<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />

Difficulty Level: At Grade Created by: CK-12

Learning Objectives

At the end of this lesson, students will be able to:

• Write quadratic equations in standard form.
• Factor quadratic expressions for different coefficient values.
• Factor when a=1\begin{align*}a = -1\end{align*}.

Vocabulary

Terms introduced in this lesson:

Teaching Strategies and Tips

In this lesson, students learn to factor quadratic polynomials according to the signs of a,b,\begin{align*}a, b,\end{align*} and c\begin{align*}c\end{align*}:

• a=1,b>0,c>0\begin{align*}a = 1, b > 0, c > 0\end{align*}. See Examples 1-4.

Factor.

a. x2+15x+26\begin{align*}x^2 + 15x + 26\end{align*}. Answer: (x+13)(x+2)\begin{align*}(x + 13)(x + 2)\end{align*}

b. x2+13x+40\begin{align*}x^2 + 13x + 40\end{align*}. Answer: (x+8)(x+5)\begin{align*}(x + 8)(x + 5)\end{align*}

c. x2+20x+75\begin{align*}x^2 + 20x + 75\end{align*}. Answer: (x+15)(x+5)\begin{align*}(x + 15)(x + 5)\end{align*}

• a=1,b<0,c>0\begin{align*}a = 1, b < 0, c > 0\end{align*}. See Examples 5 and 6.

Factor.

a. x217x+42\begin{align*}x^2 - 17x + 42\end{align*}. Answer: (x14)(x3)\begin{align*}(x - 14)(x - 3)\end{align*}

b. x221x+90\begin{align*}x^2 - 21x + 90\end{align*}. Answer: (x15)(x6)\begin{align*}(x - 15)(x - 6)\end{align*}

c. x214x+48\begin{align*}x^2 - 14x + 48\end{align*}. Answer: (x6)(x8)\begin{align*}(x - 6)(x - 8)\end{align*}

• a=1,c<0\begin{align*}a = 1, c < 0\end{align*}. See Examples 7-9.

Factor.

a. x215x54\begin{align*}x^2 - 15x - 54\end{align*}. Answer: (x18)(x+3)\begin{align*}(x - 18)(x + 3)\end{align*}

b. x2+7x60\begin{align*}x^2 + 7x - 60\end{align*}. Answer: (x+12)(x5)\begin{align*}(x + 12)(x - 5)\end{align*}

c. x216x192\begin{align*}x^2 - 16x - 192\end{align*}. Answer: (x24)(x+8)\begin{align*}(x - 24)(x + 8)\end{align*}

• a=1\begin{align*}a = -1\end{align*}. See Example 10.

Factor.

a. x24x+60\begin{align*}-x^2 - 4x + 60\end{align*}. Answer: (x6)(x+10)\begin{align*}-(x - 6)(x + 10)\end{align*}

b. x2+14x40\begin{align*}-x^2 + 14x - 40\end{align*}. Answer: (x10)(x4)\begin{align*}-(x - 10)(x - 4)\end{align*}

c. x225x156\begin{align*}-x^2 - 25x - 156\end{align*}. Answer: (x+12)(x+13)\begin{align*}-(x + 12)(x + 13)\end{align*}

• Allow students to infer that if c>0(a=1)\begin{align*}c > 0 (a = 1)\end{align*}, then the factorization will be either of the form (+)\begin{align*}(\underline{\;\;\;\;} + \underline{\;\;\;\;})\end{align*}(+)\begin{align*}(\underline{\;\;\;\;} + \underline{\;\;\;\;})\end{align*} or ()\begin{align*}(\underline{\;\;\;\;} - \underline{\;\;\;\;})\end{align*}()\begin{align*}(\underline{\;\;\;\;} - \underline{\;\;\;\;})\end{align*} (same signs). If c<0\begin{align*}c < 0\end{align*} (a=1)\begin{align*}(a = 1)\end{align*}, then use the form ()\begin{align*}(\underline{\;\;\;\;} - \underline{\;\;\;\;})\end{align*}()\begin{align*}(\underline{\;\;\;\;} - \underline{\;\;\;\;})\end{align*} (different signs).
• See summary at the end of the lesson for a list of procedures and examples for each case.

Emphasize that factoring is the reverse of multiplication.

• Use an example such as (x+3)(x+7)=x2+10x+21\begin{align*}(x + 3)(x + 7) = x^2 + 10x + 21\end{align*} in which the binomials are expanded one step at a time to motivate factoring.
• Demonstrate that factoring is equivalent to putting squares and rectangles back together into larger rectangles.

Example:

Multiply.

(x+3)(x+7)\begin{align*}(x + 3)(x + 7)\end{align*}.

Solution. The diagram shows that (x+3)(x+7)=x2+10x+21\begin{align*}(x + 3)(x + 7) = x^2 + 10x + 21\end{align*}. Observe that it also shows how to factor x2+10x+21\begin{align*}x^2 + 10x + 21\end{align*}.

Suggest that students stop listing the possible products for c\begin{align*}c\end{align*} after the correct choice is evident.

Error Troubleshooting

General Tip: For quadratic trinomials with a=1\begin{align*}a = -1\end{align*}, remind students to factor 1\begin{align*}-1\end{align*} from every term. Remind students to include it in their final answer.

Date Created:

Feb 22, 2012

Aug 22, 2014
Save or share your relevant files like activites, homework and worksheet.
To add resources, you must be the owner of the section. Click Customize to make your own copy.