Intermediate Algebra Practice Questions — ACT

For the complex number $z = \frac{3 + 5i}{2 - i}$ , where $i = \sqrt{-1}$ , what is the real part of $z$ ?

\frac{1}{5}

\frac{11}{5}

\frac{3}{2}

\frac{1}{3}

-\frac{1}{5}

Show answer & explanation

Answer: $\frac{1}{5}$

Answer

The real part of the complex number is

\frac{1}{5}

.

To find the real part of the complex fraction, we multiply both the numerator and the denominator by the complex conjugate of the denominator,

2 + i

. This yields a denominator of

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

. The numerator becomes

(3+5i)(2+i) = 6 + 3i + 10i + 5i^2 = 1 + 13i

. Writing the result in standard form gives

\frac{1}{5} + \frac{13}{5}i

, so the real part is

\frac{1}{5}

.

Step-by-Step Solution

1

Identify the complex conjugate of the denominator, which is

2 + i

, and set up the multiplication for both the numerator and the denominator of

z

.

\frac{(3 + 5i)(2 + i)}{(2 - i)(2 + i)}

Multiplying the numerator and denominator by the conjugate of the denominator allows us to simplify the expression and eliminate the imaginary unit from the denominator.

2

Expand the numerator using binomial multiplication and simplify using the identity

i^2 = -1

.

6 + 3i + 10i + 5i^2 = 6 + 13i - 5 = 1 + 13i

This determines the simplified complex expression in the numerator.

3

Expand the denominator and simplify using the identity

i^2 = -1

.

(2 - i)(2 + i) = 4 - i^2 = 4 - (-1) = 5

This simplifies the denominator into a real number.

4

Combine the simplified numerator and denominator to write

z

in standard form

a + bi

and identify the real part

a

.

z = \frac{1 + 13i}{5} = \frac{1}{5} + \frac{13}{5}i

, so the real part is

\frac{1}{5}

.

Expressing the complex number in standard form separates the real and imaginary components.

Key Concept

Complex Division and Simplification

Estimated Time:1m 30s

Question 2Question →

For the imaginary unit $i$ , what is the simplified form of the expression $(2 + 3i)(1 - 2i)$ ?

-4 - i

8 + i

8 - i

-4 + i

8 - 7i

Show answer & explanation

Answer: $8 - i$

Answer

8 - i

The correct answer is

8 - i

. To simplify the product of

(2 + 3i)(1 - 2i)

, we distribute the terms to get

2(1) + 2(-2i) + 3i(1) + 3i(-2i) = 2 - 4i + 3i - 6i^2

. Since

i^2 = -1

, the last term

-6i^2

simplifies to

-6(-1) = 6

. Combining the real parts (

2 + 6 = 8

) and the imaginary parts (

-4i + 3i = -i

) yields

8 - i

.

Step-by-Step Solution

1

Expand the product of the two binomials

(2 + 3i)(1 - 2i)

using the distributive property.

2 - 4i + 3i - 6i^2

Each term of the first binomial must be multiplied by each term of the second binomial.

2

Substitute

-1

for

i^2

in the expression.

2 - 4i + 3i - 6(-1) = 2 - 4i + 3i + 6

By definition, the imaginary unit

i

satisfies the equation

i^2 = -1

.

3

Combine the real terms and combine the imaginary terms to obtain the final simplified form.

8 - i

Adding the real parts (

2 + 6 = 8

) and the imaginary parts (

-4i + 3i = -i

) simplifies the expression to a single complex number.

Key Concept

Multiplying complex binomials and simplifying using the definition of the imaginary unit

i^2 = -1

.

Estimated Time:1m 0s

Question 3Question →

The elevation $y$ (in meters) of a roller coaster track is modeled by the equation $y = (x - 3)^2 - 4$ , where $x$ represents the horizontal distance (in meters) from the start of the ride. A straight support beam is designed such that the height of the track is 7 meters less than 7 times the horizontal distance. The support beam connects to the roller coaster track at two points, $(x_1, y_1)$ and $(x_2, y_2)$ . What is the value of $x_1 y_1 + x_2 y_2$ ?

126

-42

924

336

2664

Show answer & explanation

Answer: 924

Answer

The value of the expression is 924, which corresponds to the coordinates of the two connection points

(1, 0)

and

(12, 77)

.

By translating the verbal relationship to the equation

y = 7x - 7

, we can equate it to the track's equation:

(x - 3)^2 - 4 = 7x - 7

. Expanding and simplifying yields

x^2 - 13x + 12 = 0

, which factors as

(x - 1)(x - 12) = 0

. This gives the horizontal intersection coordinates

x_1 = 1

and

x_2 = 12

. Substituting these coordinates back into the linear equation gives the vertical coordinates

y_1 = 0

and

y_2 = 77

. Evaluating the final expression yields

(1)(0) + (12)(77) = 924

.

