to factorize the expression a2 + 5ab - 84b2.

Question

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

The question asks to factorize the expression $a^2 + 5ab - 84b^2$ .

Identify the coefficients.
This is a quadratic expression in terms of $a$ and $b$ . We need to find two terms that multiply to the product of the first and last terms ( $a^2 \times -84b^2 = -84a^2b^2$ ) and add up to the middle term ( $5ab$ ).
Find two numbers that multiply to $-84$ and add to 5.
We need two numbers whose product is $-84$ and whose sum is 5. These numbers are 12 and $-7$ .
So, the terms are $12ab$ and $-7ab$ .
Rewrite the middle term using these two terms.
$a^2 + 5ab - 84b^2 = a^2 + 12ab - 7ab - 84b^2$
Factor by grouping.
Group the terms and factor out the common factors from each group.
$(a^2 + 12ab) - (7ab + 84b^2)$
Factor $a$ from the first group and $7b$ from the second group.
$a(a + 12b) - 7b(a + 12b)$
Factor out the common binomial.
$(a + 12b)(a - 7b)$
The factored form of $a^2 + 5ab - 84b^2$ is $(a + 12b)(a - 7b)$ .

To factorize the expression a2 + 5ab - 84b2.