum of the unknown temperatures

x

and

y

.

x + y = 165.0

The sum of the five known temperatures is

88 + 92 + 75 + 85 + 90 = 430.0

. Thus,

x + y = 595.0 - 430.0 = 165.0

.

3

Set up and solve the system of equations with the range constraint to find

x

and

y

.

x = 71.5

and

y = 93.5

We are given that

y

is the maximum and

x

is the minimum, so the range is

y - x = 22.0

. Solving the system

y + x = 165.0

and

y - x = 22.0

by adding the equations gives

2y = 187.0 \implies y = 93.5

. Substituting back gives

x = 71.5

.

4

Sort the seven temperatures in ascending order and identify the median value.

Sorted list:

71.5, 75, 85, 88, 90, 92, 93.5

; Median =

88.0

For an odd number of data points (7), the median is the 4th value when the list is sorted. The 4th value in the sorted list is

88.0

.

Key Concept

Calculating the median of a dataset containing unknown values by deriving and solving a system of linear equations based on the mean and the range.

Alternative Method

Instead of setting up and solving the system of equations algebraically, one can test the median by using the fact that the sum of the deviations from the mean (

85

) must equal

0

. The deviations of the known numbers from

85

are:

(88-85) + (92-85) + (75-85) + (85-85) + (90-85) = 3 + 7 - 10 + 0 + 5 = 5

. Therefore, the sum of the deviations of

x

and

y

from

85

must be

-5

:

(x-85) + (y-85) = -5

, which simplifies to

x + y - 170 = -5

, or

x + y = 165

. Since

y - x = 22

, we can quickly find

x = 71.5

and

y = 93.5

, then sort the list to find the median.

Estimated Time:3m 0s

Question 1725Question →

When the expression $6x - 2x(x - 3y) - \left(2x - \frac{1}{2}y\right)^2 - \frac{3}{4}x(6x - 8)$ is simplified by combining like terms, what is the coefficient of the $x^2$ term?

-\frac{9}{2}

-5

-6

-\frac{17}{2}

-\frac{21}{2}

Show answer & explanation

Answer: $-\frac{21}{2}$

Answer

-\frac{21}{2}

Expanding the entire expression yields the

x^2

terms

-2x^2

,

-4x^2

, and

-\frac{9}{2}x^2

. Summing these coefficients gives

-2 - 4 - \frac{9}{2} = -\frac{21}{2}

. Therefore, the coefficient of the

x^2

term is

-\frac{21}{2}

.

Step-by-Step Solution

1

Expand the first parenthetical expression by distributing the term

-2x

.

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

. The expression becomes:

6x - 2x^2 + 6xy - \left(2x - \frac{1}{2}y\right)^2 - \frac{3}{4}x(6x - 8)

.

Distribution is required to eliminate parentheses before terms can be combined.

2

Expand the squared binomial

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

using the identity

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

, then distribute the negative sign.

\left(2x - \frac{1}{2}y\right)^2 = 4x^2 - 2xy + \frac{1}{4}y^2

. Distributing the negative gives

-4x^2 + 2xy - \frac{1}{4}y^2

.

This simplifies the second parenthetical term of the expression.

3

Expand the third parenthetical term by distributing

-\frac{3}{4}x

.

-\frac{3}{4}x(6x - 8) = -\frac{9}{2}x^2 + 6x

.

This simplifies the final parenthetical term of the expression.

4

Combine the coefficients of all the

x^2

terms.

The

x^2

terms are

-2x^2

,

-4x^2

, and

-\frac{9}{2}x^2

. Combining their coefficients gives:

-2 - 4 - \frac{9}{2} = -6 - \frac{9}{2} = -\frac{12}{2} - \frac{9}{2} = -\frac{21}{2}

.

Combining like terms simplifies the expression to find the final coefficient of

x^2

.

Key Concept

Simplifying algebraic expressions by distributing coefficients and combining like terms.

Estimated Time:1m 30s

Question 1726Question →

For all real values of $x$ such that $x \neq 3$ , which of the following expressions is equivalent to $\frac{x}{x - 3} - \frac{9}{(x - 3)^2}$ ?

