Intermediate Algebra Practice Questions — ACT — Page 2

Question 21Question →

For the imaginary unit $i$ , if the complex number $z$ is defined by $z = \frac{(1 + 2i)^2}{2 - i}$ , what is the real part of $z$ ?

-2

-\frac{6}{5}

\frac{6}{5}

-\frac{10}{3}

-\frac{3}{2}

Show answer & explanation

Answer: $-2$

Answer

The real part of the complex number

z

is

-2

.

To find the real part of the complex number, we first simplify the expression by expanding the squared binomial in the numerator, which yields

-3 + 4i

. Next, we rationalize the fraction by multiplying both the numerator and the denominator by the complex conjugate of the denominator,

2 + i

. This multiplication yields

\frac{-10 + 5i}{5}

. Dividing both the real and imaginary terms by

5

results in the standard form

-2 + i

. Thus, the real part of this complex number is

-2

.

Step-by-Step Solution

1

Expand the squared binomial in the numerator of the expression for

z

.

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

Apply the algebraic identity

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

.

2

Simplify the expanded numerator using the definition of the imaginary unit.

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

Since

i^2 = -1

, the term

4i^2

simplifies to

-4

. Combining this with

1

gives the real part

-3

.

3

Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator to rationalize it.

z = \frac{-3 + 4i}{2 - i} \cdot \frac{2 + i}{2 + i} = \frac{(-3 + 4i)(2 + i)}{(2 - i)(2 + i)}

Multiplying the denominator by its complex conjugate,

2 + i

, eliminates the imaginary unit from the denominator.

4

Expand and simplify the numerator and denominator.

z = \frac{-6 - 3i + 8i + 4i^2}{4 - i^2} = \frac{-6 + 5i - 4}{4 - (-1)} = \frac{-10 + 5i}{5}

Using the distributive property in the numerator gives

-6 + 5i + 4i^2

. Since

i^2 = -1

, this simplifies to

-10 + 5i

. In the denominator,

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

.

5

Divide both terms of the simplified numerator by the denominator to express

z

in standard form

a + bi

.

z = -2 + i

Dividing the real part

-10

by

5

yields the real part

-2

, and dividing the imaginary part

5i

by

5

yields the imaginary part

i

.

Key Concept

Division of complex numbers using the complex conjugate

Estimated Time:2m 0s

Question 22Question →

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $z$ is defined by $z = \frac{11 + 3i}{3 - i}$ . What is the real part of $z$ ?

Show answer & explanation

Answer: 3

Answer

The real part of the complex number is 3.

By multiplying both the numerator and the denominator of

\frac{11 + 3i}{3 - i}

by the conjugate of the denominator,

3 + i

, we obtain

\frac{(11+3i)(3+i)}{(3-i)(3+i)} = \frac{33 + 11i + 9i + 3i^2}{9 - i^2} = \frac{30 + 20i}{10} = 3 + 2i

. The real part of this complex number is the term without

i

, which is 3.

Step-by-Step Solution

1

Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator, which is

3 + i

.

z = \frac{(11 + 3i)(3 + i)}{(3 - i)(3 + i)}

To eliminate the imaginary unit from the denominator.

2

Expand and simplify the numerator using the distributive property and substituting

-1

for

i^2

.

(11 + 3i)(3 + i) = 33 + 11i + 9i + 3i^2 = 33 + 20i + 3(-1) = 30 + 20i

To combine the real and imaginary terms of the numerator.

3

Expand and simplify the denominator using the difference of squares property and substituting

-1

for

i^2

.

(3 - i)(3 + i) = 9 - i^2 = 9 - (-1) = 10

To find the real number denominator.

4

Divide each term in the simplified numerator by the denominator.

z = \frac{30 + 20i}{10} = 3 + 2i

To express the complex number in the standard form

a + bi

.

5

Extract the real part of the resulting complex number

3 + 2i

.

3

The real part of a complex number in the form

a + bi

is

a

.

Key Concept

Division of complex numbers using the complex conjugate

Alternative Method

Instead of simplifying the fraction directly, assume the resulting complex number is

x + yi

, where

x

represents the real part and

y

represents the imaginary part. We can set up the equation

x + yi = \frac{11 + 3i}{3 - i}

and multiply both sides by

3 - i

to get

(x + yi)(3 - i) = 11 + 3i

. Expanding the left side gives

(3x + y) + (3y - x)i = 11 + 3i

. Equating the real and imaginary parts yields a system of linear equations:

3x + y = 11

and

-x + 3y = 3

. Multiplying the second equation by 3 and adding it to the first equation gives

10y = 20

, which means

y = 2

. Substituting

y = 2

back into the first equation yields

3x + 2 = 11

, which simplifies to

3x = 9

, or

x = 3

. The real part is therefore 3.

Estimated Time:1m 30s

Question 23Question →

For all real numbers $x$ such that $x \neq -2$ and $x \neq 3$ , the expression $\frac{2x}{x - 3} - \frac{5}{x + 2}$ is equivalent to which of the following?

