How to factor quadratics

When you multiply (x + 2)(x + 3), you get x² + 5x + 6. Factoring goes the other way: you start with x² + 5x + 6 and find the brackets. You can always check a factoring answer by multiplying it back out. If you get the original expression, you are right.

Before anything else, look for a greatest common factor: a number or variable that goes into every term. Pulling it out first makes everything after it smaller and easier. Students who skip this step often get stuck on a problem that would have been simple.

When the x² term has no number in front, look for two numbers that multiply to c and add to b. Those two numbers go in the brackets.

The signs tell you where to look. If c is positive, both numbers have the same sign as b. If c is negative, one number is positive and one is negative, and the bigger one has the same sign as b.

When there is a number other than 1 in front of x², and no GCF to pull out, use grouping, also called the ac method. Instead of looking for numbers that multiply to c, you look for numbers that multiply to a × c.

1. Multiply a and c.
2. Find two numbers that multiply to ac and add to b.
3. Split the middle term into those two pieces.
4. Group the first two terms and the last two terms, and take the GCF out of each group.
5. Both groups now share a bracket. Take that bracket out as a factor.
6. Check by multiplying back out.

Two patterns come up so often that it pays to spot them on sight. A difference of squares is one square minus another, with no middle term. A perfect square trinomial has a first and last term that are squares and a middle term that is twice their product.

• A sum of squares like x² + 9 does not factor using real numbers. Do not write (x + 3)², which multiplies out to x² + 6x + 9.

Factoring is how you solve many quadratic equations. The key fact is the zero product property: if two things multiply to 0, at least one of them must be 0. This only works when one side of the equation is 0, so move everything to one side first.

• Do not divide both sides by x. In x² = 5x, dividing by x gives only x = 5 and loses the answer x = 0. Instead write x² − 5x = 0, so x(x − 5) = 0, and x = 0 or x = 5.
• Plug each answer back into the original equation to check it.

Not every quadratic factors nicely. If you have tried for a minute and cannot find the two numbers, use the quadratic formula, which works on every quadratic: x = (−b ± √(b² − 4ac))/(2a).

The part under the square root, b² − 4ac, is called the discriminant, and it tells you what to expect before you start. If a, b and c are whole numbers and the discriminant is a perfect square like 1, 4, 9 or 16, the quadratic factors. If it is positive but not a perfect square, the answers have square roots in them, so use the formula. If it is negative, there are no real solutions.