Step-by-Step Solution

1

Translate the verbal description of the support beam's height into an algebraic equation.

y = 7x - 7

The description states that the height (

y

) is '7 meters less than 7 times the horizontal distance (

x

)', which translates to

7x - 7

.

2

Set the two equations for

y

equal to each other to find the

x

-coordinates of the intersection points.

(x - 3)^2 - 4 = 7x - 7 \Rightarrow x^2 - 6x + 5 = 7x - 7 \Rightarrow x^2 - 13x + 12 = 0

At the points of intersection, the elevation of the track and the support beam must be equal.

3

Solve the quadratic equation for

x

by factoring.

(x - 1)(x - 12) = 0 \Rightarrow x_1 = 1

and

x_2 = 12

Factoring the quadratic expression allows us to identify its roots directly.

4

Substitute the

x

-values back into the linear equation to find the corresponding

y

-coordinates.

For

x_1 = 1

,

y_1 = 7(1) - 7 = 0

. For

x_2 = 12

,

y_2 = 7(12) - 7 = 77

. The intersection points are

(1, 0)

and

(12, 77)

.

The intersection points must satisfy both equations in the system.

5

Calculate the required expression

x_1 y_1 + x_2 y_2

.

(1)(0) + (12)(77) = 0 + 924 = 924

This evaluates the target expression using the coordinates found in the previous steps.

Key Concept

Solving systems of linear and non-linear (quadratic) equations using substitution and factoring.

Question 4Question →

If the product of the complex numbers $4 - 2i$ and $k + 6i$ is a real number, where $k$ is a real constant and $i = \sqrt{-1}$ , what is the value of $k$ ?

Show answer & explanation

Answer: 12

Answer

12

The product of the complex numbers

(4 - 2i)(k + 6i)

expands to

4k + 24i - 2ki - 12i^2

. Substituting

i^2 = -1

simplifies the expression to

(4k + 12) + (24 - 2k)i

. For this expression to represent a real number, the imaginary part must be zero:

24 - 2k = 0

, which yields

k = 12

.

Step-by-Step Solution

1

Multiply the two complex numbers

(4 - 2i)

and

(k + 6i)

using the FOIL method.

4k + 24i - 2ki - 12i^2

To find the product of the two complex expressions.

2

Substitute

i^2 = -1

into the expression and group the real and imaginary parts.

(4k + 12) + (24 - 2k)i

To simplify the expression into standard complex form

a + bi

.

3

Set the imaginary part of the resulting complex number to

0

and solve for

k

.

24 - 2k = 0 \implies k = 12

A complex number is real if and only if its imaginary part is equal to zero.

Key Concept

Complex multiplication and the definition of a real number in the complex plane

Question 5Question →

What is the sum of the squares of the two real solutions to the quadratic equation $\frac{1}{2}x^2 - 3x + 2 = 0$ ?

Show answer & explanation

Answer: 28

Answer

The sum of the squares of the two real solutions is 28.

Multiplying the equation

\frac{1}{2}x^2 - 3x + 2 = 0

by 2 yields

x^2 - 6x + 4 = 0

. By Vieta's formulas, the sum of the solutions is

x_1 + x_2 = 6

and the product of the solutions is

x_1 x_2 = 4

. Using the algebraic identity

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

, the sum of the squares is

6^2 - 2(4) = 36 - 8 = 28

. Alternatively, using the quadratic formula on the simplified equation gives solutions of

3 + \sqrt{5}

and

3 - \sqrt{5}

. Squaring these values yields

14 + 6\sqrt{5}

and

14 - 6\sqrt{5}

, which sum to 28.

Step-by-Step Solution

1

Multiply the quadratic equation by 2 to clear the fraction.

x^2 - 6x + 4 = 0

Simplifying the fractional coefficients makes the equation easier to analyze and solve.

2

Identify the sum and product of the roots using Vieta's formulas.

x_1 + x_2 = 6

and

x_1 x_2 = 4

For a quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is

-b/a

and the product is

c/a

.

3

Express the sum of the squares of the roots using the algebraic identity.

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

This identity allows the direct calculation of the sum of squares without needing to find the individual roots.

4

Substitute the values of the sum and product into the identity.

6^2 - 2(4) = 36 - 8 = 28

Plugging in the sum of 6 and product of 4 yields the final numerical answer.

Key Concept

Vieta's formulas state that for a quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is

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

and the product of the roots is

x_1 x_2 = \frac{c}{a}

. Symmetric functions of roots like

x_1^2 + x_2^2

can be expressed in terms of these values.

