Question 1

Which of the following is the correct definition of the imaginary unit $$i$$?

Accepted Answer

$$i = \sqrt{-1}$$

Question 2

Simplify the expression $$(-5 + i) - (-11 - 6i)$$ and write your answer in standard form $$a + bi.$$

Accepted Answer

$$6 + 7i$$

Question 3

Multiply $$(4i)(3 - 5i)$$ and write your answer in standard form $$a + bi.$$

Accepted Answer

$$20i - 12$$

Question 4

Divide $$\dfrac{4i}{3 - 9i}$$ and write your answer in standard form $$a + bi.$$

Accepted Answer

$$-\dfrac{12}{90} + \dfrac{4}{90}i$$

Question 5

What is the principal square root of $$-25$$?

Accepted Answer

$$5i$$

Question 6

Solve the quadratic equation $$x^2 = 8x - 15$ by factoring.

Accepted Answer

$$x = 3 or$$ $$x = 5$$

Question 7

Which of the following best describes the solutions to the equation $$x^2 + 6x + 13 = 0$?

Accepted Answer

Two complex conjugate solutions

Question 8

Given the quadratic equation $$3x^2 + 12x + 3 = 0$, what is the value of the discriminant and what does it indicate about the solutions?

Accepted Answer

$$b^2 - 4ac = 108$, two real solutions

Question 9

Solve $$x^2 - 12x + 85 = 0$ using the quadratic formula.

Accepted Answer

$$x = 6 \pm 7$$

Question 10

A rectangular park is 6 miles long and 2 miles wide. How long is the diagonal route across the park? Express your answer in radical form and as a decimal to the nearest tenth.

Accepted Answer

$$\sqrt{40} \approx 6.3$$ miles

Question 11

Which of the following best describes the solutions to $$x^2 + 3x - 1 = 0$ based on the discriminant?

Accepted Answer

Two real solutions

Question 12

Solve the equation $$\sqrt{x + 1} = x - 1$$ and check for extraneous solutions.

Accepted Answer

$$x = 2$$

Question 13

Solve the absolute value equation $$|2x - 3| = 11.$$

Accepted Answer

$$x = 7 or$$ $$x = -4$$

Question 14

If the discriminant $$b^2 - 4ac$ of a quadratic equation is negative, what can you conclude about the solutions?

Accepted Answer

There are two complex conjugate solutions

Question 15

Solve the equation $$x^{2/3} = 27$.

Accepted Answer

$$x = 729$$

Question 16

Which of the following is the correct application of the Pythagorean Theorem for a right triangle with legs of length $$a$$ and $$b$$ and hypotenuse $$c$$?

Accepted Answer

$$a^2$$ + $$b^2$$ = $$c^2$$

Question 17

Solve the absolute value inequality $$|2x - 3| \u003c 5$$ and express the solution in interval notation.

Accepted Answer

$$(-1, 4)$$

Question 18

You have grades of 86%, 88%, 92%, and 84% on your first four exams. If the final exam counts as two grades, what is the minimum score you must earn on the final to have an average of at least 90%?

Accepted Answer

95%

Question 19

Which of the following is the correct standard form for a complex number?

Accepted Answer

$$a + bi$$

Complex Numbers, Quadratic Equations, and Linear Inequalities

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