\frac{x^2 - 3x - 9}{x^2 + 9}

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

\frac{x^2 - 3x - 9}{x^2 - 6x + 9}

\frac{x - 9}{x^2 - 7x + 12}

\frac{x^2 - 3x - 9}{x^2 - 9}

Show answer & explanation

Answer: $\frac{x^2 - 3x - 9}{x^2 - 6x + 9}$

Answer

\frac{x^2 - 3x - 9}{x^2 - 6x + 9}

To subtract the rational expressions, we must first find a common denominator. The denominators are

x-3

and

(x-3)^2

, so the least common denominator is

(x-3)^2

. We multiply the numerator and denominator of the first term by

x-3

, resulting in

\frac{x(x-3)}{(x-3)^2} - \frac{9}{(x-3)^2}

. Expanding the first numerator gives

x^2 - 3x

. Subtracting the second numerator gives

x^2 - 3x - 9

in the numerator. Expanding the denominator

(x-3)^2

yields

x^2 - 6x + 9

. Therefore, the simplified equivalent expression is

\frac{x^2 - 3x - 9}{x^2 - 6x + 9}

.

Step-by-Step Solution

1

Identify the least common denominator (LCD) for the two rational expressions.

The LCD of

x - 3

and

(x - 3)^2

is

(x - 3)^2

.

To perform subtraction between two rational expressions, they must have a common denominator.

2

Rewrite the first expression with the common denominator by multiplying its numerator and denominator by

x - 3

.

\frac{x(x - 3)}{(x - 3)^2} = \frac{x^2 - 3x}{(x - 3)^2}

.

Multiplying the numerator and denominator by the same non-zero quantity preserves the value of the expression.

3

Subtract the numerators of the two expressions while keeping the common denominator.

\frac{x^2 - 3x - 9}{(x - 3)^2}

.

Once denominators are identical, subtract the numerators directly over the common denominator.

4

Expand the binomial in the denominator

(x - 3)^2

to match the standard polynomial form of the options.

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

, yielding

\frac{x^2 - 3x - 9}{x^2 - 6x + 9}

.

Expanding the denominator allows direct comparison with the polynomial choices provided.

Key Concept

Simplifying rational expressions by finding a common denominator

Question 1727Question →

An algebraic expression of the form $x^4 - 10x^2y^2 + 9y^4 - x^2 + 9y^2$ is defined for all real numbers $x$ and $y$ . When this expression is factored completely over the integers, which of the following is one of its factors?

x - 9y^2

x - y - 1

x - y

x + 3y

x^2 - y^2 + 1

Show answer & explanation

Answer: $x + 3y$

Answer

The binomial

x + 3y

is a factor of the expression.

To factor the polynomial completely, we group the terms as

(x^4 - 10x^2y^2 + 9y^4) - (x^2 - 9y^2)

. The first trinomial is in quadratic form and factors to

(x^2 - 9y^2)(x^2 - y^2)

. The expression is then rewritten as

(x^2 - 9y^2)(x^2 - y^2) - (x^2 - 9y^2)

, which has a GCF of

(x^2 - 9y^2)

. Factoring out the GCF yields

(x^2 - 9y^2)(x^2 - y^2 - 1)

. Finally, the difference of squares

(x^2 - 9y^2)

factors into

(x - 3y)(x + 3y)

. The fully factored expression is

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

, which contains the factor

x + 3y

.

Step-by-Step Solution

1

Group the terms of the polynomial into two parts.

(x^4 - 10x^2y^2 + 9y^4) - (x^2 - 9y^2)

Grouping terms allows us to find and extract common algebraic factors from distinct parts of the polynomial.

2

Factor the trinomial from the first group as a quadratic form in terms of

x^2

and

y^2

.

(x^2 - 9y^2)(x^2 - y^2)

The trinomial

x^4 - 10x^2y^2 + 9y^4

can be written as

(x^2)^2 - 10(x^2)(y^2) + 9(y^2)^2

, which factors as

(x^2 - 9y^2)(x^2 - y^2)

.