Alternative Method

Find the roots of the equation directly using the quadratic formula. After simplifying to

x^2 - 6x + 4 = 0

, the roots are

x = \frac{6 \pm \sqrt{(-6)^2 - 4(1)(4)}}{2} = 3 \pm \sqrt{5}

. Squaring both solutions gives

(3 + \sqrt{5})^2 = 14 + 6\sqrt{5}

and

(3 - \sqrt{5})^2 = 14 - 6\sqrt{5}

. Adding these squares together yields

(14 + 6\sqrt{5}) + (14 - 6\sqrt{5}) = 28

.

Estimated Time:1m 30s

Question 6Question →

For the imaginary unit $i$ , which of the following is equivalent to the complex expression $\frac{2 + i^{15}}{1 - 2i^9}$ ?

\frac{4}{5} + \frac{3}{5}i

i

\frac{3}{5}i

\frac{4}{5}

-\frac{4}{3} - i

Show answer & explanation

Answer: $\frac{4}{5} + \frac{3}{5}i$

Answer

\frac{4}{5} + \frac{3}{5}i

Simplifying the powers of

i

yields

i^{15} = -i

and

i^9 = i

, giving the expression

\frac{2 - i}{1 - 2i}

. Multiplying the numerator and denominator by the complex conjugate

1 + 2i

results in the fraction

\frac{(2 - i)(1 + 2i)}{(1 - 2i)(1 + 2i)}

. Expanding both parts and substituting

i^2 = -1

gives

\frac{4 + 3i}{5}

, which simplifies to

\frac{4}{5} + \frac{3}{5}i

.

Step-by-Step Solution

1

Simplify the powers of the imaginary unit

i

in the expression.

i^{15} = i^{12} \cdot i^3 = 1 \cdot (-i) = -i

and

i^9 = i^8 \cdot i = 1 \cdot i = i

. The expression becomes

\frac{2 - i}{1 - 2i}

.

Reducing powers of

i

simplifies the expression and makes it easier to work with binomials.

2

Multiply the numerator and denominator by the complex conjugate of the denominator,

1 + 2i

.

\frac{2 - i}{1 - 2i} \cdot \frac{1 + 2i}{1 + 2i} = \frac{(2 - i)(1 + 2i)}{(1 - 2i)(1 + 2i)}

Multiplying by the conjugate rationalizes the denominator, converting it to a real number.

3

Expand and simplify the numerator and denominator using the property

i^2 = -1

.

Numerator:

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

. Denominator:

(1 - 2i)(1 + 2i) = 1 - 4i^2 = 1 - 4(-1) = 5

.

Expanding the binomial products allows combining real and imaginary parts.

4

Write the resulting fraction in standard complex form

a + bi

.

\frac{4 + 3i}{5} = \frac{4}{5} + \frac{3}{5}i

Standard form separates the real part and the imaginary part clearly.

Key Concept

Simplifying complex expressions by reducing powers of

i

and rationalizing the denominator using the complex conjugate.

Question 7Question →

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 8Question →

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 9Question →

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 10Question →

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 11Question →

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 12Question →

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 13Question →

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 14Question →

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 15Question →

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 16Question →

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 17Question →

For what real value of $x$ is the rational expression $\frac{2x + 1}{x - 7}$ undefined? Fill in the blank with the correct number.

Fill in the blanks below

The rational expression

\frac{2x + 1}{x - 7}

is undefined when $x = .

Show answer & explanation

Answer

The rational expression is undefined when the variable x equals 7.

A rational expression is undefined when its denominator is equal to

0

. For the expression

\frac{2x + 1}{x - 7}

, setting the denominator

x - 7 = 0

and solving for

x

yields

x = 7

.

Step-by-Step Solution

1

Set the denominator of the rational expression equal to 0.

x - 7 = 0

A rational expression is undefined when its denominator is equal to 0 because division by zero is undefined in the real number system.

2

Solve the linear equation for

x

by adding 7 to both sides.

x = 7

Isolating the variable

x

gives the value that makes the denominator zero.

Key Concept

Identifying values that make a rational expression undefined

Question 18Question →

For what real values of $x$ is the rational expression $g(x) = \frac{x^3 - 3x^2}{\frac{2}{x-2} - \frac{x+4}{x^2 - 4}}$ undefined? Determine these three values and enter them in the blanks below in order from least to greatest.

Fill in the blanks below

The values of

x

for which

g(x)

is undefined, ordered from least to greatest, are

x =

,

x =

, and

x =

.

Show answer & explanation

Answer

The values of

x

for which the expression is undefined, ordered from least to greatest, are

-2

,

0

, and

2

.

