Lesson 6: Rewriting Quadratic Expressions in Factored Form (Part 1)

Rewriting Quadratic Expressions in Factored Form (Part 1)

Student Summary

Previously, you learned how to expand a quadratic expression in factored form and write it in standard form by applying the distributive property.

For example, to expand $(x+4)(x+5)$ , we apply the distributive property to multiply $x$ by $(x+5)$ and 4 by $(x+5)$ . Then, we apply the property again to multiply $x$ by $x$ , $x$ by 5, 4 by $x$ , and 4 by 5.

To keep track of all the products, we could make a diagram like this:

	$x$	$4$
$x$
$5$

Next, we could write the products of each pair inside the spaces:

	$x$	$4$
$x$	$x^2$	$4x$
$5$	$5x$	$4 \boldcdot 5$

The diagram helps us see that $(x+4)(x+5)$ is equivalent to $x^2 +5x +4x + 4 \boldcdot 5$ , or in standard form, $x^2 +9x + 20$ .

The linear term, or the term with a single factor of $x$ in the standard form of a quadratic expression, is $9x$ and has a coefficient of 9, which is the sum of 5 and 4.
The constant term, 20, is the product of 5 and 4.

We can use these observations to reason in the other direction: starting with an expression in standard form and writing it in factored form.

For example, suppose we wish to write $x^2 - 11x + 24$ in factored form.

Let’s start by creating a diagram and writing in the terms $x^2$ and 24.

We need to think of two numbers that multiply to make 24 and add up to -11.

	$x$
$x$	$x^2$
		$24$

After some thinking, we see that -8 and -3 meet these conditions. The product of -8 and -3 is 24. The sum of -8 and -3 is -11.

So, $x^2 - 11x + 24$ written in factored form is $(x-8)(x-3)$ .

	$x$	$\text-8$
$x$	$x^2$	$\text-8x$
$\text-3$	$\text-3x$	$24$

Visual / Anchor Chart

Standards

Building On

6.G.1

6.G.A.1

Addressing

A-SSE.2

HSA-SSE.A.2

Building Toward

A-REI.4.b

HSA-REI.B.4.b

A-REI.4.b

A-SSE.2

A-SSE.3.a

HSA-REI.B.4.b

HSA-SSE.A.2

HSA-SSE.B.3.a

A-REI.4.b

A-SSE.3.a

HSA-REI.B.4.b

HSA-SSE.B.3.a