3

Substitute this factorization back into the grouped expression and factor out the greatest common factor.

(x^2 - 9y^2)(x^2 - y^2 - 1)

Both parts of the grouped expression share

(x^2 - 9y^2)

as a GCF, leaving

(x^2 - y^2 - 1)

when factored out.

4

Factor the difference of squares term completely over the integers.

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

The term

(x^2 - 9y^2)

is a difference of squares of the form

a^2 - b^2

, which factors into

(a - b)(a + b)

.

Key Concept

Factoring high-degree polynomials using quadratic form substitution, grouping, and difference of squares.

Question 1728Question →

The sum of $9$ and the product of a number and $6$ less than that number is equal to $3$ more than the number. What is the sum of all possible values of this number?

7

1

5

-2

-7

Show answer & explanation

Answer: 7

Answer

7

The correct answer is

7

. The verbal description translates to

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

. Distributing the left side gives

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

. Subtracting

x

and

3

from both sides results in

x^2 - 7x + 6 = 0

. Factoring the quadratic expression gives

(x - 6)(x - 1) = 0

, which yields the solutions

x = 6

and

x = 1

. Summing these values gives

6 + 1 = 7

.

Step-by-Step Solution

1

Translate the verbal statement into an algebraic equation.

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

, where

x

represents the unknown number.

Establishing the relationship between the algebraic expressions defined by the problem.

2

Expand the left side of the equation and combine like terms to set the equation to zero.

x^2 - 6x + 9 = x + 3 \implies x^2 - 7x + 6 = 0

.

Quadratic equations must be set to zero before they can be solved by factoring.

3

Factor the quadratic equation over the integers.

(x - 6)(x - 1) = 0

.

Finding two numbers that multiply to

6

and add up to

-7

allows us to factor the trinomial.

4

Apply the zero-product property to find the individual roots.

x = 6

and

x = 1

.

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

5

Sum the possible values of the number.

6 + 1 = 7

.

The question asks for the sum of all possible values of the number.

Key Concept

Solving Quadratic Equations by Factoring

Question 1729Question →

Simplify each of the algebraic expressions on the left by distributing and combining like terms, then match it with its equivalent simplified expression on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The expressions match as follows: the first expression matches

-2x^3 + 6x^2y + 4y^3

; the second expression matches

x^3 - 6x^2y + 6xy^2 + 8y^3

; the third expression matches

0

; and the fourth expression matches

y^2

.

Each expression on the left reduces to its matching counterpart on the right by carefully expanding terms (including cubes, squares, difference of squares, and difference of cubes) and combining like terms while correctly distributing negative signs.

Step-by-Step Solution

1

Simplify the first expression by expanding each term individually.

E_1 = -x^3 + 6x^2y - 12xy^2 + 8y^3 - x^3 + 4xy^2 + 8xy^2 - 4y^3

Expanding the binomial cube, the difference of squares product, and distributing the monomial allows us to identify like terms.

2

Combine like terms in the first expression.

E_1 = -2x^3 + 6x^2y + 4y^3

Combining the

x^3

,

x^2y

,

xy^2

, and

y^3

terms simplifies the expression. The

xy^2

terms sum to zero.

3

Simplify the second expression by expanding each term individually.

E_2 = 2x^3 - 4x^2y + 2xy^2 - x^3 + 8y^3 - 2x^2y + 4xy^2

Using algebraic expansion rules (including the difference of cubes product) exposes all individual terms.

4

Combine like terms in the second expression.

E_2 = x^3 - 6x^2y + 6xy^2 + 8y^3

Adding coefficients of like variable terms gives the simplified form.

5

Simplify the third expression by expanding the cubic terms.

E_3 = (x^3 + 3x^2y + 3xy^2 + y^3) - (x^3 - 3x^2y + 3xy^2 - y^3) - (6x^2y + 2y^3) = 6x^2y + 2y^3 - 6x^2y - 2y^3

Expanding the binomial cubes and distributing the negative signs shows that all terms cancel.

6

Combine like terms in the third expression.

E_3 = 0

All terms cancel out, leaving a final value of zero.

