# Lesson 19

### Problem 1

1. The quadratic equation $$x^2 + 7x + 10 = 0$$ is in the form of $$ax^2 + bx + c = 0$$. What are the values of $$a$$, $$b$$, and $$c$$?
2. Some steps for solving the equation by completing the square have been started here. In the third line, what might be a good reason for multiplying each side of the equation by 4?

\displaystyle \begin {align}\\ x^2 + 7x + 10 &= 0 &\hspace{0.1in}& \text {Original equation}\\\\ x^2 + 7x &= \text-10 &\hspace{0.1in}& \text {Subtract 10 from each side}\\\\ 4x^2 + 4(7x) &= 4(\text-10) &\hspace{0.1in}& \text {Multiply each side by 4}\\\\ (2x)^2 + 2(7)2x + \underline{\hspace{0.3in}}^2 &= \underline{\hspace{0.3in}}^2 - 4(10) &\hspace{0.1in}& \text {Rewrite } 4x^2 \text{ as } (2x)^2\\ &\text{} &\hspace{0.1in}& \text{and }4(7x) \text{ as } 2(7)2x\\\\ (2x+\underline{\hspace{0.3in}})^2 &= \underline{\hspace{0.3in}}^2 - 4(10)\\\\ 2x+\underline{\hspace{0.3in}} &= \pm \sqrt { \underline{\hspace{0.3in}}^2 - 4(10)}\\\\ 2x &= \underline{\hspace{0.3in}} \pm \sqrt { \underline{\hspace{0.3in}}^2 - 4(10)}\\\\ x &=\\ \end {align}

3. Complete the unfinished steps, and explain what happens in each step in the second half of the solution.
4. Substitute the values of $$a$$, $$b$$, and $$c$$ into the quadratic formula, $$\displaystyle x = {\text-b \pm \sqrt{b^2-4ac} \over 2a}$$, but do not evaluate any of the expressions. Explain how this expression is related to solving $$x^2+7x+10=0$$ by completing the square.

### Solution

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### Problem 2

Consider the equation $$x^2-39=0$$.

1. Does the quadratic formula work to solve this equation? Explain or show how you know.
2. Can you solve this equation using square roots? Explain or show how you know.

### Solution

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### Problem 3

Clare is deriving the quadratic formula by solving $$ax^2+bx+c=0$$ by completing the square.

She arrived at this equation.

$$(2ax+b)^2=b^2-4ac$$

Briefly describe what she needs to do to finish solving for $$x$$ and then show the steps.

### Solution

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### Problem 4

Tyler is solving the quadratic equation $$x^2 + 8x +11=4$$

Study his work and explain the mistake he made. Then, solve the equation correctly.

\displaystyle \begin{align} x^2 + 8x+11&= 4\\ x^2+8x+16&=4\\(x + 4)^2 &= 4\\ x = \text-8 \quad &\text { or } \quad x = 0\\ \end{align}\\

### Solution

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(From Unit 7, Lesson 12.)

### Problem 5

Solve the equation by using the quadratic formula. Then, check if your solutions are correct by rewriting the quadratic expression in factored form and using the zero product property.

1. $$2x^2-3x-5=0$$
2. $$x^2-4x=21$$
3. $$3-x-4x^2=0$$

### Solution

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(From Unit 7, Lesson 16.)

### Problem 6

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 $$6 = 4 + 12t - 16t^2$$ without graphing. Show your reasoning.
2. What do the solutions say about the tennis ball?

### Solution

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(From Unit 7, Lesson 17.)

### Problem 7

Consider the equation $$y=2x(6-x)$$.

1. What are the $$x$$-intercepts of the graph of this equation? Explain how you know.
2. What is the $$x$$-coordinate of the vertex of the graph of this equation? Explain how you know.
3. What is the $$y$$-coordinate of the vertex? Show your reasoning.
4. Sketch the graph of this equation.

### Solution

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(From Unit 6, Lesson 11.)