# Lesson 8

Rewriting Quadratic Expressions in Factored Form (Part 3)

• Let’s look closely at some special kinds of factors.

### 8.1: Math Talk: Products of Large-ish Numbers

Find each product mentally.

$$9 \boldcdot 11$$

$$19 \boldcdot 21$$

$$99 \boldcdot 101$$

$$109\boldcdot101$$

### 8.2: Can Products Be Written as Differences?

1. Clare claims that $$(10+3)(10-3)$$ is equivalent to $$10^2 - 3^2$$ and $$(20+1)(20-1)$$ is equivalent to $$20^2-1^2$$. Do you agree? Show your reasoning.
1. Use your observations from the first question and evaluate $$(100+5)(100-5)$$. Show your reasoning.
2. Check your answer by computing $$105 \boldcdot 95$$.
2. Is $$(x+4)(x-4)$$ equivalent to $$x^2-4^2$$? Support your answer:

With a diagram:

$$x$$     $$4$$
$$x$$
$$\text-4$$

Without a diagram:

3. Is $$(x+4)^2$$ equivalent to $$x^2+4^2$$? Support your answer, either with or without a diagram.

1. Explain how your work in the previous questions can help you mentally evaluate $$22 \boldcdot 18$$ and $$45 \boldcdot 35$$.
2. Here is a shortcut that can be used to mentally square any two-digit number. Let’s take $$83^2$$, for example.

• 83 is $$80+3$$.
• Compute $$80^2$$ and $$3^2$$, which give 6,400 and 9. Add these values to get 6,409.
• Compute $$80 \boldcdot 3$$, which is 240. Double it to get 480.
• Add 6,409 and 480 to get 6,889.
Try using this method to find the squares of some other two-digit numbers. (With some practice, it is possible to get really fast at this!) Then, explain why this method works.

### 8.3: What If There is No Linear Term?

Each row has a pair of equivalent expressions.

Complete the table.

If you get stuck, consider drawing a diagram. (Heads up: one of them is impossible.)

factored form standard form
$$(x-10)(x+10)$$
$$(2x+1)(2x-1)$$
$$(4-x)(4+x)$$
$$x^2-81$$
$$49-y^2$$
$$9z^2-16$$
$$25t^2-81$$
$$(c + \frac25)(c-\frac25)$$
$$\frac{49}{16}-d^2$$
$$(x+5)(x+5)$$
$$x^2-6$$
$$x^2+100$$

### Summary

Sometimes expressions in standard form don’t have a linear term. Can they still be written in factored form?

Let’s take $$x^2-9$$ as an example. To help us write it in factored form, we can think of it as having a linear term with a coefficient of 0: $$x^2 + 0x -9$$. (The expression $$x^2-0x-9$$ is equivalent to $$x^2-9$$ because 0 times any number is 0, so $$0x$$ is 0.)

We know that we need to find two numbers that multiply to make -9 and add up to 0. The numbers 3 and -3 meet both requirements, so the factored form is $$(x+3)(x-3)$$.

To check that this expression is indeed equivalent to $$x^2-9$$, we can expand the factored expression by applying the distributive property: $$(x+3)(x-3) = x^2 -3x + 3x + (\text-9)$$. Adding $$\text-3x$$ and $$3x$$ gives 0, so the expanded expression is $$x^2-9$$.

In general, a quadratic expression that is a difference of two squares and has the form:

$$a^2-b^2$$

can be rewritten as:

$$\displaystyle (a+b)(a-b)$$

Here is a more complicated example: $$49-16y^2$$. This expression can be written $$7^2-(4y)^2$$, so an equivalent expression in factored form is $$(7+4y)(7-4y)$$.

What about $$x^2+9$$? Can it be written in factored form?

Let’s think about this expression as $$x^2+0x+9$$. Can we find two numbers that multiply to make 9 but add up to 0? Here are factors of 9 and their sums:

• 9 and 1, sum: 10
• -9 and -1, sum: -10
• 3 and 3, sum: 6
• -3 and -3, sum: -6

For two numbers to add up to 0, they need to be opposites (a negative and a positive), but a pair of opposites cannot multiply to make positive 9, because multiplying a negative number and a positive number always gives a negative product.

Because there are no numbers that multiply to make 9 and also add up to 0, it is not possible to write $$x^2+9$$ in factored form using the kinds of numbers that we know about.

### Glossary Entries

• coefficient

In an algebraic expression, the coefficient of a variable is the constant the variable is multiplied by. If the variable appears by itself then it is regarded as being multiplied by 1 and the coefficient is 1.

The coefficient of $$x$$ in the expression $$3x + 2$$ is $$3$$. The coefficient of $$p$$ in the expression $$5 + p$$ is 1.

• constant term

In an expression like $$5x + 2$$ the number 2 is called the constant term because it doesn't change when $$x$$ changes.

In the expression $$5x-8$$ the constant term is -8, because we think of the expression as $$5x + (\text-8)$$. In the expression $$12x-4$$ the constant term is -4.

• linear term

The linear term in a quadratic expression (in standard form) $$ax^2 + bx + c$$, where $$a$$, $$b$$, and $$c$$ are constants, is the term $$bx$$. (If the expression is not in standard form, it may need to be rewritten in standard form first.)

• zero product property

The zero product property says that if the product of two numbers is 0, then one of the numbers must be 0.