7

Simplify the fourth expression by expanding.

E_4 = x^4 - 2x^2y + y^2 - x^4 + y^2 - y^2 + 2x^2y

Squaring the binomial, using the difference of squares, and distributing the negative variable simplifies the individual components.

8

Combine like terms in the fourth expression.

E_4 = y^2

The

x^4

and

x^2y

terms cancel, and the

y^2

terms simplify to

y^2

.

Key Concept

Simplifying Expressions and Combining Like Terms

Question 1730Question →

Let $z$ be the complex number resulting from the product $(2 - 3i)(3 + i)$ , where $i = \sqrt{-1}$ . If $\bar{z}$ represents the complex conjugate of $z$ , what is the value of the product $z \cdot \bar{z}$ ?

32

58

81

95

130

Show answer & explanation

Answer: 130

Answer

130

The correct answer is obtained by first expanding

(2 - 3i)(3 + i)

using FOIL to get

6 + 2i - 9i - 3i^2

. Since

i^2 = -1

, this simplifies to

6 - 7i - 3(-1) = 9 - 7i

. The complex conjugate is

9 + 7i

. Multiplying these gives

9^2 - (7i)^2 = 81 - 49(-1) = 130

.

Step-by-Step Solution

1

Multiply the complex binomials to find

z

z = (2 - 3i)(3 + i) = 6 + 2i - 9i - 3i^2 = 6 - 7i - 3(-1) = 9 - 7i

To express the complex number in standard form

a + bi

, we expand the product using the distributive property and substitute

i^2 = -1

.

2

Determine the complex conjugate of

z

, denoted as

\bar{z}

\bar{z} = 9 + 7i

The complex conjugate of a complex number

a + bi

is

a - bi

, which is found by reversing the sign of the imaginary part.

3

Calculate the product of

z

and

\bar{z}

z \cdot \bar{z} = (9 - 7i)(9 + 7i) = 9^2 + 7^2 = 81 + 49 = 130

The product of a complex number

a + bi

and its conjugate

a - bi

is always a real number equal to

a^2 + b^2

.

Key Concept

Multiplying complex numbers and finding the product of a complex number and its conjugate.

Question 1731Question →

When the expression $-\frac{1}{3}x(x - 2y)^2 - \frac{1}{2}x(xy - 3y^2) - (x^3 - 2x^2y)$ is fully simplified, what is the coefficient of the $x^2y$ term?

\frac{10}{3}

\frac{5}{6}

-\frac{7}{6}

\frac{17}{6}

\frac{3}{2}

Show answer & explanation

Answer: $\frac{17}{6}$

Answer

The coefficient of the

x^2y

term is

\frac{17}{6}

.

By expanding all components of the expression: the first term yields

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

, the second term yields

-\frac{1}{2}x^2y + \frac{3}{2}xy^2

, and the third term yields

-x^3 + 2x^2y

. Summing the coefficients of the

x^2y

term gives

\frac{4}{3} - \frac{1}{2} + 2 = \frac{17}{6}

.

Step-by-Step Solution

1

Expand the first term of the expression.

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

Applying binomial expansion to

(x-2y)^2

and distributing

-\frac{1}{3}x

.

2

Expand the second term of the expression.

-\frac{1}{2}x(xy - 3y^2) = -\frac{1}{2}x^2y + \frac{3}{2}xy^2

Distributing the term

-\frac{1}{2}x

over the parenthetical terms.

3

Distribute the negative sign in the third term.

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

Distributing the negative sign across all terms inside the parentheses.

4

Combine the coefficients of the

x^2y

terms.

\frac{4}{3} - \frac{1}{2} + 2 = \frac{8}{6} - \frac{3}{6} + \frac{12}{6} = \frac{17}{6}

Finding a common denominator of 6 to sum the coefficients of the

x^2y

term.

Key Concept

Simplifying expressions by distributing terms, expanding binomials, and combining like terms with fractional coefficients.

Question 1732Question →

For all real numbers $x > 1$ , the expression below is equivalent to which of the following?

