Question

How do you complete the square to solve `x^2 + 5x + 6 = 0`?

Answer 1

You can complete the square by first getting the form `x^2 + kx = h`.

`x^2 + 5x = -6`

Then just add and subtract a certain value that is equal to `(k/2)^2`. Just remember that the function is `x^2 + kx`, so `k` may be negative, but the added `(k/2)^2` will always be positive.

`x^2 + 5x + (5/2)^2 = (5/2)^2 - 6`

`(x+5/2)^2 = 25/4 - 24/4 = 1/4`

`(x+5/2)^2 - 1/4 = 0`

graph{(x+5/2)^2 - 1/4 [-10, 10, -5, 5]}

If you wanted to solve this:

`(x+5/2)^2 = 1/4`

`x+5/2 = pmsqrt(1/4)`

Thus:
`x+5/2 = pmsqrt(1/4)`

`x = pm(sqrt(1/4)) - 5/2`

`x = pm(1/2) - 5/2`

`x = 1/2 - 5/2 = -2`

`x = -1/2 - 5/2 = -3`

`(-2)^2 + 5(-2) + 6 = 4 - 10 + 6 = 0`
`(-3)^2 + 5(-3) + 6 = 9 - 15 + 6 = 0`

You could just factor, though...

`(x+2)(x+3) = x^2 + 2x + 3x + 6 = x^2 + 5x + 6`