Factoring simple quadratics review

Factoring quadratics is very similar to multiplying binomials, just going the other way. For example, x^2+3x+2 factors to (x+1)(x+2) because (x+1)(x+2) multiplies to x^2+3x+2. This article reviews the basics of how to factor quadratics into the product of two binomials.

Example

Factor as the product of two binomials.

x^{2} + 3 x + 2

Our goal is to rewrite the expression in the form:

(x + a) (x + b)

Expanding

(x + a) (x + b)

gives us a clue.

\begin{aligned} x^{2} + 3 x + 2 & = (x + a) (x + b) \\ = x^{2} + a x + b x + a b \\ = x^{2} + (a + b) x + a b \end{aligned}

(a + b) = 3

and

a b = 2

After playing around with different possibilities for

a

and

b

, we discover that

a = 1

b = 2

satisfies both conditions.

Plugging these in, we get:

(x + 1) (x + 2)

And we can multiply the binomials to check our solution if we'd like:

\begin{aligned} (x + 1) (x + 2) \\ = & x^{2} + 2 x + x + 2 \\ = & x^{2} + 3 x + 2 \end{aligned}

Yep, we get our original expression back, so we know we factored correctly to get our answer:

(x + 1) (x + 2)

Want to see another example? Check out this video.

Practice

Factor the quadratic expression as the product of two binomials.

x^{2} - x - 42 =

Want more practice? Check out this exercise.

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