Lesson 7

Rewriting Quadratic Expressions in Factored Form (Part 2)

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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.

Solution

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Problem 2

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

Solution

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

Solution

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Problem 4

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

A:

\(x^2 -2x-35\)

B:

\(x^2 +12x+35\)

C:

\(x^2 +2x-35\)

D:

\(x^2 -12x+35\)

1:

\((x+5)(x+7)\)

2:

\((x-5)(x-7)\)

3:

\((x+5)(x-7)\)

4:

\((x-5)(x+7)\)

Solution

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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. \(x^2 -28x-60\)

Solution

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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

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(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.

Use the graph to help you:

  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.
Function on coordinate plane.

Solution

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(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

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(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.

Solution

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(From Unit 5, Lesson 7.)