Overview

Quadratic identities appear constantly in simplification and factoring. They also power clever substitutions in symmetric problems.

Key Ideas

• $(x+y)^2 = x^2 + 2xy + y^2$.
• $(x-y)^2 = x^2 - 2xy + y^2$.
• $x^2 - y^2 = (x-y)(x+y)$.

Core Skills

Recognize Patterns

Look for expressions that match $a^2 \pm 2ab + b^2$ or $a^2-b^2$ and factor without expanding.

Use Substitution

If $x+y$ or $xy$ is given, use identities to compute $x^2+y^2$ or $(x-y)^2$.

Choose Expand vs Factor

Expand when you need coefficients, factor when you need zeros or cancellation.

Worked Example

If $x+y=7$ and $xy=10$, find $x^2+y^2$.

Use $(x+y)^2 = x^2 + 2xy + y^2$. Then $x^2 + y^2 = (x+y)^2 - 2xy = 49 - 20 = 29$.

More Examples

Example 1: Difference of Squares

Factor $9x^2-25$.

$(3x-5)(3x+5)$.

Example 2: Square of a Difference

Expand $(2x-3)^2$.

$4x^2-12x+9$.

Example 3: Compute $(x-y)^2$

If $x+y=11$ and $xy=24$, find $(x-y)^2$.

$(x-y)^2=(x+y)^2-4xy=121-96=25$.

Strategy Checklist

• Scan for $a^2-b^2$ or perfect square trinomials.
• Use $(x+y)^2$ and $(x-y)^2$ to connect sums and products.
• Factor before expanding when the goal is solving.

Common Pitfalls

• Expanding when factoring is faster, or vice versa.
• Losing the sign on the middle term in $(x-y)^2$.
• Forgetting that $(x+y)^2$ includes the $2xy$ term.

Practice Problems