Question: I was looking up some shortcuts to solve quadratic equations. I got a technique that applies when the sum of the numerators and denominators are equal, but I am unable to understand the reasoning behind it. Here I'm showing an example:

$$ \frac{3x + 4}{6x + 7} = \frac{5x + 6}{2x + 3} $$

The solution goes as follows:

"Minute observation of the question helps us to identify that this question falls in a special category of quadratic equations, where the sum of the numerators (N) and the sum of the denominators (D) are found to be equal to 8x + 10.''

For the first root,

$ N\_1 + N\_2 = D\_1 + D\_2 = 0$

or, $ 8x + 10 = 0 $

or, $ x = -5/4 $

For the second root

$ N\_1 - D\_1 = N\_2 - D\_2 = 0 $

or, $ 3x + 3 = 0 $

or, $ x = -1 $

Can someone explain the reasoning/proof behind this?

Answer: 
Your equation has the form 
$$
N\_1/D\_1 = N\_2/D\_2
$$
that implies 
$$
N\_1 D\_2 = N\_2 D\_1
$$
Add $N\_2 D\_2$ to both sides, 
$$
(N\_1+N\_2) D\_2 = N\_2 (D\_1+D\_2) \qquad \qquad (\*)
$$
but $(N\_1+N\_2) = (D\_1+D\_2) $, so you cam simplify the terms in the parenthesis, and remain with 
$$
D\_2 = N\_2 \quad \Rightarrow \quad D\_2 -N\_2 = 0
$$
Similarly you can show that an equivalent condition is $D\_1-N\_1 = 0$.

Note that equation $(\*)$ is satisfied also if $(N\_1+N\_2)=(D\_1+D\_2)=0$, which gives the other solution.
Componendo and dividendo ([Brilliant](https://brilliant.org/wiki/componendo-and-dividendo/)) is another method.

Using the third rule with $k=1$, we have:

$$\frac{3x+4+(6x+7)}{3x+4-(6x+7)} = \frac{5x+6+(2x+3)}{5x+6-(2x+3)}$$
$$\Rightarrow \frac{9x+11}{-3x-3} = \frac{7x+9}{3x-3}$$
$$\Rightarrow -9x-11 = 7x+9$$
$$\Rightarrow x = -\frac{5}{4}$$

which is true in general, when we have:

$$\frac{N\_1 + D\_1}{N\_1 - D\_1} = \frac{N\_2 + D\_2}{N\_2 - D\_2}$$

and $N\_1 - D\_1 = N\_1 + N\_2 - D\_1 - N\_2 = D\_1 + D\_2 - D\_1 - N\_2 = D\_2 - N\_2$, so $N\_1 - D\_2 = -(N\_2 - D\_2)$.

Using the fourth rule with $k=1$, we have:
$$\frac{3x+4+(5x+6)}{3x+4+(2x+3)} = \frac{5x+6+(3x+4)}{5x+6+(6x+7)}$$
$$\Rightarrow \frac{8x+10}{5x+5} = \frac{8x+10}{11x+13}$$
$$\Rightarrow 11x+13=5x+5$$
$$\Rightarrow x = -1$$

This can be proven by cross multiplying as well.