Question 1

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 2

Given the quadratic equation $$3x^2 + 4x + 1 = 0$, what is the value of the discriminant?

Accepted Answer

$$b^2 - 4ac$$ = $$4^2 - 4$$ \cdot 3 \cdot 1 = 16 - 12 = 4$$

Question 3

For the equation $$2x^2 - 3x = -3$, what is the correct standard form before applying the quadratic formula?

Accepted Answer

$$2x^2 - 3x + 3 = 0$

Question 4

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

Accepted Answer

There are two imaginary solutions.

Question 5

A chemical reaction is modeled by the equation $$x^2 - 10x + 25 = 0$. Without solving, determine the number and type of solutions.

Accepted Answer

One real solution

Question 6

Which method is most appropriate for solving $$x^2 + 2x + 1 = 0$?

Accepted Answer

Factoring

Question 7

A projectile's height is modeled by $$2x^2 + 8x + 16 = 0$. What is the discriminant and what does it tell you about the solutions?

Accepted Answer

$$b^2 - 4ac$$ = $$8^2 - 4$$ \cdot 2 \cdot 16 = 64 - 128 = -64$$; two imaginary solutions

Question 8

Which of the following is NOT a step in using the quadratic formula?

Accepted Answer

Divide both sides by $$x$$

Question 9

Given $$x^2 + 2x - 3 = 0$, which of the following represents the correct solutions using the quadratic formula?

Accepted Answer

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

Question 10

A medical dosage calculation leads to the equation $$x^2 - 4x + 4 = 0$. What is the value of the discriminant and what does it indicate?

Accepted Answer

$$b^2 - 4ac$$ = $$(-4)^2 - 4$$ \cdot 1 \cdot 4 = 16 - 16 = 0$$; one real solution

Question 11

Which scenario would require you to use the quadratic formula rather than factoring?

Accepted Answer

When the quadratic does not factor easily

Question 12

For the equation $$4x^2 + 5x + 5 = 0$, what is the discriminant and what does it tell you about the solutions?

Accepted Answer

$$b^2 - 4ac$$ = $$5^2 - 4$$ \cdot 4 \cdot 5 = 25 - 80 = -55$$; two imaginary solutions

Question 13

A quadratic equation $$x^2 + 4x + 1 = 0$ models the path of a ball. What are the solutions using the quadratic formula?

Accepted Answer

$$x = \frac{-4 \pm \sqrt{16 - 4}}{2}$$

Question 14

If a quadratic equation has a discriminant of zero, which of the following is true?

Accepted Answer

There is one real solution

Question 15

Which of the following best describes the role of the discriminant in the quadratic formula?

Accepted Answer

It determines the number and type of solutions

Question 16

A quadratic equation $$x^2 - 6x + 9 = 0$ is used in a business profit model. What is the discriminant and what does it indicate?

Accepted Answer

$$b^2 - 4ac$$ = $$(-6)^2 - 4$$ \cdot 1 \cdot 9 = 36 - 36 = 0$$; one real solution

Question 17

Which of the following equations would have two real solutions?

Accepted Answer

$$x^2 - 5x + 6 = 0$

Question 18

A quadratic equation $$x^2 + x + 1 = 0$ is used in an engineering calculation. What is the discriminant and what does it indicate?

Accepted Answer

$$b^2 - 4ac$$ = $$1^2 - 4$$ \cdot 1 \cdot 1 = 1 - 4 = -3$$; two imaginary solutions

Quadratic Equations: Solving Methods and the Discriminant

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