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

\frac{2x^2 - x - 15}{x^2 - x - 6}

\frac{x + 15}{x^2 - x - 6}

\frac{2x^2 - x + 15}{x^2 - x - 6}

\frac{2x^2 - x + 15}{x^2 - 6}

Show answer & explanation

Answer: \frac{2x^2 - x + 15}{x^2 - x - 6}

Answer

The expression is equivalent to

\frac{2x^2 - x + 15}{x^2 - x - 6}

.

To subtract the rational expressions, we find the common denominator, which is

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

. We rewrite each fraction with this denominator, which gives

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

and

\frac{5(x - 3)}{x^2 - x - 6} = \frac{5x - 15}{x^2 - x - 6}

. Subtracting the second numerator from the first gives

(2x^2 + 4x) - (5x - 15) = 2x^2 - x + 15

. Placing this over the common denominator gives the simplified expression

\frac{2x^2 - x + 15}{x^2 - x - 6}

.

Step-by-Step Solution

1

Identify the common denominator.

The common denominator is

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

.

To subtract rational expressions, we need a common denominator.

2

Rewrite each rational expression with the common denominator.

The first term becomes

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

, and the second term becomes

\frac{5(x - 3)}{(x - 3)(x + 2)} = \frac{5x - 15}{x^2 - x - 6}

.

Multiplying the numerator and denominator of each term by the missing factor keeps the values of the expressions unchanged.

3

Subtract the numerators.

(2x^2 + 4x) - (5x - 15) = 2x^2 + 4x - 5x + 15 = 2x^2 - x + 15

.

Subtracting the second numerator requires distributing the negative sign to both terms of the expression

(5x - 15)

.

4

Combine the result over the common denominator.

\frac{2x^2 - x + 15}{x^2 - x - 6}

Write the simplified numerator over the common denominator.

Key Concept

Subtraction of rational expressions involves finding a common denominator, expanding the numerators, and distributing negative signs carefully.

Question 24Question →

If $\sqrt{3x - 2} = 4$ , what is the value of $x$ ?

Show answer & explanation

Answer: 6

Answer

The value of

x

is 6.

Squaring both sides of the equation

\sqrt{3x - 2} = 4

eliminates the radical, leading to

3x - 2 = 16

. Adding 2 to both sides results in

3x = 18

. Dividing both sides by 3 gives

x = 6

. Substituting 6 back into the original equation yields

\sqrt{3(6) - 2} = \sqrt{18 - 2} = \sqrt{16} = 4

, which verifies that the solution is correct.

Step-by-Step Solution

1

Square both sides of the equation to remove the radical.

3x - 2 = 16

Squaring a square root removes the radical sign since

(\sqrt{a})^2 = a

for non-negative values.

2

Add 2 to both sides of the equation.

3x = 18

Adding the constant term moves it to the other side to isolate the term containing the variable.

3

Divide both sides by 3.

x = 6

Dividing by the coefficient of

x

yields the final solution.

Key Concept

Solving basic radical equations by isolating the radical and squaring both sides.

Question 25Question →

For the imaginary unit $i$ , where $i^2 = -1$ , which of the following is equivalent to the expression $\frac{5}{2 - i}$ ?

2 - i

10 + 5i

2 + i

\frac{10 + 5i}{3}

2 + 5i

Show answer & explanation

Answer: $2 + i$

Answer

The simplified expression is

2 + i

.

To simplify the expression, multiply both the numerator and denominator by the complex conjugate of the denominator, which is

2 + i

. This results in

\frac{5(2+i)}{(2-i)(2+i)} = \frac{10+5i}{4-i^2}

. Since

i^2 = -1

, the denominator becomes

4 - (-1) = 5

. Dividing both terms in the numerator by

5

gives

2 + i

.

Step-by-Step Solution

1

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

2 + i

.

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

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

2

Expand the numerator and the denominator, substituting

-1

for

i^2

.

\frac{10 + 5i}{4 - (-1)} = \frac{10 + 5i}{5}

Using the distributive property for the numerator and the difference of squares identity for the denominator, along with the definition

i^2 = -1

.

3

Divide each term in the numerator by the denominator.

2 + i

Distributing the division by

5

to both the real and imaginary parts of the numerator simplifies the expression to standard form.

Key Concept

Rationalizing the denominator of a complex fraction by multiplying by the complex conjugate of the denominator.

Estimated Time:45s

Question 26Question →

For the imaginary unit $i$ , where $i^2 = -1$ , let $z$ be the complex number defined by $z = \frac{10 + ki}{2 - i}$ , where $k$ is a real constant. If the imaginary part of $z$ is $4$ , what is the value of $k$ ?

Show answer & explanation

Answer: 5

Answer

The value of

k

is

5

.

