1. fastest way to factor quadratic

What is the fastest possible way to factor a quadratic? Can someone explain to me how to use the guess and check method, and any limitations it may have (e.g. coefficient in front of $\displaystyle x^2$ must be 1)

For example how would someone factor:
$\displaystyle 6x^2+23x+20$

2. factoered from must be

$\displaystyle (ax+b)(cx+d)$ where $\displaystyle a,b,c,d \in \Re$

in your example $\displaystyle 6x^2+23x+20$

$\displaystyle a\times c = 6$ and $\displaystyle b\times c = 20$

now starting guessing and checking:

1) $\displaystyle a=2, b= -4, c=3, d= -5$ so $\displaystyle (2x-4)(3x-5)$ now expand and check.

keep guessing until you get the correct answer. Your example has many combinations to try, good luck...

3. Another thing to think about is the signs. If you take a quadratic $\displaystyle ax^2 \pm bx \pm c$, where a is positive...

If the quadratic as two plus signs, then the factored binomials both have plus signs:
$\displaystyle ax^2\;{\color{red}+}\;bx\;{\color{red}+}\;c = (?\;{\color{red}+}\;?)(?\;{\color{red}+}\;?)$

If the quadratic as a plus sign followed by a minus sign, then the factored binomials both have minus signs:
$\displaystyle ax^2\;{\color{red}-}\;bx\;{\color{red}+}\;c = (?\;{\color{red}-}\;?)(?\;{\color{red}-}\;?)$

Otherwise, the factored binomials will have one plus sign and one minus sign:
$\displaystyle ax^2\;{\color{red}+}\;bx\;{\color{red}-}\;c = (?\;{\color{red}+}\;?)(?\;{\color{red}-}\;?)$
OR
$\displaystyle ax^2\;{\color{red}-}\;bx\;{\color{red}+}\;c = (?\;{\color{red}+}\;?)(?\;{\color{red}-}\;?)$

Remembering this may cut down the number of guess-and-checks you have to try when factoring.

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