# Lesson 7

Rewriting Quadratic Expressions in Factored Form (Part 2)

### Problem 1

Find two numbers that...

1. multiply to -40 and add to -6.
2. multiply to -40 and add to 6.
3. multiply to -36 and add to 9.
4. multiply to -36 and add to -5.

If you get stuck, try listing all the factors of the first number.

### Problem 2

Create a diagram to show that $$(x-5)(x+8)$$ is equivalent to $$x^2+3x-40$$.

### Problem 3

Write a $$+$$ or a $$-$$ sign in each box so the expressions on each side of the equal sign are equivalent.

1. $$(x \; \boxed{\phantom{30}} \;18)(x \; \boxed{\phantom{30}} \; 3)=x^2-15x-54$$
2. $$(x \; \boxed{\phantom{30}} \; 18)(x \; \boxed{\phantom{30}} \; 3)=x^2+21x+54$$
3. $$(x \; \boxed{\phantom{30}} \; 18)(x \; \boxed{\phantom{30}} \; 3)=x^2+15x-54$$
4. $$(x \; \boxed{\phantom{30}} \; 18)(x \; \boxed{\phantom{30}} \; 3)=x^2-21x+54$$

### Problem 4

Match each quadratic expression in standard form with its equivalent expression in factored form.​​​​​​

### Problem 5

Rewrite each expression in factored form. If you get stuck, try drawing a diagram.

1. $$x^2 -3x-28$$
2. $$x^2 +3x-28$$
3. $$x^2 +12x-28$$
4. $$x^2 -28x-60$$

### Problem 6

Which equation has exactly one solution?

A:

$$x^2=\text-4$$

B:

$$(x+5)^2=0$$

C:

$$(x+5)(x-5)=0$$

D:

$$(x+5)^2=36$$

### Solution

(From Unit 7, Lesson 5.)

### Problem 7

The graph represents the height of a passenger car on a ferris wheel, in feet, as a function of time, in seconds since the ride starts.

1. Find $$H(0)$$.
2. Does $$H(t)=0$$ have a solution? Explain how you know.
3. Describe the domain of the function.
4. Describe the range of the function.

### Solution

(From Unit 4, Lesson 11.)

### Problem 8

Elena solves the equation $$x^2=7x$$ by dividing both sides by $$x$$ to get $$x=7$$. She says the solution is 7.

Lin solves the equation $$x^2=7x$$ by rewriting the equation to get $$x^2-7x=0$$. When she graphs the equation $$y=x^2-7x$$, the $$x$$-intercepts are $$(0,0)$$ and $$(7,0)$$. She says the solutions are 0 and 7.

Do you agree with either of them? Explain or show how you know.

### Solution

(From Unit 7, Lesson 5.)

### Problem 9

A bacteria population, $$p$$, can be represented by the equation $$p = 100,\!000 \boldcdot \left(\frac{1}{4} \right)^d$$, where $$d$$ is the number of days since it was measured.

1. What was the population 3 days before it was measured? Explain how you know.
2. What is the last day when the population was more than 1,000,000? Explain how you know.