Multiplying the numerator and denominator of

z = \frac{10 + ki}{2 - i}

by the complex conjugate

2 + i

gives

z = \frac{(20 - k) + (10 + 2k)i}{5}

. The imaginary part is

\frac{10 + 2k}{5}

. Setting this expression equal to

4

and solving for

k

yields

k = 5

.

Step-by-Step Solution

1

Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator, which is

2 + i

.

z = \frac{(10 + ki)(2 + i)}{(2 - i)(2 + i)}

To eliminate the imaginary unit from the denominator and express the complex number in standard form.

2

Expand both the numerator and the denominator, using the property

i^2 = -1

.

z = \frac{20 + 10i + 2ki + ki^2}{4 - i^2} = \frac{(20 - k) + (10 + 2k)i}{5}

To separate the real terms and imaginary terms in the numerator and simplify the denominator to a real number.

3

Express the complex number in standard form

a + bi

to identify the imaginary part.

z = \frac{20 - k}{5} + \left(\frac{10 + 2k}{5}\right)i

The imaginary part of a complex number is the coefficient of

i

, which is

\frac{10 + 2k}{5}

.

4

Set the imaginary part equal to

4

and solve the linear equation for

k

.

\frac{10 + 2k}{5} = 4 \implies 10 + 2k = 20 \implies 2k = 10 \implies k = 5

To find the specific value of the constant

k

that makes the imaginary part of

z

equal to

4

.

Key Concept

Rationalizing complex numbers and identifying real and imaginary components

Question 27Question →

For all real values of

x

where the expression is defined, consider the equation:

\frac{2x}{\sqrt{x + 3}} = \sqrt{x + 3} - 1

Which of the following represents the complete set of real solutions to this equation?

\{1\}

\{1, 6\}

\{6\}

\{-3, 4\}

\{2\}

Show answer & explanation

Answer: $\{1\}$

Answer

The set containing only 1

The correct answer is the set containing only 1. To solve the equation, we first multiply both sides by the denominator

\sqrt{x+3}

, which yields

2x = x + 3 - \sqrt{x+3}

. Isolating the radical gives

\sqrt{x+3} = 3-x

. Squaring both sides produces the quadratic equation

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

, which simplifies to

x^2-7x+6=0

. Solving this quadratic gives candidate solutions of 1 and 6. Substituting 6 back into the original equation results in an invalid statement (

4 = 2

), making it extraneous. Substituting 1 yields a valid statement (

1 = 1

), meaning the only real solution is 1.

Step-by-Step Solution

1

Determine the domain of the equation.

x > -3

The expression inside the square root must be non-negative (

x+3 \ge 0

), and since it is in the denominator, it cannot be zero (

x+3 \neq 0

).

2

Clear the denominator by multiplying both sides by

\sqrt{x + 3}

.

2x = x + 3 - \sqrt{x + 3}

Multiplying both sides by the denominator simplifies the rational expression into a form where we can isolate the radical.

3

Isolate the radical term.

\sqrt{x + 3} = 3 - x

Grouping all non-radical terms on one side prepares the equation for squaring to eliminate the radical.

4

Analyze constraints on the variable.

x \le 3

Since the principal square root on the left side is non-negative, the right side

3 - x

must also be non-negative, which restricts any valid solutions to

x \le 3

.

5

Square both sides and simplify to form a quadratic equation.

x^2 - 7x + 6 = 0

Squaring both sides eliminates the radical:

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

.

6

Solve the quadratic equation by factoring.

x = 1

or

x = 6

Factoring

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

gives the potential solutions.

7

Verify the solutions in the original equation.

x = 1

is valid;

x = 6

is extraneous.

Substituting

x = 6

yields

\frac{12}{3} = 4

on the left and

\sqrt{9}-1 = 2

on the right, which are not equal. Substituting

x = 1

yields

\frac{2}{2} = 1

on the left and

\sqrt{4}-1 = 1

on the right, which are equal.

Key Concept

Solving equations containing both rational and radical expressions requires clearing denominators, isolating the radical, squaring both sides, and verifying candidate solutions to eliminate extraneous solutions.

Question 28Question →

Which of the following is the complete set of real solutions to the equation $\sqrt{5x - 9} = x - 3$ ?

\{9\}

\{2, 9\}

\{0, 5\}

\{1.5, 2\}

\{2\}

Show answer & explanation

Answer: $\{9\}$

Answer

\{9\}

The correct option is the set containing only the value 9. To solve the equation, we square both sides to get

5x - 9 = x^2 - 6x + 9

, which simplifies to the quadratic equation

x^2 - 11x + 18 = 0

. Factoring this yields the potential solutions 9 and 2. Substituting these back into the original equation, we find that 9 satisfies the equation while 2 results in a contradiction, making it extraneous.

Step-by-Step Solution

1

Isolate the radical and square both sides of the equation.

5x - 9 = (x - 3)^2

Squaring both sides eliminates the square root to allow solving for the variable.

2

Expand the right-hand side using the binomial squaring rule.

