Question

How do you solve the equation by completing the square: `x^2 - 8x - 3 = 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^2-8x`, this means that `2ax=-8x`, and so `a=-4`.
Adding `19` at both sides, we have
`x^2-8x+16=19`
Which is the form we wanted, because now we have
`(x-4)^2=19`
Which leads us to
`x-4=\pm\sqrt{19}` and finally `x=\pm\sqrt{19}+4`