Which two numbers?

The idea

To factorise x² + (a + b)x + ab, find two numbers a and b whose sum is the coefficient of x and whose product is the constant term, split the x-term accordingly, and write (x + a)(x + b).

Split 7x

To factorise x² + 7x + 12 without tiles, we split the 7x into two pieces, a x and b x, so that a + b = 7. The unit tiles must also fit: ab = 12.

Each pair below adds up to 7. Which pair also multiplies to 12?

• 1 and 6
• 2 and 5
• 3 and 4

Find the pair

Choose a and b. The sum row and the product row must both turn green. Then the splitting is written out for you.

Sum and product

To factorise x² + (a + b)x + ab, find two numbers a and b whose sum is the coefficient of x and whose product is the constant term. Split the x-term, then write (x + a)⁠(x + b).

x² + 7x + 12 = x² + 3x + 4x + 12 = (x + 3)⁠(x + 4) If the x-term is negative, both numbers are negative: x² − 5x + 6 has a + b = −5 and ab = 6, so a = −2 and b = −3.

Notes

To factorise x² + (a + b)x + ab, find a and b with sum = the coefficient of x and product = the constant term, then write (x + a)⁠(x + b).

Check yourself

For x² + 9x + 20, find two numbers with sum 9 and product 20. What is the larger one?

Answer: 5

4 + 5 = 9 and 4 × 5 = 20. The larger number is 5, and x² + 9x + 20 = (x + 4)⁠(x + 5).

x² + 8x + 12 = ?

x² − 7x + 10 = ?

x² + 2x − 15 = ?

• (x + 2)⁠(x + 6) — correct. Yes: 2 + 6 = 8 and 2 × 6 = 12.
• (x + 3)⁠(x + 4). 3 × 4 = 12, but 3 + 4 = 7, not 8.
• (x + 1)⁠(x + 12). 1 × 12 = 12, but 1 + 12 = 13, not 8.
• (x + 2)⁠(x + 5). That gives +7x. The x-term here is −7x, so both numbers are negative.
• (x − 2)⁠(x − 5) — correct. Yes: (−2) + (−5) = −7 and (−2) × (−5) = +10.
• (x − 2)⁠(x + 5). That gives x² + 3x − 10. The product (−2) × 5 is −10, but we need +10.
• (x − 5)⁠(x + 3). That gives x² − 2x − 15. The sum −5 + 3 is −2, but we need +2.
• (x + 5)⁠(x + 3). The product 5 × 3 = 15 is positive, but we need −15, so one number is negative.
• (x + 5)⁠(x − 3) — correct. Yes: 5 + (−3) = 2 and 5 × (−3) = −15.