5x - 9 = x^2 - 6x + 9

The square of a binomial

(a - b)^2

is

a^2 - 2ab + b^2

.

3

Move all terms to one side to set the quadratic equation to zero, then factor.

x^2 - 11x + 18 = 0 \Rightarrow (x - 9)(x - 2) = 0

Rearranging terms simplifies the equation into a standard quadratic form that can be factored.

4

Solve for the potential roots and check for extraneous solutions in the original equation.

x = 9

(valid) and

x = 2

(extraneous)

Checking

x = 9

gives

\sqrt{5(9) - 9} = 9 - 3 \Rightarrow 6 = 6

(true). Checking

x = 2

gives

\sqrt{5(2) - 9} = 2 - 3 \Rightarrow 1 = -1

(false).

Key Concept

Solving radical equations and verifying for extraneous solutions

Question 29Question →

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $w$ is defined as $w = \frac{6 + 4i}{2i}$ . What is the imaginary part of $w$ ?

Show answer & explanation

Answer: -3

Answer

The imaginary part of

w

is

-3

.

Dividing each term in the numerator of

\frac{6 + 4i}{2i}

by the denominator

2i

yields

\frac{6}{2i} + \frac{4i}{2i}

, which simplifies to

\frac{3}{i} + 2

. Since

i^2 = -1

, the term

\frac{3}{i}

can be rationalized to

-3i

. Thus, the complex number in standard form is

2 - 3i

. The imaginary part is the real coefficient of

i

, which is

-3

.

Step-by-Step Solution

1

Divide each term in the numerator by the denominator.

w = \frac{6}{2i} + \frac{4i}{2i}

This separates the quotient into two simpler terms that can be simplified individually.

2

Simplify both terms.

w = \frac{3}{i} + 2

Reduce the fractions by dividing out common factors in both the numerators and the denominators.

3

Rationalize the denominator of the imaginary term.

\frac{3}{i} \cdot \frac{i}{i} = \frac{3i}{i^2} = \frac{3i}{-1} = -3i

Multiply the numerator and denominator by

i

to eliminate the imaginary unit from the denominator, using the property

i^2 = -1

.

4

Combine the real and imaginary parts into standard form

a + bi

.

w = 2 - 3i

Group the real constant and the simplified imaginary term together.

5

Identify the imaginary part of the complex number.

-3

The imaginary part of a complex number

a + bi

is the real coefficient

b

of the imaginary unit

i

.

Key Concept

Simplifying a quotient of complex numbers by dividing by a pure imaginary number.

Question 30Question →

If $x$ is a real number that satisfies the equation $\sqrt{2x + 7} + \sqrt{x + 3} = 1$ , what is the value of $x$ ?

Show answer & explanation

Answer: -3

Answer

The only real solution to the equation is

-3

.

The value

-3

is the only real number that satisfies the original equation. Substituting

-3

back into the original equation yields

\sqrt{2(-3) + 7} + \sqrt{-3 + 3} = \sqrt{1} + 0 = 1

, which is true.

Step-by-Step Solution

1

Isolate the first radical term.

\sqrt{2x + 7} = 1 - \sqrt{x + 3}

This allows for squaring both sides to eliminate one radical.

2

Square both sides and simplify.

2x + 7 = x + 4 - 2\sqrt{x + 3}

Squaring removes the radical on the left side, though it creates a middle term on the right side.

3

Isolate the remaining radical term.

x + 3 = -2\sqrt{x + 3}

Grouping the non-radical terms on one side prepares the equation for a second squaring step.

4

Square both sides again to eliminate the remaining radical.

x^2 + 6x + 9 = 4(x + 3)

Squaring both sides eliminates the radical completely, converting the expression into a polynomial equation.

5

Solve the quadratic equation.

x = -3

and

x = 1

Rearranging to

x^2 + 2x - 3 = 0

and factoring as

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

gives the candidate solutions.

6

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

The only valid solution is

x = -3

.

Substituting

x = 1

yields

5 = 1

(invalid), while substituting

x = -3

yields

1 = 1

(valid).

Key Concept

Solving radical equations by isolating radicals and squaring, then testing for extraneous solutions.

Question 31Question →

What is the sum of all real values of

x

that satisfy the equation

\frac{x^2 - 3x}{x^2 - 3x + 2} + \frac{x^2 - 3x - 1}{x^2 - 3x - 3} = \frac{11}{3}

?

Show answer & explanation

Answer: 3

Answer

The sum of all real values of

x

that satisfy the equation is

3

.

Substituting

y = x^2 - 3x

simplifies the original rational equation into the quadratic form

y^2 - y - 12 = 0

. Solving for

y

yields the values

4

and

-3

. Substituting back

x^2 - 3x

for

y

produces two quadratic equations. The first,

x^2 - 3x - 4 = 0

, has real solutions of

4

and

-1

. The second,

x^2 - 3x + 3 = 0

, has a negative discriminant and produces no real solutions. Summing the valid real solutions gives

4 + (-1) = 3

.

