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Factoring x² + bx + c, step by step

Find two numbers that multiply to the last term and add to the middle one.

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The question

x² + 12x + 11

x² + 12x + 11
multiply to11· add to12
1·11
1·11
11and1
x² + 12x + 11=(x + 11)(x + 1)
  1. Factoring this means finding two numbers to put in (x + ▢)(x + ▢). Two conditions decide them, and both have to hold: they MULTIPLY to 11, the last term, and they ADD to 12, the number in front of x.
  2. Now list the pairs that multiply to 11. There are only 1, so this is a short search rather than a guess: 1 and 11.
  3. Which of them adds to 12? 11 × 1 = 11 and 11 + 1 = 12. So the two numbers are 11 and 1.
  4. Put them in the brackets: x² + 12x + 11 = (x + 11)(x + 1). Multiply it back out if you want to check — the x terms should add up to 12x again.

How it works.

  1. 01

    Factoring this means finding two numbers to put in (x + ▢)(x + ▢). Two conditions decide them, and both have to hold: they MULTIPLY to 11, the last term, and they ADD to 12, the number in front of x.

  2. 02

    Now list the pairs that multiply to 11. There are only 1, so this is a short search rather than a guess: 1 and 11.

  3. 03

    Which of them adds to 12? 11 × 1 = 11 and 11 + 1 = 12. So the two numbers are 11 and 1.

  4. 04

    Put them in the brackets: x² + 12x + 11 = (x + 11)(x + 1). Multiply it back out if you want to check — the x terms should add up to 12x again.

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