\frac{\sqrt{x^8} - x^2 \cdot x^3}{x^2(1-x) - (x^2 - x)^2}

-x^2(x+1)

0

x

x^2 + x

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

Show answer & explanation

Answer: $x$

Answer

The expression is equivalent to

x

.

The correct answer is

x

. Simplifying the numerator yields

x^4 - x^5 = x^4(1-x)

. Simplifying the denominator by expanding both parts yields

(x^2 - x^3) - (x^4 - 2x^3 + x^2) = x^3 - x^4 = x^3(1-x)

. Dividing the numerator by the denominator and canceling the common non-zero term

(1-x)

leaves

\frac{x^4}{x^3}

, which simplifies to

x

.

Step-by-Step Solution

1

Simplify the radical term in the numerator.

\sqrt{x^8} = x^4

Since

x > 1

, we can take the square root of

x^8

directly by dividing the exponent by 2:

(x^8)^{1/2} = x^{8/2} = x^4

.

2

Simplify the multiplication of exponential terms in the numerator.

x^2 \cdot x^3 = x^5

By exponent rules, when multiplying terms with the same base, we add the exponents:

x^{2+3} = x^5

.

3

Combine the terms in the numerator and factor out the greatest common factor.

Numerator =

x^4 - x^5 = x^4(1-x)

Factoring out

x^4

prepares the numerator for potential cancellation with the denominator.

4

Expand and simplify the terms in the denominator.

Denominator =

x^3(1-x)

Expanding the first term gives

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

. Expanding the squared binomial gives

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

. Subtracting them yields

(x^2 - x^3) - (x^4 - 2x^3 + x^2) = x^3 - x^4 = x^3(1-x)

.

5

Divide the simplified numerator by the simplified denominator.

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

Since

x > 1

, the factor

(1-x)

is non-zero, allowing us to cancel it. Finally, dividing the remaining terms gives

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

.

Key Concept

Simplifying rational expressions by factoring and applying exponent rules, including radical simplification and binomial expansion.

Question 1733Question →

The rational expression $\frac{x - 3}{x^2 - x - 12}$ is undefined for two real values of $x$ . What are these two values, where the smaller value is entered in the first blank and the larger value is entered in the second blank?

Fill in the blanks below

The smaller value is and the larger value is .

Show answer & explanation

Answer

The expression is undefined when the smaller value is -3 and the larger value is 4.

A rational expression is undefined when the denominator is equal to

0

. For the expression

\frac{x - 3}{x^2 - x - 12}

, setting the denominator equal to zero gives

x^2 - x - 12 = 0

. Factoring the quadratic expression yields

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

. Setting each factor to zero gives the solutions

x = 4

and

x = -3

. Thus, the smaller value for which the expression is undefined is

-3

and the larger value is

4

.

Step-by-Step Solution

1

Identify the condition that makes a rational expression undefined.

A rational expression is undefined when its denominator is equal to zero.

Division by zero is undefined in real numbers.

2

Set the denominator of the expression equal to zero.

x^2 - x - 12 = 0

This equation will yield the values of

x

for which the denominator is zero.

3

Factor the quadratic equation.

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

Factoring allows us to find the roots by setting each linear factor to zero.

4

Solve for

x

by setting each factor equal to zero.

x = 4

or

x = -3

If

x - 4 = 0

, then

x = 4

. If

x + 3 = 0

, then

x = -3

.

5

Identify the smaller and larger values.

The smaller value is

-3

and the larger value is

4

.

Comparing the two values,

-3 < 4

.

Key Concept

Identifying values that make a rational expression undefined by finding the roots of the denominator.

Estimated Time:1m 15s

Question 1734Question →

To find the domain of the rational expression

f(x) = \frac{x+3}{x^2 - 1} \div \frac{x^2 - 9}{x^2 - 3x - 4}

a student must identify all values of

x

for which the expression is undefined. The expression is undefined for exactly five real values of

x

. What are these five values in increasing order?

