Factorize the expression x2 + xy - 72y2.

Question

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I will solve question 7(d) from the image.

The question asks to factorize the expression $x^2 + xy - 72y^2$ .

Identify the coefficients.
This is a quadratic expression in two variables. We are looking for two terms that multiply to the product of the first and last terms ( $x^2 \times -72y^2 = -72x^2y^2$ ) and add up to the middle term ( $xy$ ).
Find two terms that multiply to $-72y^2$ and add to $y$ .
We need two numbers that multiply to $-72$ and add to 1. These numbers are 9 and $-8$ .
So, the terms are $9xy$ and $-8xy$ .
Rewrite the middle term using these two terms.
$x^2 + xy - 72y^2 = x^2 + 9xy - 8xy - 72y^2$
Factor by grouping.
Group the terms and factor out the common factors from each group.
$(x^2 + 9xy) - (8xy + 72y^2)$
Factor $x$ from the first group and $8y$ from the second group.
$x(x + 9y) - 8y(x + 9y)$
Factor out the common binomial.
$(x + 9y)(x - 8y)$
The factored form of $x^2 + xy - 72y^2$ is $(x + 9y)(x - 8y)$ .

Answer