EZ Math

Algebra 2 · Factoring polynomials

How to Factor a Difference of Squares

Problem

x2−49x^2 - 49

Answer

(x−7)(x+7)(x - 7)(x + 7)

A difference of squares is the easiest polynomial to factor. You only need to recognise two perfect squares with a minus sign between them.

Step-by-step solution

  1. Check the pattern a^2 - b^2.

    x2−49=x2−72x^2 - 49 = x^2 - 7^2

    Why: Both terms are perfect squares with a minus sign between them.

  2. Write the two brackets.

    (x−7)(x+7)(x - 7)(x + 7)

    Why: The rule is a^2 - b^2 = (a - b)(a + b).

  3. Check by expanding.

    x2+7x−7x−49=x2−49x^2 + 7x - 7x - 49 = x^2 - 49

    Why: The middle terms cancel, which confirms the factors.

Common mistakes

  • Writing (x - 7)^2. Squaring keeps a middle term, so it is not a difference of squares.
  • Keeping the minus sign in both brackets. One bracket must add.
  • Missing that 49 is 7 squared.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

  1. x2−16x^2 - 16

    Show answer

    (x−4)(x+4)(x - 4)(x + 4)

  2. x2−100x^2 - 100

    Show answer

    (x−10)(x+10)(x - 10)(x + 10)

  3. 4x2−94x^2 - 9

    Show answer

    (2x−3)(2x+3)(2x - 3)(2x + 3)

  4. x2−1x^2 - 1

    Show answer

    (x−1)(x+1)(x - 1)(x + 1)

  5. 9x2−259x^2 - 25

    Show answer

    (3x−5)(3x+5)(3x - 5)(3x + 5)