I've began practicing factoring expressions of the form $Ax^2+Bx+C$. One method I've learned is decomposition where B can be split into two terms say, D and E where DE=AC. which results in $Ax^2+Dx+Ex+C$. The $Ax^2+Dx$ and $Ex+C$ are then factored, leaving a common factor which with further factorizing, leads to having the original expression factorized.

How might this method be derived? What line of reasoning led to it? I'm really trying to get a more intuitive understanding of why this method works but don't have much algebra knowledge beyond high school.

Hint $$Ax^2+Dx=\frac{1}{E}\left(AEx^2+DEx\right)=\frac{1}{E}\left(AEx^2+ACx\right)=\frac{Ax}{E}\left(Ex+C\right)$$

and then

$$(Ax^2+Dx)+(Ex+C)=(Ex+C)\left(\frac{Ax}{E}+1\right)$$

also,

$$Ex+C=\frac{1}{D}(DEx+DC)=\frac{1}{D}(ACx+DC)=\frac{C}{D}(Ax+D)$$

and you can get $D$ and $E$ solving:

$$D+E=B\\ D\cdot E=A\cdot C$$

• One should probably take care about whether the coefficients are zero in stating this method. For instance, if $E=0$ then the above manipulations are no longer valid. – Semiclassical May 31 '17 at 17:45
• @Semiclassical: Fure sure, I'm assuming that everything is ok. But, $E=0$ or $D=0$ just happen if $A=0$ or $C=0$ and on that case we can solve separately. – Arnaldo May 31 '17 at 17:48