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 \(105 \boldcdot 95\).

  3. 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:

  4. 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.