Question

How do you solve `x^2- 1/4x - 3/64=64` by completing the square?

Answer 1

`x=(1+-10sqrt(41))/8`.

Explanation:

To complete the square, you can apply the formula `x^2+ax=(x+1/2a)^2-(1/2a)^2`.

You have to substitute `a=(-1/4)/2=-1/8` to solve this problem.

`x^2-1/4x-3/64=64`
`x^2-1/4x=64+3/64`
`(x-1/8)^2-(-1/8)^2=64+3/64`
`(x-1/8)^2=64+3/64+(-1/8)^2`
`(x-1/8)^2=64+4/64=1025/16`

Notice that `1025=5^2*41`.
The root of the equation is
`x-1/8=+-sqrt(1025/16)`
`x-1/8=+-(5sqrt(41))/4`
`x=(1+-10sqrt(41))/8`.