Factor the following expression: 4a^2 + 12ab + 9b^2 - 25c^2
Understand the Problem
The question requires factoring the given expression. The expression can be seen as a difference of squares if we recognize the first three terms as a perfect square trinomial.
Answer
$(x+y+3)(x-y+3)$
Answer for screen readers
$(x+y+3)(x-y+3)$
Steps to Solve
- Recognize the perfect square trinomial
The first three terms, $x^2 + 6x + 9$, form a perfect square trinomial.
- Factor the perfect square trinomial
Factor $x^2 + 6x + 9$ as $(x+3)^2$.
- Rewrite the expression
The expression becomes $(x+3)^2 - y^2$.
- Recognize the difference of squares
The expression $(x+3)^2 - y^2$ is now in the form of a difference of squares, $a^2 - b^2$, where $a = (x+3)$ and $b = y$.
- Apply the difference of squares factorization
The difference of squares $a^2 - b^2$ factors as $(a+b)(a-b)$. Therefore, $(x+3)^2 - y^2$ factors as $((x+3) + y)((x+3) - y)$.
- Simplify the factors
Simplify the factors to obtain $(x+3+y)(x+3-y)$, which can be written as $(x+y+3)(x-y+3)$.
$(x+y+3)(x-y+3)$
More Information
The difference of squares factorization is a useful technique in algebra that allows us to quickly factor expressions in the form of $a^2 - b^2$. Recognizing patterns like perfect square trinomials and difference of squares can greatly simplify factoring problems.
Tips
A common mistake is to incorrectly factor the perfect square trinomial or to not recognize the difference of squares pattern. Another mistake is to not fully simplify the expression after applying the difference of squares factorization.
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