Question

How do you solve using completing the square method `x^2+x-6=0`?

Answer 1

Using the completing the squares method as requested.
`x=2` or `x=-3`

Explanation:

Given
`color(white)("XXX")x^2+x-6=0`

Adding `6` to both sides to clear the constant from the left side
`color(white)("XXX")x^2+x=6`

If `x^2+x` are the first two terms of an expanded squared binomial
`(x+a)^2=x^2+2ax+a^2`
then `2a=1` and `a=1/2`

Therefore, in order to complete the square, we will need to add `a^2=(1/2)^2` to both sides:
`color(white)("XXX")x^2+xcolor(red)(+(1/2)^2)=6color(red)(+(1/2)^2)`

Rewriting as a squared binomial and simplifying the right side
`color(white)("XXX")(x+1/2)^2=6 1/4=25/4`

Taking the square root of both sides:
`color(white)("XXX")x+1/2= +-sqrt(25/4)=+-5/2`

Subtracting `1/2` from both sides
`color(white)("XXX")x=5/2-1/2=2color(white)("XX")orcolor(white)("XX")x=-5/2-1/2=-3`