Doubt in Factoring quadratic equations

Question

I was factorising the equation $x^2 - 3x + 10$ and realised that I can do it as $x^2 -5x + 2x + 10$, since $-5x +2x = -3x$ and $5 \cdot 2 = 10$. However, this doesn't work. As we don't get the common terms in the brackets $x(x-5) + 2(x+5)$

Is this mistake because we need $-10$ instead of $10$? That is, while checking out the multiplication of factors of $a$ and $c$, we need to make sure that it equals $ac$ along with its sign ($+$ or $-$)?

Answer 1

You are right that this doesn't work, but this is because you have $x-5$ in one bracket and $x+5$ in the other. Recall the distributive law, shown in this image with numbers:

Multiplying two numbers can be thought of the area of a rectangle having the numbers as side lengths. When we have rectangles of different dimensions, there is no way to combine them to make a larger rectangle with the same width or height, unless the sides have a common factor.

The same applies for variables, so that there is no way to combine a rectangle with sides $x, x-5$ and another one with sides $2, x+5$ as in your example.

Answer 2

It works but you can no more guess. So or the roots of $Ax^2+Bx+C=0$ $$x_{1,2}=\frac{-B\pm \sqrt{B^2-4AC}}{2A}$$

Also see: Getting back to the quadratic equation again