How to factor quadratic $ax^2+bx+c$?

Solution 1:

You might appreciate the series of Khan Academy lectures (+ practice problems, etc).

Each lesson is given via a short 12 minute video, which you can replay if needed: the link I'm including below should take you to "Algebra" lessons; but you might also want to look at "Developmental math I and II".

This is a great way to learn via video demonstrations, practice, repetition, etc., and if you stumble, say, with negative exponents, e.g., you can go directly to a lesson addressing the issue at hand. I believe there is diagnostic testing available (this is all free, no fees for watching videos, doing practice sets, tests, etc.), and such testing can really be helpful in learning where you need to direct your energies to move to the next level.

Edit: I've also come across this site, devoted to tutoring/demonstrating how to factor polynomials: it's developmental in approach, in that it starts slowly, with very common patterns used when factoring, and progresses in difficulty, building on what you've already learned, so best to approach the tutorial in the order given.

Solution 2:

Given

$$A: x^2 + x - 2$$

you're trying to do the 'magic' in your head in order to get backwards to

$$B: (x+2)(x-1)$$

What is it that you are trying to do backwards. It's the original multiplication of $(x+2)(x-1)$. Note that

• the -2 in $A$ comes from multiplying the +2 and -1 in $B$
• the +1 (it's kind of invisible it's the coefficient of $x$ ) in $B$ comes from:
  • $x$ in the first part times -1 in the second, plus
  • +2 in the first part times $x$ in the second

or $(-1)+2 = +1$.

So that's how the multiplication works going forward. Now you have to think of that to go backwards. In $x^2 + x - 2$:

• where does the -2 come from? From two things that multiply to get -2. What could those possibly be? Usually we assume integers so the only possibilities are the two pairs 2, -1, and -2, 1.
• of those two pairs, they have to -add- to the coefficient for $x$ or just plain positive 1. So the answer has to be the pair 2 and -1.

Another example might help: given

$$x^2-5x+6$$

what does this factor to? (that is, find $(x-a)(x-b)$ which equals $x^2 -5x + 6$).

So the steps are:

• what are the factors of 6? (you should get 2 pairs, all negative.
• for those pairs, which pair adds up to -5?

The main difficulty is keeping track in your head of what is multiplying, what is adding, and what is positive and negative.

The pattern for any sort of problem solving skill like this that seems like magic (but really is not) is to:

• Do more examples to get a speedier feel for it.
• Check your work. Since you're going backwards, once you get a possible answer, you can do the non-magic (multiplying) to see if you can get the original item in the question.

Solution 3:

As Ross pointed out, and as was previously discussed, we know that

$$ (x+a)(x+b) = x(x+b) + a(x+b) = x^2 + bx + ax + ab = x^2 + (a+b)x + ab. $$

Therefore, to factor a quadratic expression $x^2 + cx + d$, all one has to do is find two numbers that multiply to $d$ and add to $c$. Let $m$ and $n$ be those two numbers that add to $c$ and multiply $d$. Then:

$$ x^2 + cx + d = (x+m)(x+n). $$

Note that it doesn't matter which order we pick the two numbers since $a+b = b+a$ and $ab = ba$.

In your case, to factor $x^2 + x - 2 = x^2 + 1x - 2$, we need to find two numbers that add to $1$ and multiply to $-2$. Just by picking and trying different numbers, we find the two numbers are $2$ and $-1$.

A good way to go about finding the numbers is to determine all the factors of the constant term, which for us is $-2$. Since $-2 = 2 \cdot (-1) = (-2) \cdot 1$, we only have to check whether $2 + (-1)$ or $(-2) + 1$ equals one. Since it is the first pair of numbers with this property, these are the two numbers we were looking for. Therefore,

$$ x^2 + x - 2 = (x+2)(x+(-1)) = (x+2)(x-1). $$

The reason this works: Suppose that we want to factor $x^2 + cx + d$, and we have found the numbers $m$ and $n$ such that $m + n = c$ and $mn = d$. Then:

$$ \begin{align} x^2 + cx + d &= x^2 + (m+n)x + mn \\ &= x^2 + mx + nx + mn \\ &= x(x+m) + n(x+m) \\ &= (x+n)(x+m) \end{align} $$