Factor x^2 - 4x + 7

Step by step

The question

Question: factor completely x² − 4x + 7.

Completely means: until no part can be factored any more.

1 Common factor first

Always ask this first.

Here no number and no x goes into every term.

So there is no common factor.

2 Count the terms

The bracket has 3 terms.

With 3 terms we look for two numbers that multiply to c and add to b.

3 Prime or not

We need two numbers.

They multiply to 7, and they add to −4.

We list every pair that makes 7.

We add each pair.

We need 4.

None of them gives 4.

No pair works.

So x² − 4x + 7 does not factor with whole numbers.

We call it prime.

So x² − 4x + 7 is prime.

Let's look back.

First: common factor first.

Then: count the terms.

Last: prime or not.