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Difference of Two Squares

Factorisation · Class VIII

Check that a^2 - b^2 always equals (a - b)(a + b).

7
type a value
-1515
3
type a value
-1515
Live values
  • a^2 - b^240
  • (a - b)(a + b)40
  • Equal? (1 yes)1
xy
  • y = x^2 - b^2
  • y = (x - b)(x + b)

Formulas in this lab

  • a^2 - b^2
    a2−b2a^{2} - b^{2}
  • (a - b)(a + b)
    (a−b)(a+b)(a-b)(a+b)
  • Equal? (1 yes)
    [identity][\text{identity}]
Tip: Both curves overlap perfectly , that is the identity at work, with zeros at x = +/- b.

Frequently asked questions

▶What is the difference of squares identity?

It states that $a^2 - b^2 = (a - b)(a + b)$. So 7^2 - 3^2 = 49 - 9 = 40, and (7 - 3)(7 + 3) = 4 * 10 = 40. The two sides always match for any a and b.

▶How do I use the difference of squares lab?

Slide a and b between -15 and 15. The lab shows a^2 - b^2 next to (a - b)(a + b), both giving the same number. Try a = 10 and b = 4 to see both as 84.

▶Where is the difference of squares useful?

It speeds up mental maths. To compute 53 * 47, write it as (50 + 3)(50 - 3) = 50^2 - 3^2 = 2500 - 9 = 2491. The lab lets you verify this trick for any pair, useful in quick competitive exam calculations.

▶Difference of squares vs sum of squares: can we factor both?

Difference of squares factors neatly: a^2 - b^2 = (a - b)(a + b). But sum of squares a^2 + b^2 does not factor with real numbers; you cannot break it into nice linear factors. The lab focuses on the difference case only, which is the workable one.