Difference of Squares

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Difference of Squares: meaning, method and solved example (Algebra 1) – Maths PDF

In short: Difference of squares names a binomial where one perfect square is subtracted from another, like 49x² − 81, which splits into a sum times a difference.

Algebra 1AlgebraA-SSE.A.2

What is Difference of Squares?

Spot a perfect square with another perfect square taken away, say \(49x^2 – 81\), and a shortcut opens up: it is a difference of squares, and it splits into two conjugate factors with no guessing.

The rule is \(a^2 – b^2 = (a + b)(a – b)\). Expanding shows why it works: the cross products \(+ab\) and \(-ab\) wipe each other out. Add the squares instead and you get \(a^2 + b^2\), which stays prime over the real numbers.

How to spot it

  • A difference of squares hides in two perfect squares separated by a minus sign.
  • Numbers like 4, 9, 25, 49 or 81 appear, and so do even powers such as \(x^4\) or \(y^6\).
  • An equation such as \(x^2 – 36 = 0\) has no middle term at all.
  • A quick-math question asks for \(53^2 – 47^2\) without a calculator.

The method

  1. Confirm that both terms are perfect squares joined by subtraction.
  2. Take square roots, variables included, to get a and b.
  3. Write the conjugate pair \((a + b)(a – b)\).
  4. Multiply it out as a check; the middle terms must cancel.
a² − b²b²(a + b)(a − b)7² − 3² = 49 − 9 = 4010 × 4 = 40a + b = 10a − b = 4cut and rearrange: same area, two ways to write it
With a = 7 and b = 3, the L-shaped region and the 10 by 4 rectangle both have area 40.

Difference of Squares: solved example

Problem

A mural panel is a square with sides of \(7x\) cm, and a 9 cm square window is cut from it. The leftover wall area is \(49x^2 – 81\). Rewrite that area as a product of two binomials.

Solution

Both pieces are squares: \(49x^2\) is \((7x)^2\) and 81 is \(9^2\), with subtraction between them.

Taking \(a = 7x\) and \(b = 9\) gives \((7x + 9)(7x – 9)\). Multiplying back, \(49x^2 – 63x + 63x – 81\) returns \(49x^2 – 81\).

Try it

Question 1

Compute \(53^2 – 47^2\) by hand, using the pattern.

Show the answer

\((53 + 47)(53 – 47) = 100 \cdot 6 = 600\).

Question 2

Explain why \(x^2 + 16\) resists this factoring.

Show the answer

It adds the squares, and the cross terms would not cancel, so no real factors appear.

Difference of Squares vs. Perfect Square Trinomial

The shapes can look alike, yet they differ in a crucial way. This pattern has two terms and a minus sign, with conjugate factors \((a + b)(a – b)\). The trinomial kind has three terms and one repeated factor, \((a + b)^2\). Count the terms before choosing.

Perfect Square Trinomial

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