Difference of Squares
Learn to recognize and factor expressions in the form a squared minus b squared.
Definition
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Worked Examples
Factor
Check if it's a difference of squares
is a perfect square, is a perfect square, and there's a minus sign between them → Yes, it's a difference of squares
Identify and
, and → ,
Apply the formula
→
Verify by expanding
\checkmark → Correct!
Answer:
Common Mistakes
Trying to factor as
Why it's wrong: The pattern only works for SUBTRACTION. When you expand , you get , not .
Correct: The sum of squares cannot be factored over the real numbers. Only the DIFFERENCE of squares can be factored.
Forgetting to check if each factor can be factored further
Why it's wrong: Some expressions like can be factored multiple times: first to , then factors again.
Correct: Always check: is the result still a difference of squares? If yes, factor again until no more factoring is possible.
Not recognizing perfect squares like , , or
Why it's wrong: To use this pattern, you must identify that both terms are perfect squares.
Correct: Memorize perfect squares: . For variables: , .
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Practice Problems
18 problemsWhich expression is a difference of squares?
Why It Matters
- Mental Math: Calculate instantly as
- Simplifying: Factor complex expressions quickly without trial and error
- Problem Solving: Many geometry and physics problems involve this pattern
- Foundation: This pattern appears throughout calculus and higher mathematics
Real World Applications
Mental Math Multiplication
Calculate products of numbers equidistant from a round number using the difference of squares pattern.
Example:
To calculate : Both numbers are 2 away from 25, so
A store sells items for 48 dollars and 52 dollars. A customer buys one of each.
Use the difference of squares to find the total mentally.
Step 1: Write the mathematical expression
Write as and simplify:
Area and Geometry
The difference of squares appears when calculating the area between two squares.
Example:
A large square has side and a small square inside has side . The area of the border is . Using our pattern (with and ):
A picture frame is made by cutting a square hole (side 10 cm) from a larger square (side 14 cm).
Find the area of the frame using difference of squares.
Step 1: Write the mathematical expression
Calculate using the pattern:
Key Takeaways
- 1The difference of squares formula is
- 2Both terms must be perfect squares with a minus sign between them
- 3The factors and are called conjugates
- 4Always check if the result can be factored again (like )
- 5The sum of squares cannot be factored over real numbers
Frequently Asked Questions
Glossary
- Difference of squares
- An expression of the form that factors as
- Perfect square
- A number or expression that is the square of an integer or monomial (e.g., 16, , )
- Conjugates
- A pair of binomials that differ only in the sign between their terms: and
- Factor completely
- To break down an expression into factors that cannot be factored further
Formula Card
Difference of Squares
Factor the difference of two perfect squares into conjugate binomials
Perfect Square Trinomial (Plus)
Factor a trinomial that is the square of a binomial sum
Perfect Square Trinomial (Minus)
Factor a trinomial that is the square of a binomial difference