Question 1

Which of the following is the correct first step when solving $$9x^2 - 0x - 4 = 0$ by completing the square?

Accepted Answer

Divide both sides by $$9 to $$make the coefficient of $$x^2$$ equal to $$1.$$

Question 2

Given the equation $$x^2 - x$$ = \frac{1}{4}$$, what value should be added to both sides to complete the square?

Accepted Answer

$$\left(\frac{1}{2}\$$right)^2$$ = \frac{1}{4}$$

Question 3

What are the real solutions to $$x^2 - 6x + 4 = 0$?

Accepted Answer

Question 4

When solving $$x^3$$ + $$x^2 - 4x - 4 = 0$ by factoring by grouping, which of the following is a correct factorization?

Accepted Answer

$$(x+1)(x^2-4) = 0$

Question 5

For the equation $$\sqrt{2x+1} + 2 = x$$, what is the first step to isolate the radical?

Accepted Answer

Subtract $$2$$ from both sides to get $$\sqrt{2x+1} = x-2$$

Question 6

What is the solution to $$x^{3/4} - 2 = 0$?

Accepted Answer

$$x = 16$$

Question 7

If $$x^{2/3} - 8 = -4$, what are the real solutions for x$?

Accepted Answer

$$x = \pm 8$$

Question 8

Given $$x^4$$ - $$8x^2 - 9 = 0$, what substitution simplifies the equation for factoring?

Accepted Answer

Let $$v = x^2$

Question 9

What are all the solutions to $$x^4$$ - $$8x^2 - 9 = 0$?

Accepted Answer

$$x = \pm 3, \pm i$$

Question 10

When completing the square for $$x^2$$ - \frac{2}{3}x = -\frac{4}{9}$$, what is the value added to both sides?

Accepted Answer

$$\frac{1}{9}$$

Question 11

Which of the following equations has only complex (non-real) solutions?

Accepted Answer

$$x^2 + 1 = 0$

Question 12

If you want real solutions for $$\sqrt{2x+1} = x-2$$, what must be true about $$x$$?

Accepted Answer

$$x-2 \geq 0$$ and $$2x+1 \geq 0$$

Question 13

What is the solution to $$3x^4 - 27x = 0$?

Accepted Answer

$$x = 0, 3, -3$$

Question 14

Which step is necessary when completing the square for $$ax^2 + bx + c = 0$ if a$$ 
eq 1$$?

Accepted Answer

Divide both sides by $$a to $$make the coefficient of $$x^2$$ equal to $$1.$$

Question 15

What is the solution to $$2x^2 - 7x + 3 = 0$?

Accepted Answer

$$x = \frac{7 \pm \sqrt{25}}{4}$$

Question 16

If $$x^2 - x$$ = \frac{3}{8}$$, what is the completed square form?

Accepted Answer

$$(x - \frac{1}{2})^2$ = \frac{7}{8}$$

Question 17

What is the solution to $$x^2 - x$$ = \frac{3}{8}$$?

Accepted Answer

$$x = \frac{1}{2} \pm \frac{\sqrt{7}}{\sqrt{8}}$$

Question 18

Which of the following is a necessary condition for a quadratic equation to have real solutions when completing the square?

Accepted Answer

The value under the square root must be non-negative.

Question 19

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

Accepted Answer

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

Solving Quadratic and Higher-Degree Equations: Methods and Examples

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