# Factoring Quadratic Equations -- Splitting the Middle Term

• August 9th 2011, 03:33 PM
happyface
Factoring Quadratic Equations -- Splitting the Middle Term
Is there a good way to always know how to split the middle term of a quadratic equation like x^2-9x+20=0?

It turns out that -5x and -4x "works" but is there a better way to split the term that is less tedious?
• August 9th 2011, 03:38 PM
Plato
Re: Factoring Quadratic Equations -- Splitting the Middle Term
Quote:

Originally Posted by happyface
Is there a good way to always know how to split the middle term of a quadratic equation like x^2-9x+20=0?
It turns out that -5x and -4x "works" but is there a better way to split the term that is less tedious?

Note that the constant term is $+$, that means the two factors must have the same sign. But it must be $-$ because the middle term is negative. So what are they?
• August 9th 2011, 03:43 PM
happyface
Re: Factoring Quadratic Equations -- Splitting the Middle Term
Quote:

Originally Posted by Plato
Note that the constant term is $+$, that means the two factors must have the same sign. But it must be $-$ because the middle term is negative. So what are they?

I see! Negative 4 and 5 equal 9 when added, and 20 when multiplied.

There are a few cases (like ax^2 + bx + c) where it is not so obvious where to split the middle term. That is especially where I get stumped, since c does not always equal the product of the "split" term.

Thats actually what I meant to ask about.
• August 9th 2011, 03:51 PM
Plato
Re: Factoring Quadratic Equations -- Splitting the Middle Term
Quote:

Originally Posted by happyface
I see! Negative 4 and 5 equal 9 when added, and 20 when multiplied. There are a few cases (like ax^2 + bx + c) where it is not so obvious where to split the middle term. That is especially where I get stumped, since c does not always equal the product of the "split" term.

BUT $\frac{c}{a}$ is the product and $-\frac{b}{a}$ is the sum.
• August 9th 2011, 04:50 PM
happyface
Re: Factoring Quadratic Equations -- Splitting the Middle Term