How do you simplify (x^2-12x+36)/(4x-24)?

Oct 21, 2015

$\frac{\left(x - 6\right)}{4}$

Explanation:

Look for potential of parts of the numerator also occurring in the denominator so that you can cancel out.

Consider the numerator: ${x}^{2} - 12 x + 36$

To get anything at all that starts to looks like the denominator we have to factorise. Lets try it!

Try: $\left(x + 6\right) \left(x - 6\right)$

That does not work. Why is that?
Look at the constants: $\left(- 6\right) \times \left(+ 6\right)$ gives $- 36$. That is wrong so we need to change something.

Try: $\left(x - 6\right) \left(x - 6\right) = {x}^{2} - 12 x + 36$ Now we have found the factors.

Substitute this into our original equation giving:

$\frac{\left(x - 6\right) \left(x - 6\right)}{4 x - 24}$

We will have found what we want if we factor out 4 from the denominator giving:

$\frac{\left(x - 6\right) \left(x - 6\right)}{4 \left(x - 6\right)}$

Write as:

$\frac{\left(x - 6\right)}{4} \times \frac{\left(x - 6\right)}{\left(x - 6\right)}$

But:

$\frac{\left(x - 6\right)}{\left(x - 6\right)} = 1$

giving:

$\frac{\left(x - 6\right)}{4}$

Oct 21, 2015

The answer is $\frac{x - 6}{4}$.

Explanation:

The numerator $\left({x}^{2} - 12 x + 36\right)$ is in the form ${a}^{2} - 2 a b + {b}^{2}$, where $a = x \mathmr{and} b = 6$.

Rewrite the numerator.

$\frac{\left({x}^{2}\right) - 2 \left(x\right) \left(6\right) + \left({6}^{2}\right)}{4 x - 24}$

Apply the square of a difference ${\left(a - b\right)}^{2} = {a}^{2} - 2 a b + {b}^{2}$.

${\left(x - 6\right)}^{2} / \left(4 x - 24\right)$

Factor $4$ out of the denominator.

${\left(x - 6\right)}^{2} / \left(4 \left(x - 6\right)\right)$

Rewrite the numerator.

$\frac{\left(x - 6\right) \left(x - 6\right)}{4 \left(x - 6\right)}$

Cancel $\left(x - 6\right)$ from the numerator and denominator.

((cancel(x-6))(x-6))/(4cancel((x-6))$=$

$\frac{x - 6}{4}$