Lesson 7

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.

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

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

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

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

Which equation has exactly one solution?

\(x^2=\text-4\)

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

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

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

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.

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.

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.