Question

factor. x^2-12x+32

Answer 1

There is a very good way to factor this kind of mathematical expressions and only applyable for a quadratic mathematical expression (any hgher grade has more complex processes).
We will take that expression and make it equal zero:
\[x^2 -12x +32 =0\]
Why?
well, when you make it equal zero, you find the "roots" of the function represented by such mathematical expression and take into a form: (x-a)(x-b) where a and b represent the roots of the function.
So, let's take the general formula and solve that for "x":
\[x= \frac{ -b \pm \sqrt{b^2 - 4ac} }{ 2a }\]
\[x= \frac{ 12 \pm \sqrt{12^2 - 4(1)(32)} }{ 2 }\]
doing some magical arithmetics and simplifications:
\[x= \frac{ 12 \pm \sqrt{16} }{ 2 }\]
\[x= \frac{ 12 \pm 4 }{ 2 }\]
therefore:
\[x_1 = 8\]
\[x_2 = 4\]
so, the factorized form of the expression would be:
\[(x-8)(x-4)\]