Question

Solve X^2 + 8x - 3 = 0 using the completing-the-square-method

Answer 1

We want to write
`x^2+8x-3` in the form
`(x-a)^2+b`. To do that, if we expand the second expression and set it equal to the first, we get

`x^2-2ax+a^2+b=x^2+8x-3`

So we need to have

`\begin{cases}-2a=8\\a^2+b=-3\end{cases}`

The first condition tells us that
`a=-4`, so
`(-4)^2+b=-3\implies b=-19`. Then

`x^2+8x-3=(x+4)^2-19=0\implies(x+4)^2=19\implies x+4=\pm√(19)`

`\implies x=-4\pm√(19)`