Factoring Quadratic Expressions: x² + bx + c

Step 1 of 4

Warm Up

Success criteria

I can find two integers from a sum and product • use signs to narrow the search • spot ‘no integer factorization’ • take out a GCF first • check by expanding.

Warm Up

Expand (x + 2)(x + 6).

ax² + 12bx² + 8x + 12cx² + 12x + 8dx² + 8x + 8

Look back at your expansion: 8 is 2 + 6 and 12 is 2 × 6. That's the pattern you'll use today — in reverse.

Warm Up

Which pair of numbers multiplies to −12 and adds to 1?

a3 and −4b−3 and 4c2 and −6d−2 and 6

Step 2 of 4

Learn

Why sum and product?

Expand (x + m)(x + n): x² + nx + mx + mn = x² + (m + n)x + mn.

So in x² + bx + c, the middle coefficient b is the SUM m + n and the constant c is the PRODUCT mn. Factoring means finding m and n.

The pattern

(x + m)(x + n) = x² + (m + n)x + mnexpanding
b = m + n, c = mnmatch coefficients

Algebra workspace · Sum-and-Product Detective

Tap two number chips. The workspace checks the sum and product live. When both match, the factors appear.

Predict first

For SUM 7 and PRODUCT 10, what signs will the two numbers have?

Make a prediction to open the workspace.

Sign shortcuts

c positive → same signs (match the sign of b). c negative → opposite signs (the bigger number takes the sign of b). No pair at all → the trinomial does not factor over the integers.

Worked example

Factor x² + 9x + 20.

Targets
Sum b = 9, product c = 20.
Factor pairs of 20
1 & 20 (sum 21), 2 & 10 (sum 12), 4 & 5 (sum 9) ✓
Write factors
m = 4, n = 5.
(x + 4)(x + 5)
Check
x² + 5x + 4x + 20 = x² + 9x + 20 ✓

Answer: (x + 4)(x + 5)

Try it

Factor x² + 7x + 12.

a(x + 2)(x + 6)b(x + 3)(x + 4)c(x + 1)(x + 12)d(x − 3)(x − 4)

Worked example

Factor x² − 3x − 18.

Targets
Sum −3, product −18. Negative product → opposite signs; negative sum → the larger number is negative.
Search
−18 & 1 (−17), −9 & 2 (−7), −6 & 3 (−3) ✓
Write factors
Each number becomes the constant in a bracket.
(x − 6)(x + 3)
Check
x² + 3x − 6x − 18 = x² − 3x − 18 ✓

Answer: (x − 6)(x + 3)

GCF first

If every term shares a factor, take it out first: 2x² + 10x + 12 = 2(x² + 5x + 6) = 2(x + 2)(x + 3).

Try it

Factor x² − 2x − 15.

a(x − 5)(x + 3)b(x + 5)(x − 3)c(x − 5)(x − 3)d(x − 15)(x + 1)

Factoring Quadratic Expressions: x² + bx + c

Warm Up

Learn

Why sum and product?

Practice

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Apply it

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Mastery Check

Lesson 2.3 mastery check