Example Card: Factoring the Difference of Two Squares
Factor the difference of two squares in Grade 9 algebra using a²-b²=(a+b)(a-b): identify perfect square terms, apply the pattern, and verify by expanding to check your work.
Key Concepts
Now let’s look for a different pattern, the difference of two squares. This example highlights our second key idea for today's lesson.
Example Problem Determine if $25x^8 49y^2$ is a difference of two squares. If so, factor it.
Step by Step 1. We start with the expression $25x^8 49y^2$. We need to check if both terms are perfect squares. 2. We factor each term to see if it can be written as a square. $$ (5 \cdot 5)(x^4 \cdot x^4) (7 \cdot 7)(y \cdot y) $$ 3. We can now rewrite the expression as a difference of two squares, in the form $a^2 b^2$. $$ (5x^4)^2 (7y)^2 $$ 4. With $a = 5x^4$ and $b = 7y$, we can apply the difference of two squares formula, $a^2 b^2 = (a+b)(a b)$. $$ (5x^4 + 7y)(5x^4 7y) $$.
Common Questions
What is the difference of two squares and how do you factor it?
The difference of two squares is an expression of the form a² - b², which factors into (a + b)(a - b). For example, x² - 16 = (x + 4)(x - 4) because x² and 16 are both perfect squares.
How do you recognize a difference of two squares problem?
Look for a binomial with two terms separated by subtraction where both terms are perfect squares. If either condition fails — it's not subtraction, or a term is not a perfect square — the pattern does not apply directly.
How can you verify your factoring of a difference of two squares?
Multiply the factors back together using FOIL or the sum-difference identity. (a + b)(a - b) should return a² - b². If the original expression is reproduced, the factoring is correct.