Question

How do you factor `v^2+16v+63`?

Answer 1

`v^2+16v+63 = (v+7)(v+9)`

Explanation:

Given:

`v^2+16v+63`

Here are a few methods...

`color(white)()`
Fishing for factors

Note that:

`(v+a)(v+b) = v^2+(a+b)v+abv`

So if we can find two numbers `a, b` with sum `16` and product `63` then we can factor the given quadratic.

We might try:

`8, 8 -> 8*8 = color(red)(cancel(color(black)(64)))`

`7, 9 -> 7*9 = 63`

Hence:

`v^2+16v+63 = (v+7)(v+9)`

`color(white)()`
Completing the square

The difference of squares identity can be written:

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

Complete the square, then use this with `a=(v+8)` and `b=1` as follows:

`v^2+16v+63 = v^2+16v+64-1" "` (where `64 = (16"/"2)^2`)

`color(white)(v^2+16v+63) = (v+8)^2-1^2`

`color(white)(v^2+16v+63) = ((v+8)-1)((v+8)+1)`

`color(white)(v^2+16v+63) = (v+7)(v+9)`

One advantage of this method is that it will always work, regardless of whether the coefficients of the factors are integers, rational numbers, irrational numbers or complex numbers.

`color(white)()`
Quadratic formula

This method is overkill for this particular problem, but is as powerful as completing the square.

Note that:

`v^2+16v+63`

is of the form:

`av^2+bv+c`

with `a=1`, `b=16` and `c=63`.

Its zeros are given by the quadratic formula:

`v = (-b+-sqrt(b^2-4ac))/(2a)`

`color(white)(v) = (-16+-sqrt(16^2-4(1)(63)))/(2*1)`

`color(white)(v) = (-16+-sqrt(256-252))/2`

`color(white)(v) = (-16+-sqrt(4))/2`

`color(white)(v) = (-16+-2)/2`

`color(white)(v) = -8+-1`

That is:

`v = -7" "` or `" "v = -9`

So both `(v+7)` and `(v+9)` are factors of `v^2+16+63`

Since the leading coefficient is `1`, we can deduce that:

`v^2+16v+63 = (v+7)(v+9)`