To factor x² + bx + c, find two numbers multiplying to c and adding to b.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Reading the signs
A positive c means both numbers share a sign; a negative c means they differ.
Which sign
With a positive c, the shared sign is the sign of b.
Always expand back
Multiplying the factors out must return the original trinomial exactly.
Find two numbers
To factor x² + 5x + 6, find two numbers multiplying to 6 and adding to 5: 2 and 3. So it is (x + 2)(x + 3). The product is the constant term and the sum is the coefficient of x.
What the signs tell you
If the constant is positive, both numbers share the sign of the middle coefficient. If the constant is negative, the numbers have opposite signs and the larger takes the middle term's sign. Working out the signs first narrows the search.
When the leading coefficient is not 1
For 2x² + 7x + 3, look for numbers multiplying to 2 × 3 = 6 and adding to 7: 1 and 6. Split the middle term and factor by grouping. The extra step is what the leading coefficient costs.
Not everything factors nicely
x² + x + 1 has no factorisation over the integers. That is not a failure of technique — some quadratics simply have irrational or complex roots. Knowing when to stop looking and switch to the formula saves a lot of time.
Step 2: Try It Yourself
Tap and try it out.
8 + 8 + 8 + 8 + 8 = 40
5 × 8 = 40
5 × 8 = (5 × 3) + (5 × 5) = 15 + 25 = 40
The array is cut into 5 × 3 and 5 × 5. Two easier facts that add to the same total.
Step 3: Watch an Example
One step at a time.
Watch Dev Factor x² + 7x + 12
Dev needs two numbers multiplying to 12 and adding to 7.
- Step 1
Pairs multiplying to 12: 1 and 12, 2 and 6, 3 and 4.
Step 4: Your Turn
Practice makes it stick.
The Pair
Problem 1 of 2
Two numbers multiply to 12 and add to 7. What is the larger?
A Negative Constant
Problem 2 of 2
Two numbers multiply to −10 and add to 3. What is the positive one?
Find the Pair
1 of 8
Multiply to 6, add to 5. Larger number?
2 of 8
Multiply to 15, add to 8. Larger number?
3 of 8
Multiply to 9, add to 6. Larger number?
4 of 8
Multiply to −12, add to 1. Positive number?
5 of 8
x² + 5x + 6 factors to (x + 2)(x + ?). What is the missing number?
6 of 8
A positive constant with a positive middle term. Are both numbers positive? 1 for yes, 0 for no.
7 of 8
Multiply to 20, add to 9. Larger number?
8 of 8
Order the steps for factoring x² + bx + c.
- 1Find the pair adding to b
- 2Write the two brackets
- 3Expand to check
- 4List pairs multiplying to c
Step 5: Quick Check
Show what you know.
Question 1 of 1
Multiply to 18, add to 9. Larger number?
What You Learned
- Factoring x² + bx + c needs two numbers multiplying to c and adding to b.
- A positive c means matching signs; a negative c means opposite signs.
- Expanding the factors back is the only check that matters.