How to simplify a fraction

Question

I'm having some difficulty understanding how the expression

$$x = \frac{12 \pm 4\sqrt{7}}{8}$$

is simplified to

$$x = \frac{3 \pm \sqrt{7}}{2}$$

Where does the $4$ go? It just disappears?

Answer 1

Recall that multiplying by 1 never makes any difference (it is the multiplicative identity). For all x: $x \times 1 = x$; that is, instead of writing $x \times 1$ we can write $x$, which is the same thing but shorter.

Note also that $\frac 4 4 = 1$. The standard way of reducing fractions is to factor and see if anything cancels, that is, if any factors divide out to 1, which then can in fact be dropped from the writing because multiplying by 1 makes no difference. Thus:

$x = \frac {12 \pm 4 \sqrt 7}{8} = \frac {4 \times (3 \pm \sqrt 7)}{4 \times 2} = \frac 4 4 \times \frac {3 \pm \sqrt 7}{2} = 1 \times \frac {3 \pm \sqrt 7}{2} = \frac {3 \pm \sqrt 7}{2}$

Answer 2

Expanding the reasoning with a middle step:

$$x=\frac{12 \pm 4 \sqrt 7 }{8} = \frac{ 4 \left(3 \pm \sqrt 7 \right)}{4\cdot 2}=\frac{3 \pm \sqrt 7}{2}$$

As multiplying and dividing both numerator and denominator by the same number leaves the same result. For example, let $a,b,n \in \mathbb R$, then $$\frac{a}{b} = \frac{na}{nb}$$

This is the reason behind $\frac{1}{2}= \frac{2}{4} = \frac{50}{100}$, for example.