Q: Given the quadratic equation $$2x^2 - 4x + 1 = 0$, what is the value of the discriminant?

$$D = 4 - 4 \times 2 \times 1 = -4$$

Question 1

Which of the following is the definition of a linear equation in one variable?

Accepted Answer

$$ax + b = 0$$ where $$a$$ and $$b$$ are real numbers

Question 2

Solve the linear equation $$3x - 9 = -24.$$

Accepted Answer

$$x = -11$$

Question 3

Which of the following is the standard form of a quadratic equation?

Accepted Answer

$$ax^2 + bx + c = 0$

Question 4

What is the solution to the quadratic equation $$x^2 - 5x = 0$ by factoring?

Accepted Answer

$$x = 0 or$$ $$x = 5$$

Question 5

Given the quadratic equation $$2x^2 - 4x + 1 = 0$, what is the value of the discriminant?

Accepted Answer

$$D = 4 - 4 	imes 2 	imes 1 = -4$$

Question 6

If the discriminant of a quadratic equation is negative, what can you conclude about its solutions?

Accepted Answer

There are no real solutions; the solutions are complex conjugates.

Question 7

Solve the radical equation $$\sqrt{x} = 3.$$

Accepted Answer

$$x = 9$$

Question 8

What is the principal square root of $$-49 in $$the complex number system?

Accepted Answer

$$7i$$

Question 9

Simplify the expression $$(4 + 7i) + (2 + 7i).$$

Accepted Answer

$$6 + 14i$$

Question 10

What is the product of $$(6 - 2i)(3 + i)$$?

Accepted Answer

$$18 - 6i + 3i - 2i^2$ = 18 - 3i + 2 ($$since $$i^2 = -1$), so = 20 - 3i$

Question 11

If $$i is $$the imaginary unit, what is the value of $$i^{23}$$?

Accepted Answer

$$i$$

Question 12

Which of the following statements about complex conjugates is true?

Accepted Answer

The product of a complex number and its conjugate is always a real number.

Question 13

Solve the equation $$x^2 + 8 = 0$ in the complex number system.

Accepted Answer

$$x = \pm 2\sqrt{2}i$$

Question 14

A total of $$15,000 is to be divided between Jon and Wendy, with Jon to receive 2$$,500 less than Wendy. How much will Jon receive?

Accepted Answer

$$7,250$$

Question 15

Which of the following is a necessary step when solving a radical equation such as $$\sqrt{x + 5} = 3$$?

Accepted Answer

Isolate the radical and then square both sides.

Question 16

What is the solution to the equation $$x^2 - 4x + 4 = 0$?

Accepted Answer

$$x = 2 ($$double root)

Question 17

If $$x = 3 + 5i$$ and $$w = -2 - 4i$$, what is $$x + w in $$standard form?

Accepted Answer

$$1 + i$$

Question 18

Which of the following is the correct quadratic formula for solving $$ax^2 + bx + c = 0$?

Accepted Answer

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

Question 19

A square piece of cardboard has a side of 3 cm. If squares of side $$x$$ are cut from each corner and the edges are turned up to form a box with a volume of $$147 cm$^3$$, what equation models the situation?

Accepted Answer

$$(3 - 2x)^2$ x = 147$$

College Algebra: Linear, Quadratic, Complex, and Radical Equations Study Guide

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