Difference of Two Squares Calculator

Difference of Two Squares

The difference of two squares is a special factoring pattern in algebra. Any expression of the form a^2 - b^2 can always be factored as (a + b)(a - b).

The identity:

a^2 - b^2 = (a + b)(a - b)

This works because expanding (a+b)(a-b) gives a^2 - ab + ab - b^2 = a^2 - b^2. The middle terms cancel.

With polynomial terms: If a = px (a polynomial), then:

(px)^2 - q^2 = (px + q)(px - q)

Examples:

Roots: Setting (px)^2 - q^2 = 0 gives roots x = q/p and x = -q/p. Mind the brackets: it is (px)^2, meaning p^2 x^2, not px^2. Written as px^2 - q^2 the roots would be x = ±q/√p instead, which is a different equation entirely. The two roots are always symmetric about the y-axis, because the expression has no x term to shift it off centre.

The sign you give q does not matter. It gets squared, so q = 3 and q = -3 describe the same expression x^2 - 9 and factor identically.

When does this NOT apply?

• Sum of squares: a^2 + b^2 cannot be factored over real numbers
• Expression must be a perfect square minus a perfect square
• a^2 - b^3 is NOT a difference of two squares

Multi-step factoring: Sometimes you need to apply the pattern more than once. For example: x^4 - 16 = (x^2 + 4)(x^2 - 4) = (x^2 + 4)(x+2)(x-2)

Why it matters: This pattern appears in simplifying fractions, solving equations, and in mental math. For example, 99 × 101 = (100-1)(100+1) = 100^2 - 1^2 = 10000 - 1 = 9999.