Question

Factor the expression x^2+9xy-52y^2

Answer 1

To factor the expression \( x^2 + 9xy - 52y^2 \), we will look for two numbers that multiply to \( -52y^2 \) (the product of the coefficient of \( x^2 \) and the constant term) and add up to \( 9y \) (the coefficient of \( xy \)).

1. The numbers that fit this criteria are \( 13y \) and \( -4y \), since \( 13y \times (-4y) = -52y^2 \) and \( 13y + (-4y) = 9y \).

2. We can then rewrite the expression by splitting the middle term:
\[
x^2 + 13xy - 4xy - 52y^2
\]

3. Now, group the terms:
\[
(x^2 + 13xy) + (-4xy - 52y^2)
\]

4. Factor out the common terms in each group:
\[
x(x + 13y) - 4y(x + 13y)
\]

5. Factor out the common binomial factor \( (x + 13y) \):
\[
(x + 13y)(x - 4y)
\]

Thus, the factored form of the expression \( x^2 + 9xy - 52y^2 \) is:
\[
\boxed{(x + 13y)(x - 4y)}
\]