Question 1

A diver jumps from a 10-meter high board. Her height is modeled by $$y = -4.9x^2$ + 2.5x + 10$$, where $$x is $$time in seconds. Using the Quadratic Formula, after how many seconds does she enter the water? (Round to the nearest tenth.)

Accepted Answer

$$x \approx 2.1$$ seconds

Question 2

Given the quadratic equation $$3x^2 - 5x - 1 = 0$, what is the value of the discriminant, and what does it tell you about the roots?

Accepted Answer

Discriminant is $$19$$; two real irrational roots

Question 3

Which of the following quadratic equations has exactly one real root?

Accepted Answer

$$x^2 - 8x + 16 = 0$

Question 4

A football is thrown and its height is modeled by $$h = -16t^2$ + 45t + 4. $$How long after being thrown will the football hit the ground? (Round to the nearest tenth.)

Accepted Answer

$$t \approx 2.9$$ seconds

Question 5

For the equation $$5x^2 + 8x + 11 = 0$, what are the roots?

Accepted Answer

$$x = \dfrac{-4 \pm i\sqrt{39}}{5}$$

Question 6

Which statement about the discriminant is always true?

Accepted Answer

If $$b^2 - 4ac$$ \u003c 0$$, the equation has two complex roots.

Question 7

A rectangular box has a square base of side $$x$$ inches and height $$h = 3x + 1. If $$the sides of the base are increased by 2 inches and the height by 3 inches, the volume increases by $$531 in$^3. $$Which equation models this situation?

Accepted Answer

$$(x+2)^2 (3x+4$$) - $$x^2 (3x+1) = 531$

Question 8

Which of the following equations has two real, irrational roots?

Accepted Answer

$$2x^2 - 10x + 7 = 0$

Question 9

If the discriminant of a quadratic equation is $$0$$, what can you conclude about its roots?

Accepted Answer

There is one real, rational root.

Question 10

Solve $$x^2 + 2x - 35 = 0$ using the Quadratic Formula.

Accepted Answer

$$x = -1 \pm \sqrt{36}$$,\ x = 5,\ x = -7$

Question 11

Which of the following best describes the roots of $$x^2 + 25 = 0$?

Accepted Answer

Two complex roots

Question 12

A ball is fired up a $$20-$$foot vertical chute. Its height is modeled by $$h(t) = -16t^2$ + vt. $$What is the minimum initial velocity $$v (in ft/s) $$required for the ball to reach the target at the top?

Accepted Answer

$$v = 40$$

Question 13

For which value of $$m$$ does the equation $$x^2 + x + m + 1 = 0$ have exactly one solution?

Accepted Answer

$$m = -\dfrac{1}{4}$$

Question 14

Which of the following statements is correct regarding the equation $$x^2 - 6x + 21 = 0$?

Accepted Answer

It has two complex roots.

Question 15

Given $$x^2 - 11x - 26 = 0$, what is the value of the discriminant and what does it indicate?

Accepted Answer

Discriminant is $$165$$; two real, irrational roots

Question 16

Which of the following is the correct application of the Quadratic Formula to solve $$4x^2 - 6x + 2 = 0$?

Accepted Answer

$$x = \dfrac{6 \pm \sqrt{36 - 32}}{8}$$,\ x = \dfrac{6 \pm 2}{8}$

Question 17

If $$a$$ and $$c in$$ $$ax^2 + bx + c = 0$ have different signs, what can you conclude about the roots?

Accepted Answer

The equation always has two real roots.

Question 18

Which of the following equations has a discriminant equal to zero?

Accepted Answer

$$x^2 - 8x + 16 = 0$

Question 19

Solve $$2x^2 - x - 15 = 0$ using the Quadratic Formula.

Accepted Answer

$$x = \dfrac{1 \pm \sqrt{121}}{4}$$

Question 20

Which of the following best describes the roots of $$x^2 + 8x + 13 = 0$?

Accepted Answer

Two real, irrational roots

Quadratic Equations: The Quadratic Formula and the Discriminant

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