Question

How do you solve by completing the square `x^2+3x-8=0`?

Answer 1

Solving a quadratic expression by completing the square means to manipulate the expression in order to write it in the form
`(x+a)^2=b`
So, if `b\ge 0`, you can take the square root at both sides to get
`x+a=\pm\sqrt{b}`
and conclude `x=\pm\sqrt{b}-a`.

Now, we have `(x+a)^2=x^2+2ax+a^2`. Since you equation starts with `x^2+3x`, this means that `2ax=3x`, and so `a=3/2`.
Adding `41/4` at both sides, we have
`x^2+3x+9/4=41/4`
Which is the form we wanted, because now we have
`(x+3/2)^2=41/4`
Which leads us to
`x+3/2=\pm\sqrt{41/4}=\pm \sqrt{41}/2` and finally `x=\pm\sqrt{41}/2-3/2`