Question

How do you factor `x^2 + 5x - 36`?

Answer 1

`x^2+5x-36=(x+9)(x-4)`

Explanation:

Find a pair of factors of `36` which differ by `5`.

The pair `9, 4` works in that `9xx4=36` and `9-4=5`

Hence we find:

`x^2+5x-36=(x+9)(x-4)`

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Alternative method

Alternatively, we can complete the square and use the difference of squares identity:

`a^2-b^2=(a-b)(a+b)`

with `a=(2x+5)` and `b=13` as follows.

First multiply by `2^2 = 4` to cut down on arithmetic involving fractions. Remember to divide by it at the end...

`4(x^2+5x-36)`

`=4x^2+20x-144`

`=(2x)^2+2(2x)(5)-144`

`=(2x+5)^2-25-144`

`=(2x+5)^2-169`

`=(2x+5)^2-13^2`

`=((2x+5)-13)((2x+5)+13)`

`=(2x-8)(2x+18)`

`=(2(x-4))(2(x+9))`

`=4(x-4)(x+9)`

Dividing both ends by `4` we find:

`x^2+5x-36 = (x-4)(x+9)`