Factoring

Difference of Squares

Factor x^2 - n^2 as (x + n)(x - n). Recognize the pattern: two perfect squares subtracted. Find n by taking the square root of the constant term, then write the conjugate pair.

Example

Rewrite a square difference as conjugate binomial factors.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

x2−9x^{2} - 9

Step 2 — Identify square

Identify the square: 3 squared is 9.

32=9\hl{3} ^{ 2 }= \hl{9}

Step 3 — Factored form

Use conjugate factors: x + 3 and x - 3.

(x+3)(x−3)(x+ \hl{3} )(x- \hl{3} )

Step 4 — Check

Check by expanding back to the original.

(x+3)(x−3)=x2−9(x+ 3 )(x- 3 )= x^{2} - 9
difference-of-squares x^2 - n^2 factors as (x+n)(x-n). Verify by multiplying: the outer and inner terms cancel, leaving x^2 - n^2.