# Lesson 17

Applying the Quadratic Formula (Part 1)

### Problem 1

Select all the equations that have 2 solutions.

A:

$$(x + 3)^2 = 9$$

B:

$$(x - 5)^2 = \text- 5$$

C:

$$(x + 2)^2-6 = 0$$

D:

$$(x - 9)^2+25 = 0$$

E:

$$(x + 10)^2 = 1$$

F:

$$(x - 8)^2 = 0$$

G:

$$5=(x+1)(x+1)$$

### Problem 2

A frog jumps in the air. The height, in inches, of the frog is modeled by the function $$h(t) = 60t-75t^2$$, where $$t$$ is the time after it jumped, measured in seconds.

Solve $$60t - 75t^2 = 0$$. What do the solutions tell us about the jumping frog?

### Problem 3

A tennis ball is hit straight up in the air, and its height, in feet above the ground, is modeled by the equation $$f(t) = 4 + 12t - 16t^2$$, where $$t$$ is measured in seconds since the ball was thrown.

1. Find the solutions to the equation $$0 = 4 + 12t - 16t^2$$.
2. What do the solutions tell us about the tennis ball?

### Problem 4

Rewrite each quadratic expression in standard form.

1. $$(x+1)(7x+2)$$
2. $$(8x+1)(x-5)$$
3. $$(2x+1)(2x-1)$$
4. $$(4+x)(3x-2)$$

### Solution

(From Unit 7, Lesson 10.)

### Problem 5

Find the missing expression in parentheses so that each pair of quadratic expressions is equivalent. Show that your expression meets this requirement.

1. $$(4x-1)(\underline{\hspace{1in}})$$ and $$16x^2 -8x +1$$
2. $$(9x + 2)(\underline{\hspace{1in}})$$ and $$9x^2 -16x -4$$
3. $$(\underline{\hspace{1in}})(\text-x + 5)$$ and $$\text-7x^2 +36x-5$$

### Solution

(From Unit 7, Lesson 10.)

### Problem 6

The number of downloads of a song during a week is a function, $$f$$, of the number of weeks, $$w$$, since the song was released. The equation $$f(w) = 100,\!000 \boldcdot \left(\frac{9}{10}\right)^w$$ defines this function.

1. What does the number 100,000 tell you about the downloads? What about the $$\frac{9}{10}$$?
2. Is $$f(\text-1)$$ meaningful in this situation? Explain your reasoning.

### Solution

(From Unit 5, Lesson 9.)

### Problem 7

Consider the equation $$4x^2 - 4x -15 = 0$$.

1. Identify the values of $$a$$, $$b$$, and $$c$$ that you would substitute into the quadratic formula to solve the equation.
2. Evaluate each expression using the values of $$a$$, $$b$$, and $$c$$.

​​​​$$\text- b$$

$$b^2$$

$$4ac$$

​​​​​$$b^2 - 4ac$$

$$\sqrt{b^2 - 4ac}$$

$$\text- b \pm \sqrt{b^2 - 4ac}$$

$$2a$$

$$\dfrac{\text- b \pm \sqrt{b^2 - 4ac}}{2a}$$

3. The solutions to the equation are $$x=\text-\frac 32$$ and $$x=\frac52$$. Do these match the values of the last expression you evaluated in the previous question?

### Solution

(From Unit 7, Lesson 16.)

### Problem 8

1. Describe the graph of $$y=\text-x^2$$. (Does it open upward or downward? Where is its $$y$$-intercept? What about its $$x$$-intercepts?)
2. Without graphing, describe how adding $$16x$$ to $$\text-x^2$$ would change each feature of the graph of $$y = \text-x^2$$. (If you get stuck, consider writing the expression in factored form.)

1. the $$x$$-intercepts
2. the vertex
3. the $$y$$-intercept
4. the direction of opening of the U-shape graph