Completing the Square of Sum and Difference
Master the essential algebra technique tested on every SAT Math section
1. Foundation: Perfect Square Trinomial Patterns
Completing the square is built on two fundamental identities — the square of a sum and the square of a difference.
Square of a Sum
$$(a + b)^2 = a^2 + 2ab + b^2$$
The middle term is positive $+2ab$
Example: $(x+3)^2 = x^2 + 6x + 9$
Square of a Difference
$$(a - b)^2 = a^2 - 2ab + b^2$$
The middle term is negative $-2ab$
Example: $(x-5)^2 = x^2 - 10x + 25$
Why "Completing" the Square?
When we have an expression like $x^2 + 6x$, it's almost a perfect square trinomial — it's missing the constant term. We "complete" it by figuring out what constant to add (and subtract) so the expression becomes a perfect square. We add and subtract the same value so the expression stays equal — net change is zero.
Visual: The Geometry of $(a+b)^2$
The area model shows why $(a+b)^2 = a^2 + 2ab + b^2$. The large square with side $(a+b)$ is divided into four regions:
