Question

How do you factor the trinomial `a^2 + 4ab – 45b^2`?

Answer 1

`color(blue)(=a^2 +4ab-45b^2)`

See the explanation for the logic applied thus reducing the amount of work

Explanation:

As the coefficient of `a^2` is 1 we only have to consider the factors of 45 in `45b`

We need two numbers that that have a product is 45 and a difference of 4

Consider the factors of 45 `->` {1,45}, {3, 15}, {5, 9}: last pair have a difference of 4 so they have to be the ones to use.

To give negative 45 one is positive and the other is negative. AS we have positive 4 in `4ab` the larger of the two factors is the positive one. Otherwise we would end up with `-4ab`

So we need something on the lines of `(a+?b)(a-?b)`

The `b's` give us the `b^2`

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`color(blue)("Try 1")`

`color(green)("remember that the larger needs to be positive" -> +9)`
`color(brown)((a +9b)(a-5b))color(blue)(=a^2 -5ab+9ab-45b^2`

`color(blue)(=a^2 +4ab-45b^2)`

Got it first try!
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