The rational expression is undefined where any constituent denominator is equal to zero, or where the entire main denominator is equal to zero. The constituent denominators are

x-2

and

x^2-4

, which are zero when

x = 2

or

x = -2

. The entire main denominator simplifies to

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

, which is zero when

x = 0

. Therefore, the three values of

x

that make the expression undefined are

-2

,

0

, and

2

.

Step-by-Step Solution

1

Identify the values of

x

that make the denominators of the individual rational terms equal to zero.

The denominators are

x - 2

and

x^2 - 4

. Setting these to zero gives

x = 2

and

x = \pm 2

. Thus,

x = 2

and

x = -2

are values that make the individual terms undefined.

Any rational term is undefined if its denominator is equal to zero.

2

Find the values of

x

that make the overall denominator of the main fraction equal to zero by first simplifying it.

The main denominator is

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

. Finding a common denominator yields

\frac{2(x+2) - (x+4)}{(x-2)(x+2)} = \frac{x}{(x-2)(x+2)}

. Setting this equal to zero gives

x = 0

.

The main rational expression is undefined if its entire denominator is equal to zero.

3

Combine all restricted values and list them in order from least to greatest.

The restricted values are

x = -2

,

x = 0

, and

x = 2

.

These are all the distinct real numbers that cause any division by zero in the original expression.

Key Concept

Finding the domain of nested rational expressions

Estimated Time:2m 0s

Question 19Question →

For all real numbers $x$ such that $x \neq 0$ and $x \neq -1$ , which of the following expressions is equivalent to $\frac{3}{x} + \frac{2}{x + 1}$ ?

\frac{5}{2x + 1}

\frac{5x + 1}{x(x + 1)}

\frac{5x + 3}{x(x + 1)}

\frac{5x + 3}{2x + 1}

\frac{6}{x(x + 1)}

Show answer & explanation

Answer: $\frac{5x + 3}{x(x + 1)}$

Answer

\frac{5x + 3}{x(x + 1)}

To add the rational expressions, find the common denominator

x(x + 1)

. Multiplying the first term by

\frac{x + 1}{x + 1}

and the second term by

\frac{x}{x}

gives

\frac{3(x + 1) + 2x}{x(x + 1)}

. Expanding the numerator yields

3x + 3 + 2x

, which simplifies to

5x + 3

. Thus, the equivalent expression is

\frac{5x + 3}{x(x + 1)}

.

Step-by-Step Solution

1

Identify a common denominator for the two rational expressions.

The common denominator for the denominators

x

and

x + 1

is

x(x + 1)

.

To add fractions with different denominators, they must be written with a common denominator.

2

Rewrite each fraction with the common denominator by multiplying the numerator and denominator of the first term by

(x + 1)

and the second term by

x

.

\frac{3(x + 1)}{x(x + 1)} + \frac{2x}{x(x + 1)}

This scales the fractions to have matching denominators without changing their value.

3

Add the numerators together over the common denominator and simplify.

\frac{3x + 3 + 2x}{x(x + 1)} = \frac{5x + 3}{x(x + 1)}

Combining like terms simplifies the numerator to its final form.

Key Concept

Adding rational expressions by finding a common denominator

Estimated Time:1m 0s

Question 20Question →

Solve the equation $\sqrt{2x + 7} - x = -4$ for $x$ . What is the value of the real solution?

Show answer & explanation

Answer: 9

Answer

The only real solution to the equation is 9.

Isolating the radical yields

\sqrt{2x + 7} = x - 4

. Squaring both sides results in the quadratic equation

2x + 7 = x^2 - 8x + 16

, which simplifies to

x^2 - 10x + 9 = 0

. Factoring gives

(x - 9)(x - 1) = 0

, yielding potential solutions of 9 and 1. Checking these solutions in the original equation reveals that 9 is valid, while 1 is extraneous. Therefore, the correct real solution is 9.

Step-by-Step Solution

1

Isolate the radical on one side of the equation.

\sqrt{2x + 7} = x - 4

Before squaring both sides, the radical term must be isolated to simplify the algebraic manipulation.

2

Square both sides of the equation.

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

Squaring both sides eliminates the square root.

3

Expand the squared binomial.

2x + 7 = x^2 - 8x + 16

Applying the identity

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

is necessary to write the equation in polynomial form.

4

Set the quadratic equation to zero.

x^2 - 10x + 9 = 0

Subtracting

2x

and

7

from both sides allows us to solve the quadratic equation.

5

Factor the quadratic equation.

(x - 9)(x - 1) = 0

, giving potential solutions

x = 9

or

x = 1

.

Finding the roots of the quadratic equation provides the candidate solutions.

6

Verify candidates in the original equation.

The solution

x = 9

is valid, while

x = 1

is extraneous.

Squaring both sides can introduce extraneous solutions, so each candidate must be checked in the original equation.

Key Concept

Solving radical equations and verifying for extraneous solutions

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