Study guides › Level 6 · Master

Factoring x² + bx + c

Level 6 · Master · for questions like “x² + 7x + 12 = (x + 3)(x + ?)”

💡 The idea

Multiply out (x + p)(x + q) and you get x² + (p + q)x + pq. So the two numbers add to the middle number and multiply to the last number.

🪜 The steps

1. List the pairs that multiply to the last number.
2. Pick the pair that adds to the middle number.

✏️ Worked examples

1. x² + 7x + 12 = (x + 3)(x + ?): 3 × ? = 12, so ? = 4, and 3 + 4 = 7. ✓ The answer is 4.

2. x² + 10x + 21: 3 × 7 = 21 and 3 + 7 = 10, so it's (x + 3)(x + 7).

⚠️ Watch out

• Check both conditions: multiply AND add. For 12: 2 × 6 multiplies to 12, but adds to 8, not 7.
• Multiply your answer back out to be sure.

🎯 Try these

1. x² + 9x + 20 = (x + 4)(x + ?)
2. x² + 8x + 15 = (x + ?)(x + 5)
3. x² + 13x + 36 = (x + 4)(x + ?)

Answers: 1) 5 2) 3 3) 9

This guide is also in the 10MathQ app: tap “📖 Study guide” after a question.