Difference of Squares: A Useful Formula

Article Contents

1. Formula for Difference of Squares
2. Why the Formula Works
3. Factoring Examples
4. Calculation Examples
5. Common Mistakes
6. Interactive Exercises

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1. Formula for Difference of Squares {#formula}

The difference of squares is an expression of the form $a^2-b^2$, where $a$ and $b$ can be numbers or variables.

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

Factoring Example

$$x^2-9=x^2-3^2=(x+3)(x-3)$$

Verification: $(x+3)(x-3)=x^2-3x+3x-9=x^2-9$ ✓

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2. Why the Formula Works {#why-it-works}

Using the FOIL method:

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

The terms $-ab$ and $+ab$ cancel out!

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3. Factoring Examples {#factoring-examples}

Numerical Factoring

$$25-16=5^2-4^2=(5+4)(5-4)=9\cdot 1=9$$

Variables

$$x^2-4=x^2-2^2=(x+2)(x-2)$$

More Complex

$$4x^2-9y^2=(2x{)}^2-(3y{)}^2=(2x+3y)(2x-3y)$$

With Constant

$$2x^2-18=2(x^2-9)=2(x+3)(x-3)$$

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4. Calculation Examples {#calculation-examples}

Quick Calculation

$$53^2-47^2=(53+47)(53-47)=100\cdot 6=600$$

With Complex Numbers

$$(x+1{)}^2-4=((x+1)+2)((x+1)-2)=(x+3)(x-1)$$

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5. Common Mistakes {#common-mistakes}

❌ Mistake 1: Sum Instead of Difference

$$x^2+9\ne (x+3)(x+3)$$

Sum of squares cannot be factored this way!

❌ Mistake 2: Wrong Square Root

$$x^2-7\ne (x+\sqrt{7})(x-\sqrt{7})$$

But this IS CORRECT! Remember that $\sqrt{7}\cdot \sqrt{7}=7$.

Correct: $x^2-7=(x+\sqrt{7})(x-\sqrt{7})$

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6. Interactive Exercises {#interactive-exercises}

Practice difference of squares:

• Difference of Squares
• Expanding
• Factoring

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Formula Summary

Formula	Form
Difference of squares	$a^2-b^2=(a+b)(a-b)$
Sum of squares	Cannot be factored (without complex numbers)
Perfect square	$(a+b{)}^2=a^2+2ab+b^2$

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Related Articles

• Introduction to Algebra – basic concepts
• Expanding – FOIL method
• Completing the Square – advanced technique