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.