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

$x^2+8x-3=(x+4)^2-19=0\implies(x+4)^2=19\implies x+4=\pm\sqrt{19}$

$\implies x=-4\pm\sqrt{19}$