Is there a name for this technique involving breaking a term into multiple terms?

Question

I recently saw a solution to the quadratic equation $x^2-5x-6=0$ that involved re-writing the middle term, $-5x$, into two terms, $x-6x$, so that the expression could be factored and $x$ solved for, vis-a-vie:

$$x^2+x-6x-6=0\\x(x+1)-6(x+1)=0\\(x+1)(x-6)=0\\x=-1\quad\text{or}\quad x=6$$

Is there a name for this technique, like how “completing the square” names another technique?

Answer 1

This technique is commonly referred to as "the a-c method". I did a quick Google search for examples and came across this.

Bottom line: If a quadratic can be factored into two linear factors with integer coefficients then the A-C Method will always produce those linear factors.

The first step is not just by inspection. Rather, when using this procedure one multiplies $a\cdot c$ (the coefficient of $x^2$ by the constant) then brainstorms to find factors of this product that sum to $b$ (the coefficient of $x$).

Here $a \cdot c = 1 \cdot (-6) = -6$. This product has factors of $1$ and $-6$ which sum to $b = -5$. So one "breaks" $-5x$ into $1x + (-6)x$ and then continues as you have indicated.

Can you see how this procedure would work to factor $x^2+x-6$ into the product of two linear factors?

Answer 2

@peterwhy and @blue

Thanks, both of you.

I’ve synthesised your replies into “factoring by clever inspection”