Spot a notable identity in 2 seconds flat

The three formulas

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

The three patterns to spot

When you see an expression, look for two squares among the terms. That’s the first signal.

Pattern 1: « square + term + square » with middle positive → $(a+b)^2$

$x^2 + 6x + 9$: two squares are $x^2$ and $9 = 3^2$. Middle $6x = 2 \times x \times 3$ ✓. So $= (x + 3)^2$.

Pattern 2: « square + term + square » with middle negative → $(a-b)^2$

$4x^2 - 20x + 25$: $4x^2 = (2x)^2$, $25 = 5^2$. Middle $-20x = -2 \times 2x \times 5$ ✓. So $= (2x - 5)^2$.

Pattern 3: « square minus square » (nothing between) → $(a-b)(a+b)$

$x^2 - 49 = x^2 - 7^2 = (x - 7)(x + 7)$.

$9y^2 - 16 = (3y)^2 - 4^2 = (3y - 4)(3y + 4)$.

The 3-second express test

Facing an expression, ask 3 questions:

1. Are there exactly 3 terms? → likely Pattern 1 or 2.
2. Are there exactly 2 terms separated by a « − »? → likely Pattern 3.
3. Are the outer terms perfect squares? If yes, you have your identity.

The classic mistake

$(a + b)^2 \ne a^2 + b^2$ ever. The famous $2ab$ is missing. Every time you square a sum, write out all 3 terms explicitly.

?Your turn

What is the factored form of x² − 16?

A.(x − 4)²B.(x − 4)(x + 4)C.(x + 4)²D.(x² − 4)(x² + 4)

Check

?Your turn

Expanding (2x − 3)² gives:

A.4x² + 9B.4x² − 6x + 9C.4x² − 12x + 9D.2x² − 12x + 9

Check