Fill in the blanks below

From least to greatest, the values of

x

that make the expression undefined are:
1.

x =

2.

x =

3.

x =

4.

x =

5.

x =

Show answer & explanation

Answer

The five real values of

x

that make the expression undefined, from least to greatest, are

-3

,

-1

,

1

,

3

, and

4

.

The rational expression is undefined when the denominator of the first expression is zero, when the denominator of the divisor is zero, or when the numerator of the divisor is zero. Factoring these three parts gives

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

,

(x^2 - 3x - 4) = (x-4)(x+1)

, and

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

. The values that make any of these equal to zero are

-3, -1, 1, 3, 4

.

Step-by-Step Solution

1

Identify values that make the first denominator zero.

x = 1

and

x = -1

The denominator of the first rational expression is

x^2 - 1

. Setting

x^2 - 1 = 0

yields

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

, which gives

x = 1

and

x = -1

.

2

Identify values that make the second denominator zero.

x = 4

and

x = -1

The denominator of the second rational expression (the divisor) is

x^2 - 3x - 4

. Setting

x^2 - 3x - 4 = 0

yields

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

, which gives

x = 4

and

x = -1

.

3

Identify values that make the divisor equal to zero.

x = 3

and

x = -3

Since the operation is division, dividing by zero is undefined. The divisor

\frac{x^2 - 9}{x^2 - 3x - 4}

is equal to zero when its numerator is zero. Setting

x^2 - 9 = 0

yields

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

, which gives

x = 3

and

x = -3

.

4

Combine all unique restricted values and sort them in ascending order.

-3, -1, 1, 3, 4

Combining the restricted values from steps 1, 2, and 3 gives the set

\{-3, -1, 1, 3, 4\}

. Sorting these from least to greatest yields

-3, -1, 1, 3, 4

.

Key Concept

Domain restrictions in division of rational expressions

Estimated Time:2m 30s

Question 1735Question →

Solve each quadratic equation by factoring, and match the equation to its correct solution set.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The equation

3x(4x + 5) = 5 - 2x

matches the solution set

\{-\frac{5}{3}, \frac{1}{4}\}

; the equation

x(12x + 1) = 35

matches the solution set

\{-\frac{7}{4}, \frac{5}{3}\}

; and the equation

(2x - 3)(3x + 1) = 5(1 - x)

matches the solution set

\{-1, \frac{4}{3}\}

.

Each of the quadratic equations is solved by first expanding any products, collecting all terms on the left-hand side to establish the standard form

ax^2 + bx + c = 0

, dividing by any common numerical factors, factoring the resulting quadratic expression into two linear binomials, and solving each linear equation for

x

. This correctly pairs the first equation with

\{-\frac{5}{3}, \frac{1}{4}\}

, the second equation with

\{-\frac{7}{4}, \frac{5}{3}\}

, and the third equation with

\{-1, \frac{4}{3}\}

.

Step-by-Step Solution

1

Solve

3x(4x + 5) = 5 - 2x

.

12x^2 + 17x - 5 = 0 \Rightarrow (3x + 5)(4x - 1) = 0 \Rightarrow x = -\frac{5}{3}

or

x = \frac{1}{4}

.

Distribute the term on the left, rearrange the terms to set the equation to zero, find factors of

12 \times (-5) = -60

that sum to

17

(which are

20

and

-3

), factor by grouping, and apply the Zero Product Property.

2

Solve

x(12x + 1) = 35

.

12x^2 + x - 35 = 0 \Rightarrow (3x - 5)(4x + 7) = 0 \Rightarrow x = \frac{5}{3}

or

x = -\frac{7}{4}

.

Expand the left side, subtract

35

from both sides, find factors of

12 \times (-35) = -420

that sum to

1

(which are

21

and

-20

), factor by grouping, and solve for

x

.

3

Solve

(2x - 3)(3x + 1) = 5(1 - x)

.

6x^2 - 2x - 8 = 0 \Rightarrow 3x^2 - x - 4 = 0 \Rightarrow (3x - 4)(x + 1) = 0 \Rightarrow x = \frac{4}{3}

or

x = -1

.

Expand both sides, move all terms to the left, divide the quadratic equation by

2

to simplify, factor the trinomial, and solve for the roots.

Key Concept

Rearranging non-standard quadratic equations into the standard form

ax^2 + bx + c = 0

and solving them by factoring over the integers.

Question 1736Question →

Match each quadratic equation with its correct solution set by solving the equation by factoring.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The equation

x(x - 1) = 12

matches the solution set

\{-3, 4\}

; the equation

2x^2 + 5x = 3

matches the solution set

\{-3, \frac{1}{2}\}

; the equation

3x^2 + 8 = 10x

matches the solution set

\{\frac{4}{3}, 2\}

; and the equation

2x^2 - 24 = 8x

matches the solution set

\{-2, 6\}

.

Each quadratic equation is correctly matched to its solutions by first rewriting the equation in standard form

ax^2 + bx + c = 0

, factoring the trinomial over the integers, and then applying the zero product property to find the roots.

Step-by-Step Solution

1

Set each quadratic equation to standard form

ax^2 + bx + c = 0

by expanding terms and moving all terms to one side.

The equations become:
1)

x^2 - x - 12 = 0

2)

2x^2 + 5x - 3 = 0

3)

3x^2 - 10x + 8 = 0

4)

2x^2 - 8x - 24 = 0

Before a quadratic equation can be solved by factoring, it must be set equal to zero so that the zero product property can be applied.

2

Factor each quadratic expression completely over the integers.

The factored expressions are:
1)

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

2)

(2x - 1)(x + 3) = 0

3)

(3x - 4)(x - 2) = 0

4)

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

Factoring rewrites the quadratic expressions as products of linear factors.

3

Apply the zero product property by setting each linear factor equal to zero and solving for

x

.

The solution sets are:
1)

x = 4

or

x = -3

, yielding

\{-3, 4\}

2)

x = \frac{1}{2}

or

x = -3

, yielding

\{-3, \frac{1}{2}\}

3)

x = \frac{4}{3}

or

x = 2

, yielding

\{\frac{4}{3}, 2\}

4)

x = 6

or

x = -2

, yielding

\{-2, 6\}

If the product of two or more algebraic factors is zero, then at least one of the individual factors must equal zero.

Key Concept

Solving Quadratic Equations by Factoring

Alternative Method

You can verify the solution sets by substituting the values of the roots back into the original equations to check if they yield a true statement, or by using the quadratic formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

as an alternative algebraic method.

Estimated Time:1m 30s

Question 1737Question →

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $z$ is defined as $z = (3 - 2i)^2 - 4i^{103}$ . What is the imaginary part of $z$ ?

Show answer & explanation

Answer: -8

Answer

The imaginary part of

z

is

-8

.

The expression

(3-2i)^2

expands to

9 - 12i + 4i^2 = 5 - 12i

. The term

i^{103}

simplifies to

-i

since

103

leaves a remainder of

3

when divided by

4

. Subtracting

4i^{103}

corresponds to adding

4i

, giving

z = (5-12i) + 4i = 5-8i

. The coefficient of the imaginary part is

-8

.

Step-by-Step Solution

1

Expand

(3 - 2i)^2

5 - 12i

Use the binomial expansion formula and substitute

i^2 = -1

.

2

Simplify the term

-4i^{103}

4i

Since

103

divided by

4

leaves a remainder of

3

,

i^{103} = i^3 = -i

. Therefore,

-4i^{103} = -4(-i) = 4i

.

3

Combine terms to find

z

and identify its imaginary part

-8

Add the components:

z = (5 - 12i) + 4i = 5 - 8i

. The imaginary part is the coefficient of

i

, which is

-8

.

Key Concept

Operations on complex numbers including binomial expansion, powers of the imaginary unit

i

, and identification of the imaginary part.

Question 1738Question →

For the imaginary unit $i = \sqrt{-1}$ , the complex number $w$ is defined as $w = \frac{5 + 12i}{(1 - i)^4}$ . What is the absolute value of $w$ ?

Show answer & explanation

Answer: 3.25

Answer

The absolute value of

w

is 3.25.

