Basic Difference of Squares

Factor $x^2 - 16$

Check if it's a difference of squares: $x^2$ is a perfect square, $16 = 4^2$ is a perfect square, and there's a minus sign between them = Yes, it's a difference of squares

Identify $a$ and $b$: $a^2 = x^2 \Rightarrow a = x$, and $b^2 = 16 \Rightarrow b = 4$ = $a = x$, $b = 4$

Apply the formula: $a^2 - b^2 = (a + b)(a - b)$ = $(x + 4)(x - 4)$

Verify by expanding: $(x + 4)(x - 4) = x^2 - 4x + 4x - 16 = x^2 - 16$ \checkmark = Correct!

Answer: $x^2 - 16 = (x + 4)(x - 4)$

Coefficient Greater Than 1

Factor $9y^2 - 25$

Identify the perfect squares: $9y^2 = (3y)^2$ and $25 = 5^2$ = Both are perfect squares

Identify $a$ and $b$: $a = 3y$ and $b = 5$ = $a = 3y$, $b = 5$

Apply the formula: $(3y)^2 - 5^2 = (3y + 5)(3y - 5)$ = $(3y + 5)(3y - 5)$

Verify: $(3y + 5)(3y - 5) = 9y^2 - 15y + 15y - 25 = 9y^2 - 25$ \checkmark = Correct!

Answer: $9y^2 - 25 = (3y + 5)(3y - 5)$

Factoring Multiple Times

Factor completely: $x^4 - 81$

Recognize as difference of squares: $x^4 = (x^2)^2$ and $81 = 9^2$ = $(x^2)^2 - 9^2$

Apply the formula once: $(x^2 + 9)(x^2 - 9)$ = First factorization done

Check if factors can be factored further: $x^2 + 9$ is a sum of squares (cannot factor over reals), but $x^2 - 9$ is another difference of squares! = $x^2 - 9 = (x + 3)(x - 3)$

Write the complete factorization: $x^4 - 81 = (x^2 + 9)(x + 3)(x - 3)$ = Completely factored

Answer: $x^4 - 81 = (x^2 + 9)(x + 3)(x - 3)$

Mistake: Trying to factor $a^2 + b^2$ as $(a + b)(a - b)$

Why: The pattern only works for SUBTRACTION. When you expand $(a + b)(a - b)$, you get $a^2 - b^2$, not $a^2 + b^2$.

Correct: The sum of squares $a^2 + b^2$ cannot be factored over the real numbers. Only the DIFFERENCE of squares can be factored.

Mistake: Forgetting to check if each factor can be factored further

Why: Some expressions like $x^4 - 16$ can be factored multiple times: first to $(x^2 + 4)(x^2 - 4)$, then $(x^2 - 4)$ factors again.

Correct: Always check: is the result still a difference of squares? If yes, factor again until no more factoring is possible.

Mistake: Not recognizing perfect squares like $49$, $121$, or $4x^2$

Why: To use this pattern, you must identify that both terms are perfect squares.

Correct: Memorize perfect squares: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144$. For variables: $(2x)^2 = 4x^2$, $(3y)^2 = 9y^2$.

Mental Math Multiplication

Calculate products of numbers equidistant from a round number using the difference of squares pattern.

To calculate $23 \times 27$: Both numbers are 2 away from 25, so $23 \times 27 = (25-2)(25+2) = 25^2 - 2^2 = 625 - 4 = 621$

Area and Geometry

The difference of squares appears when calculating the area between two squares.

A large square has side $x + 3$ and a small square inside has side $x - 3$. The area of the border is $(x+3)^2 - (x-3)^2$. Using our pattern (with $a = x+3$ and $b = x-3$): $= [(x+3)+(x-3)][(x+3)-(x-3)] = (2x)(6) = 12x$

Q: Can I factor $x^2 + 9$?

A: No, the sum of squares cannot be factored using real numbers. The difference of squares pattern ONLY works when there's a minus sign: $a^2 - b^2$.

Q: Why does the middle term disappear?

A: When you expand $(a+b)(a-b)$, you get $a^2 - ab + ab - b^2$. The $-ab$ and $+ab$ cancel each other out, leaving just $a^2 - b^2$.

Q: How do I recognize a perfect square?

A: For numbers: check if it's in the list 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, etc. For variables: the exponent must be even (like $x^2$, $x^4$, $y^6$), and any coefficient must itself be a perfect square (like $4x^2 = (2x)^2$).