Lesson 8 The Wow Factor Solidify Understanding

Learning Focus

Use diagrams to factor trinomial expressions when the leading coefficient is not $1$ .

How can we factor $a x^{2} + b x + c$ when $a \neq 1$ ?

Open Up the Math: Launch, Explore, Discuss

Optima’s Quilts sometimes gets orders for blocks that are multiples of a given block. For instance, Optima got an order for a block that was exactly twice as big as the rectangular block that has a side that is $1$ inch longer than the basic size, $x$ , and one side that is $3$ inches longer than the basic size.

1.

An open area model of this block is shown here along with the two equivalent expressions for area. What do you notice? What do you wonder?

$A (x) = 2 (x^{2} + 4 x + 3) = 2 x^{2} + 8 x + 6 = 2 (x + 3) (x + 1)$

2.

Try using some of the observations to factor each of the following expressions:

a.

$3 x^{2} + 21 x + 36$

b.

$2 x^{2} + 2 x - 4$

c.

$3 x^{2} - 27$

d.

$5 x^{2} + 10 x$

Because she is a great business manager, Optima offers her customers lots of options. One option is to have rectangles that have side lengths that are more than one $x$ . An open version of one of these blocks is shown below:

$A (x) = 2 x^{2} + 7 x + 3 = (2 x + 1) (x + 3)$

3.

What do you notice about the diagram and the equation for area? What do you wonder?

4.

Use your observations to complete the diagram. Be sure to fill in the lengths of the side and the missing areas.

$\begin{matrix} A (x) = & \underset{―}{} & = & \underset{―}{} \\ (factored form) & (standard form) \end{matrix}$

Try a few on your own.

5.

$A (x) = 2 x^{2} + 7 x + 3 =$

6.

$2 x^{2} - 13 x + 15$

7.

$3 x^{2} + x - 10$

As she is working on the orders, one of the employees, Anushka, stops and says, “Hey, wait! I noticed on problem 5 that if I multiply the coefficients of the first and last term, I get $6$ . Then, the middle term comes from factors of $6$ that add up to $7$ . The two factors are $6$ and $1$ .

On problem 6, when I multiply $2$ times $15$ , I get $30$ . The middle term comes from factors of $30$ that add up to $- 13$ , which are $- 10$ and $- 3$ . This is crazy!”

8.

Does the pattern work for problem 7? Explain why or why not.

9.

If you think the pattern that Anushka noticed will help, try it to write $A (x) = 2 x^{2} + 9 x - 5$ in factored form. If you don’t think it works, use whatever is working for you.

10.

There’s one more twist on the kind of blocks that Optima makes. These are the trickiest of all, because they may have more than one $x$ in the length of both sides of the rectangle!

Here’s an example: Complete the diagram using the sides that are given, and write the two expressions for area.

$\begin{matrix} A (x) = & \underset{―}{} & = & \underset{―}{} \\ (factored form) & (standard form) \end{matrix}$

11.

Anushka has partially completed this diagram for you. The area of the block is:

$A (x) = 6 x^{2} + 7 x + 2$ . You do the rest.

$\begin{matrix} A (x) = & 6 x^{2} + 7 x + 2 & = & \underset{―}{} \\ (factored form) \end{matrix}$

12.

All right, let’s put it all together for some tricky ones! They may take a little messing around to get the factored expression to match the given expression. Check your answers to be sure that you’ve got them right. Factor each of the following.

a.

$10 x^{2} + 17 x + 3$

b.

$4 x^{2} - 8 x + 3$

c.

$4 x^{2} + 4 x - 3$

d.

$9 x^{2} - 9 x - 10$

Ready for More?

Challenge your partner by making your own trinomial expressions to factor. You’ll need to start with the factored form, multiply it out, and exchange it with your partner without the answers. Your partner will work the one that you wrote, and you will work the one they wrote. When you’re done, check your work and see who has the wow factor!

Takeaways

Factoring trinomials in the form $a x^{2} + b x + c$ when $a$ is greater than $1$ .

Example: $4 x^{2} + 6 x - 18$

Lesson Summary

In this lesson, we learned to factor trinomials in the form $a x^{2} + b x + c$ when $a \neq 1$ . Sometimes the terms have a common factor that can be factored out, leaving an expression that is much easier to work with. When there is not a common factor, diagrams can be used to help think about the number and sign combinations that work to make the factored expression equivalent to the trinomial.

Retrieval

For each given quadratic equation, state the vertex, the line of symmetry, the stretch, and whether the quadratic has a maximum or a minimum.

1.

$f (x) = 5 {(x - 9)}^{2} + 16$

Vertex:

Line of symmetry:

Stretch:

Maximum or Minimum:

2.

$f (x) = - x^{2} + 10 x - 18$

Vertex:

Line of symmetry:

Stretch:

Maximum or Minimum:

Given the $x$ -intercepts of a parabola, write the equation of the line of symmetry.

3.

$x$ -intercepts: $(- 12, 0)$ and $(12, 0)$

4.

$x$ -intercepts: $(- 8, 0)$ and $(20, 0)$