Question

How do you solve `x^2-75=0`?

Answer 1

`x=+-5sqrt(3)`

Explanation:

Treat `x^2-75` as the difference of squares

We know that `a^2-b^2=(a-b)(a+b)`

So replacing `a` with `x`
and `b` with `5sqrt(3)` (so that `b^2=75`)
we have
`color(white)("XXX")x^2-75=0`
is equivalent to
`color(white)("XXX")(x-5sqrt(3))(x+5sqrt(3))=0`

which implies
either `(x-5sqrt(3))=0 color(white)("XX")orcolor(white)("XX") (x+5sqrt(3))=0`
`color(white)("XX")rarr x=5sqrt(3)color(white)("XXXXXXXX")rarrx=-5sqrt(3)`