The correct answer is 3.25 because simplifying the denominator yields

(1-i)^4 = -4

. Dividing the numerator by

-4

gives the complex number

w = -1.25 - 3i

. The absolute value of

w

is then calculated as

\sqrt{(-1.25)^2 + (-3)^2} = \sqrt{1.5625 + 9} = \sqrt{10.5625} = 3.25

. Alternatively, using properties of absolute values, the absolute value of the quotient is the quotient of the absolute values:

|w| = \frac{|5 + 12i|}{|1-i|^4} = \frac{\sqrt{5^2 + 12^2}}{(\sqrt{1^2 + (-1)^2})^4} = \frac{13}{4} = 3.25

.

Step-by-Step Solution

1

Simplify the denominator

(1 - i)^4

(1 - i)^4 = -4

Calculate

(1 - i)^2 = -2i

, then square the result to obtain

(-2i)^2 = -4

.

2

Write the complex number

w

in standard form

a + bi

w = -1.25 - 3i

Divide each term in the numerator by the simplified denominator

-4

.

3

Calculate the magnitude

|w|

|w| = 3.25

Use the definition of absolute value of a complex number,

|a + bi| = \sqrt{a^2 + b^2}

.

Key Concept

Absolute value of a complex number and operations on complex numbers

Estimated Time:2m 0s

Question 1739Question →

If $(5 - 2i) - (2 + ki) = 3 + 4i$ , where $i = \sqrt{-1}$ and $k$ is a constant, what is the value of $k$ ?

Show answer & explanation

Answer: -6

Answer

The value of the constant

k

is

-6

.

Distributing the subtraction sign across the second complex number gives

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

. Combining the real parts (

5 - 2 = 3

) and grouping the imaginary parts yields

3 + (-2 - k)i = 3 + 4i

. Since the real parts are equal, we set the coefficients of the imaginary parts equal to each other:

-2 - k = 4

. Solving for

k

gives

k = -6

.

Step-by-Step Solution

1

Distribute the negative sign to the expression

(2 + ki)

-2 - ki

To remove the parentheses, the subtraction must apply to all terms inside the parentheses.

2

Group and combine the real parts and imaginary parts on the left side of the equation

3 - (2 + k)i

Grouping like terms allows us to express the left side as a standard complex number

a + bi

.

3

Equate the imaginary parts from both sides of the equation

-2 - k = 4

For two complex numbers to be equal, their corresponding real parts and imaginary parts must be equal.

4

Solve the linear equation for

k

k = -6

Isolating

k

gives the value that satisfies the original equation.

Key Concept

Equality of complex numbers and operations of addition/subtraction on complex numbers.

Estimated Time:45s

Question 1740Question →

For all real numbers $x$ and $y$ , match each algebraic expression on the left with its simplified equivalent expression on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The expression

2(x - 3y) + 4y

simplifies to

2x - 2y

, the expression

-2(x - 3y) + 8y

simplifies to

-2x + 14y

, and the expression

2(x + 3y) - 4y

simplifies to

2x + 2y

.

Each expression on the left-hand side is expanded by applying the distributive property and then simplified by combining the terms involving

y

. This correctly matches

2(x - 3y) + 4y

to

2x - 2y

,

-2(x - 3y) + 8y

to

-2x + 14y

, and

2(x + 3y) - 4y

to

2x + 2y

.

Step-by-Step Solution

1

Simplify the expression

2(x - 3y) + 4y

.

2x - 2y

Distribute

2

to both terms inside the parentheses to get

2x - 6y

, then combine the like terms

-6y

and

4y

to get

-2y

.

2

Simplify the expression

-2(x - 3y) + 8y

.

-2x + 14y

Distribute

-2

to both terms inside the parentheses to get

-2x + 6y

, then combine the like terms

6y

and

8y

to get

14y

.

3

Simplify the expression

2(x + 3y) - 4y

.

2x + 2y

Distribute

2

to both terms inside the parentheses to get

2x + 6y

, then combine the like terms

6y

and

-4y

to get

2y

.

Key Concept

Simplifying algebraic expressions by distributing coefficients and combining like terms.

Estimated Time:1m 0s

Answer

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