Step-by-Step Solution

1

Substitute

y = x^2 - 3x

into the equation to simplify the rational terms.

\frac{y}{y+2} + \frac{y-1}{y-3} = \frac{11}{3}

Using a temporary variable simplifies the algebraic manipulation of the rational expressions.

2

Multiply the entire equation by the least common denominator,

3(y+2)(y-3)

, to eliminate all fractions.

3y(y-3) + 3(y-1)(y+2) = 11(y+2)(y-3)

, where

y \neq -2

and

y \neq 3

This clears the denominators so the equation can be solved as a polynomial.

3

Expand the terms on both sides of the equation.

(3y^2 - 9y) + (3y^2 + 3y - 6) = 11y^2 - 11y - 66

Expanding allows for combining like terms.

4

Combine like terms and move all terms to one side of the equation to set it equal to zero.

5y^2 - 5y - 60 = 0

This sets up the expression in the standard quadratic form

ay^2 + by + c = 0

.

5

Divide the entire quadratic equation by its greatest common factor,

5

, and factor the resulting expression.

y^2 - y - 12 = 0 \implies (y-4)(y+3) = 0 \implies y = 4 \text{ or } y = -3

Factoring solves for the possible values of the substituted variable

y

.

6

Substitute back

y = x^2 - 3x

for each case and solve the resulting quadratic equations for

x

.

For

y = 4

:

x^2 - 3x - 4 = 0 \implies (x-4)(x+1) = 0 \implies x = 4 \text{ or } x = -1

.
For

y = -3

:

x^2 - 3x + 3 = 0

. The discriminant is

(-3)^2 - 4(1)(3) = -3 < 0

, which means there are no real solutions.

This determines the real values of

x

that solve the original equation.

7

Verify that neither solution makes the original denominators zero, and add the valid real solutions.

4 + (-1) = 3

Since both

x = 4

and

x = -1

result in non-zero denominators, both are valid real solutions. Their sum is

3

.

Key Concept

Solving rational equations by utilizing algebraic substitution to reduce complexity and analyzing quadratic equations for real solutions.

Question 32Question →

For the imaginary unit $i$ , where $i^2 = -1$ , the complex number $z$ is defined by:

z = \frac{1 - 3i^{45}}{1 + i^{95}}

Which of the following is equivalent to $z^2$ ?

-3 + 4i

5 - 4i

3 - 4i

3

5

Show answer & explanation

Answer: 3 - 4i

Answer

3 - 4i

The correct answer is found by first simplifying the powers of

i

in the expression for

z

:

i^{45} = i

and

i^{95} = -i

, which yields

z = \frac{1 - 3i}{1 - i}

. Rationalizing the fraction by multiplying both the numerator and the denominator by the conjugate of the denominator (

1 + i

) simplifies the expression to

z = 2 - i

. Finally, squaring this result using the binomial expansion formula gives

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

.

Step-by-Step Solution

1

Simplify the high integer powers of the imaginary unit

i

by using the fact that powers of

i

repeat in a cycle of four:

i^1 = i

,

i^2 = -1

,

i^3 = -i

, and

i^4 = 1

.

i^{45} = i

and

i^{95} = -i

Since

45 = 4(11) + 1

, the remainder is

1

, so

i^{45} = i^1 = i

. Since

95 = 4(23) + 3

, the remainder is

3

, so

i^{95} = i^3 = -i

.

2

Substitute these simplified values back into the expression for

z

.

z = \frac{1 - 3i}{1 - i}

This sets up the fraction with simplified imaginary terms in both the numerator and the denominator.

3

Rationalize the denominator by multiplying the numerator and denominator of the fraction by the complex conjugate of the denominator, which is

1 + i

.

z = 2 - i

Multiplying by the conjugate eliminates the imaginary unit from the denominator:

\frac{(1 - 3i)(1 + i)}{(1 - i)(1 + i)} = \frac{1 + i - 3i - 3i^2}{1 - i^2} = \frac{4 - 2i}{2} = 2 - i

.

4

Calculate the value of

z^2

by squaring the simplified complex number

2 - i

.

z^2 = 3 - 4i

Using the binomial squaring formula

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

, we expand

(2 - i)^2

to get

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

.

Key Concept

Simplifying complex numbers by evaluating powers of

i

, rationalizing fractions with complex conjugates, and expanding complex binomials.

Estimated Time:1m 30s

Question 33Question →

Let $i$ be the imaginary unit such that $i^2 = -1$ . What is the simplified form of the expression $(1 + 2i)^2(3 - i)$ ?

-9 + 3i

-5 + 15i

19 + 7i

-13 + 15i

50i

Show answer & explanation

Answer: $-5 + 15i$

Answer

-5 + 15i

The correct answer is

-5 + 15i

. We first expand the squared binomial

(1 + 2i)^2

, which results in

1 + 4i + 4i^2

. Substituting

i^2 = -1

yields

-3 + 4i

. We then multiply this by the second binomial

(3 - i)

to get

-9 + 3i + 12i - 4i^2

. Substituting

i^2 = -1

one more time and combining like terms leads to the final simplified result of

-5 + 15i

.

