Question

How do you solve `x^2 – 10x = 15` using completing the square?

Answer 1

`x=5+-2sqrt(10)`

Explanation:

`x^2-10x = 15`
`color(white)("XXXX")`If `x^2-10x` are the first two terms of a squared binominal
`color(white)("XXXX")`then the third term must be 25, since
`color(white)("XXXX")` `color(white)("XXXX")` `(x-a)^2 = x^2-2ax+a^2`
`color(white)("XXXX")`and, in this case `a=5`
`x^2-10xcolor(blue)(+25) = 15+color(blue)(25)`

`(x-5)^2 = 40`

`color(white)("XXXX")`Taking the square root of both sides gives
`x-5 = +-2sqrt(10)`

`color(white)("XXXX")`and finally
`x = 5+-2sqrt(10)`

Answer 2

I found:

`x_1=5+sqrt(40)`
`x_2=5-sqrt(40)`

Explanation:

You can add and subtract `25` to get:
`x^2-10xcolor(red)(+25)color(red)(-25)=15`
`x^2-10x+25=15+25`
`(x-5)^2=40`
`x-5=+-sqrt(40)`
So:
`x_1=5+sqrt(40)`
`x_2=5-sqrt(40)`