Question

How do you solve `x^2-4x-1=0` using the quadratic formula?

Answer 1

`x=(2-sqrt(5))` or `x=(2+sqrt(5))`

Explanation:

For the general quadratic equation:
`color(white)("XXX")color(red)(a)x^2+color(blue)(b)x+color(green)(c)=0`
the quadratic formula tells us that
`color(white)("XXX")x=(-color(blue)(b)+-sqrt(color(blue)(b)^2-4color(red)(a)color(green)(c)))/(2(color(red)(a)))`

The given equation: `x^2-4x-1=0` can be written explicitly in the general form as:
`color(white)("XXX")x=color(red)(1)x^2+color(blue)(""(-4))+color(green)(""(-1))=0`

So the solution is
`color(white)("XXX")x=(-color(blue)(""(-4))+-sqrt(color(blue)(""(-4))^2-4color(red)(""(1))color(green)(""(-1))))/(2(color(red)(1))`

`color(white)("XXXX")=(4+-sqrt(20))/2`

`color(white)("XXXX")=2+-sqrt(5)`