Difference of Two Squares Calculator
Factor any a^2 - b^2 expression into (a+b)(a-b) with step-by-step work.
Supports symbolic (px+q)(px-q) factoring and graphs the resulting parabola.
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:
| Expression | Factored Form |
|---|---|
| x^2 - 9 | (x + 3)(x - 3) |
| 4x^2 - 25 | (2x + 5)(2x - 5) |
| 16 - x^2 | (4 + x)(4 - x) |
| 49x^2 - 1 | (7x + 1)(7x - 1) |
Roots: The equation px^2 - q^2 = 0 has roots x = q/p and x = -q/p. These are always symmetric about the y-axis.
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.