Step-by-Step Solution

1

Expand the squared binomial

(1 + 2i)^2

1 + 4i + 4i^2

Before multiplying by the second binomial, we must apply the exponent to the first binomial according to the order of operations.

2

Substitute

i^2 = -1

to simplify the expression from Step 1

-3 + 4i

Since

i^2 = -1

, the term

4i^2

becomes

4(-1) = -4

, and combining the real parts gives

1 - 4 = -3

.

3

Multiply the simplified term by

(3 - i)

using the FOIL method

-9 + 3i + 12i - 4i^2

We distribute each term of the first binomial into the second binomial:

(-3)(3) = -9

,

(-3)(-i) = 3i

,

(4i)(3) = 12i

, and

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

.

4

Simplify the resulting expression by combining like terms and substituting

i^2 = -1

-5 + 15i

Combining the imaginary parts gives

3i + 12i = 15i

. Substituting

i^2 = -1

into

-4i^2

gives

-4(-1) = +4

. Finally, combining the real parts yields

-9 + 4 = -5

.

Key Concept

Simplification of complex expressions involving binomial squaring and multiplication under the definition

i^2 = -1

.

Estimated Time:1m 30s

Question 34Question →

If $k$ is a positive real number such that $\sqrt{k} + \sqrt{4k} = 15$ , what is the value of $k$ ?

3

5

9

25

45

Show answer & explanation

Answer: 25

Answer

25

The correct answer is

25

. To find this, we simplify

\sqrt{4k}

to

2\sqrt{k}

. Substituting this into the equation gives

\sqrt{k} + 2\sqrt{k} = 15

. Combining like terms yields

3\sqrt{k} = 15

. Dividing both sides by

3

gives

\sqrt{k} = 5

. Squaring both sides of the equation results in

k = 25

. Checking the solution:

\sqrt{25} + \sqrt{4(25)} = 5 + 10 = 15

, which is true.

Step-by-Step Solution

1

Simplify the radical term

\sqrt{4k}

using the product property of radicals.

\sqrt{4k} = \sqrt{4} \cdot \sqrt{k} = 2\sqrt{k}

This allows us to express both terms in the equation using the same radical base,

\sqrt{k}

.

2

Substitute the simplified term back into the equation and combine like terms.

\sqrt{k} + 2\sqrt{k} = 15 \implies 3\sqrt{k} = 15

Combining like radical terms simplifies the equation to a single radical term.

3

Isolate the radical by dividing both sides of the equation by

3

.

\sqrt{k} = 5

Isolating the radical term is necessary before squaring both sides to solve for the variable.

4

Square both sides of the equation to solve for

k

.

k = 25

Squaring is the inverse operation of taking the square root, which isolates

k

.

Key Concept

Solving radical equations by simplifying and combining like radical terms

Question 35Question →

For the imaginary unit $i$ , where $i^2 = -1$ , what is the real part of the complex number $z = (2 - i)^3 + 9i$ ?

Show answer & explanation

Answer: 2

Answer

The real part of the complex number is 2.

Expanding the expression

(2 - i)^3

yields

2 - 11i

. Adding

9i

gives

2 - 2i

. The real part of this complex number is the term without

i

, which is 2.

Step-by-Step Solution

1

Expand the squared binomial

(2 - i)^2

.

3 - 4i

Apply the identity

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

and substitute

i^2 = -1

.

2

Multiply the result of the square by

(2 - i)

to calculate

(2 - i)^3

.

2 - 11i

Distribute the terms

(3 - 4i)(2 - i) = 6 - 3i - 8i + 4i^2

and substitute

i^2 = -1

.

3

Add

9i

to the simplified cube to find the complex number

z

.

2 - 2i

Combine the imaginary components:

-11i + 9i = -2i

.

4

Identify the real part of

z

.

2

The real part of a complex number

a + bi

is

a

.

Key Concept

Expanding complex binomials and simplifying powers of the imaginary unit

Question 36Question →

For all real numbers $y$ that satisfy the inequality $||y - 3| - 6| \leq 2$ , what is the sum of all possible integer values of $y$ ?

-15

30

45

51

65

Show answer & explanation

Answer: $30$

Answer

The sum of all possible integer values of

y

is

30

.

The compound inequality

4 \leq |y - 3| \leq 8

splits into two parts:

|y - 3| \leq 8

(which gives

-5 \leq y \leq 11

) and

|y - 3| \geq 4

(which gives

y \leq -1

or

y \geq 7

). The intersection of these intervals is

[-5, -1] \cup [7, 11]

. Summing all the integers in these intervals gives

(-5 + -4 + -3 + -2 + -1) + (7 + 8 + 9 + 10 + 11) = -15 + 45 = 30

.

Step-by-Step Solution

1

Set up the compound inequality representing the outer absolute value.

