Identity

Square of a sum and of a difference

Algebra • Expressions • Algebraic Identities

Formula

(a \pm b)^{2} = a^{2} \pm 2 ab + b^{2}

For all real

a, b

(a + b)^{2} = a^{2} + 2 ab + b^{2}

and

(a - b)^{2} = a^{2} - 2 ab + b^{2}

. Hence

a^{2} + b^{2} = (a + b)^{2} - 2 ab

and

(a - b)^{2} = (a + b)^{2} - 4 ab

How to use it

a + b = 7

and

ab = 12

, find

a^{2} + b^{2}

Step-by-step

1
$a^{2} + b^{2} = (a + b)^{2} - 2 ab$
Why: Avoids solving for $a$ and $b$ separately.
2
$= 49 - 24 = 25$

Answer

25

Watch out

(a - b)^{2} = (a + b)^{2} - 4 ab

gives

∣ a - b ∣

only; the sign needs extra information.

Also known as

perfect square identity, (a+b)^2 expansion