-2 \leq |y - 3| - 6 \leq 2

By definition,

|u| \leq a

is equivalent to

-a \leq u \leq a

for

a \geq 0

.

2

Isolate the inner absolute value term by adding

6

to all parts of the inequality.

4 \leq |y - 3| \leq 8

Isolating the absolute value allows us to split the compound inequality into two separate cases.

3

Split the compound inequality into two separate inequalities and solve each one.

|y - 3| \leq 8

and

|y - 3| \geq 4

The expression

|y - 3|

must be simultaneously less than or equal to

8

and greater than or equal to

4

.

4

Solve the first inequality,

|y - 3| \leq 8

.

-5 \leq y \leq 11

Rewriting the inequality gives

-8 \leq y - 3 \leq 8

, and adding

3

to all parts yields the interval

[-5, 11]

.

5

Solve the second inequality,

|y - 3| \geq 4

.

y \leq -1

or

y \geq 7

By definition,

|u| \geq a

is equivalent to

u \geq a

or

u \leq -a

. Thus,

y - 3 \geq 4 \implies y \geq 7

, and

y - 3 \leq -4 \implies y \leq -1

.

6

Determine the intersection of the two solution sets.

y \in [-5, -1] \cup [7, 11]

The values of

y

must lie within

[-5, 11]

and also satisfy

y \leq -1

or

y \geq 7

.

7

Identify the integers within the final intervals and calculate their sum.

Sum =

30

The integers in

[-5, -1]

are

-5, -4, -3, -2, -1

(sum =

-15

). The integers in

[7, 11]

are

7, 8, 9, 10, 11

(sum =

45

). The total sum is

-15 + 45 = 30

.

Key Concept

Solving compound and nested absolute value inequalities

Alternative Method

Instead of solving the inequality algebraically, one can test the integer values around the critical points. Since

y

must satisfy

4 \leq |y - 3| \leq 8

, we can see that the distance of

y

from

3

must be between

4

and

8

units. The integers at distance

4, 5, 6, 7, 8

to the right of

3

are

7, 8, 9, 10, 11

. The integers at distance

4, 5, 6, 7, 8

to the left of

3

are

-1, -2, -3, -4, -5

. Summing these ten integers yields

30

.

Estimated Time:3m 0s

Question 37Question →

When solving the radical equation $\sqrt{2x^2 - 4x - 6} = x - 3$ for all real values of $x$ , one of the solutions obtained from the squared equation is extraneous. What is the value of this extraneous solution?

-5

3

1

-1/2

5

Show answer & explanation

Answer: -5

Answer

The value of the extraneous solution is

-5

.

The correct answer is

-5

. Squaring both sides of the equation

\sqrt{2x^2 - 4x - 6} = x - 3

yields

2x^2 - 4x - 6 = x^2 - 6x + 9

, which simplifies to

x^2 + 2x - 15 = 0

. Factoring this quadratic gives

(x + 5)(x - 3) = 0

, leading to potential solutions of

3

and

-5

. Substituting

-5

back into the original equation results in the left side simplifying to

8

and the right side simplifying to

-8

. Since

8 \neq -8

, the value

-5

is an extraneous solution.

Step-by-Step Solution

1

Square both sides of the original radical equation to eliminate the square root.

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

Squaring both sides removes the radical on the left side.

2

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

x^2 + 2x - 15 = 0

Expanding

(x - 3)^2

gives

x^2 - 6x + 9

. Subtracting this expression from both sides yields the simplified quadratic equation.

3

Factor the quadratic equation to find the potential solutions.

(x + 5)(x - 3) = 0

, which gives

x = -5

and

x = 3

.

The factors of

-15

that add up to

2

are

5

and

-3

.

4

Substitute each potential solution back into the original equation to check for extraneous roots.

For

x = 3

,

\sqrt{2(3)^2 - 4(3) - 6} = 3 - 3

simplifies to

0 = 0

(valid). For

x = -5

,

\sqrt{2(-5)^2 - 4(-5) - 6} = -5 - 3

simplifies to

8 = -8

(invalid).

Extraneous solutions satisfy the squared equation but do not satisfy the original radical equation due to the sign difference introduced by squaring.

Key Concept

Solving radical equations and identifying extraneous solutions

Estimated Time:2m 0s

Question 38Question →

Given that $i = \sqrt{-1}$ , what is the value of the expression $i^{83}$ ?

i

-1

-i

1

0

Show answer & explanation

Answer: $-i$

Answer

The value of the expression is

-i

.

The expression

i^{83}

can be simplified by dividing the exponent 83 by 4. Because the remainder is 3, the expression is equivalent to

i^3

. Since

i^3 = i^2 \cdot i = (-1) \cdot i = -i

, the correct value is

-i

.

Step-by-Step Solution

1

Divide the exponent 83 by 4 to determine the remainder.

The quotient is 20 with a remainder of 3, which means

83 = 4 \times 20 + 3

.

Powers of the imaginary unit

i

repeat in a cycle of four:

i^1 = i

,

i^2 = -1

,

i^3 = -i

, and

i^4 = 1

.

2

Rewrite the expression using the rules of exponents and substitute the values of

i^4

and

i^3

.

i^{83} = i^{4(20) + 3} = (i^4)^{20} \cdot i^3 = (1)^{20} \cdot (-i) = -i

.

Since

i^4 = 1

, any power of

i

that is a multiple of 4 simplifies to 1, leaving only the remainder power to determine the final value.

Key Concept

Simplifying powers of the imaginary unit

i

Question 39Question →

Determine the value of $x$ for which the given rational expression is undefined.

Fill in the blanks below

The rational expression

\frac{x + 7}{2x - 10}

is undefined when

x =

.

Show answer & explanation

Answer

5

The rational expression is undefined when the denominator,

2x - 10

, equals

0

. Solving

2x - 10 = 0

gives

2x = 10

, which simplifies to

x = 5

.

Step-by-Step Solution

1

Identify the denominator of the rational expression and set it equal to zero.

2x - 10 = 0

A rational expression is undefined when its denominator is equal to zero, as division by zero is not defined in the real number system.

2

Solve the linear equation for

x

by adding 10 to both sides and then dividing by 2.

2x = 10 \Rightarrow x = 5

Isolating

x

yields the value that makes the denominator zero.

Key Concept

Rational expressions are undefined when the denominator is equal to zero.

Estimated Time:45s

Question 40Question →

Let the functions $f$ and $g$ be defined by $f(x) = |2x - 5|$ and $g(x) = \sqrt{x + 7}$ for all real numbers in their respective domains. If $a$ is a real number such that $(f \circ g)(a) = 3$ , what is the sum of all possible values of $a$ ?

Show answer & explanation

Answer: 3

Answer

The sum of all possible values of

a

is

3

.

The correct answer is

3

. The composition

(f \circ g)(a) = 3

translates to

f(g(a)) = |2\sqrt{a + 7} - 5| = 3

. This absolute value relation splits into two equations:

2\sqrt{a + 7} - 5 = 3

and

2\sqrt{a + 7} - 5 = -3

. Solving the first equation yields

\sqrt{a + 7} = 4 \Rightarrow a = 9

. Solving the second equation yields

\sqrt{a + 7} = 1 \Rightarrow a = -6

. Since both

9

and

-6

are greater than or equal to

-7

, they are within the domain of the radical function. The sum of these values is

9 + (-6) = 3

.

Step-by-Step Solution

1

Express the composition

(f \circ g)(a)

in terms of

a

.

|2\sqrt{a + 7} - 5| = 3

By definition of function composition,

(f \circ g)(a) = f(g(a))

. Substituting

g(a) = \sqrt{a + 7}

into the expression for

f(x)

yields

f(g(a)) = |2g(a) - 5| = |2\sqrt{a + 7} - 5|

.

2

Set up the two algebraic cases to eliminate the absolute value.

2\sqrt{a + 7} - 5 = 3

or

2\sqrt{a + 7} - 5 = -3

An absolute value equation of the form

|u| = c

(where

c \geq 0

) has two possible cases:

u = c

or

u = -c

.

3

Solve the first equation case for

a

.

a = 9

Adding

5

to both sides of

2\sqrt{a + 7} - 5 = 3

gives

2\sqrt{a + 7} = 8

. Dividing by

2

gives

\sqrt{a + 7} = 4

. Squaring both sides yields

a + 7 = 16

, which gives

a = 9

.

4

Solve the second equation case for

a

.

a = -6

Adding

5

to both sides of

2\sqrt{a + 7} - 5 = -3

gives

2\sqrt{a + 7} = 2

. Dividing by

2

gives

\sqrt{a + 7} = 1

. Squaring both sides yields

a + 7 = 1

, which gives

a = -6

.

5

Verify domain constraints and sum the valid solutions.

3

Both

a = 9

and

a = -6

satisfy the domain requirement for

g(x) = \sqrt{x+7}

, which is

x \geq -7

. The sum of these two valid values is

9 + (-6) = 3

.

Key Concept

Evaluating and solving equations containing composite functions, absolute values, and radical functions.

Alternative Method

Instead of expanding the composition immediately, substitute a temporary variable

u = g(a) = \sqrt{a+7}

. The equation becomes

f(u) = 3 \Rightarrow |2u - 5| = 3

. Solve this simplified absolute value equation to get

2u - 5 = 3 \Rightarrow u = 4

, and

2u - 5 = -3 \Rightarrow u = 1

. Next, substitute back

\sqrt{a+7}

for

u

: solving

\sqrt{a+7} = 4

yields

a = 9

, and solving

\sqrt{a+7} = 1

yields

a = -6

. Summing these two solutions gives

9 + (-6) = 3

.

Estimated Time:2m 0s

Intermediate Algebra

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer