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Factoring x² + bx − c, step by step

x² + 3x − 28 factors into (x + 7)(x − 4) — two numbers with opposite signs that multiply to −28.

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The question

x² + 8x − 345

x² + 8x − 345
multiply to−345· add to8
−345 < 0→one +, one −
1·3453·1155·6915·23
1·3453·1155·6915·23
23and−15
x² + 8x − 345=(x + 23)(x − 15)
  1. Factoring this means finding two numbers to put in (x + ▢)(x + ▢). Two conditions decide them, and both have to hold: they MULTIPLY to −345, the last term, and they ADD to 8, the number in front of x.
  2. −345 is negative, and a product is only negative when the two numbers have OPPOSITE signs. So one is positive and one is negative, and the bigger of the two takes the sign of 8.
  3. Now list the pairs that multiply to 345. There are only 4, so this is a short search rather than a guess: 1 and 345, 3 and 115, 5 and 69, 15 and 23.
  4. Which of them adds to 8? 23 × −15 = −345 and 23 + −15 = 8. So the two numbers are 23 and −15.
  5. Put them in the brackets: x² + 8x − 345 = (x + 23)(x − 15). Multiply it back out if you want to check — the x terms should add up to 8x 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 −345, the last term, and they ADD to 8, the number in front of x.

  2. 02

    −345 is negative, and a product is only negative when the two numbers have OPPOSITE signs. So one is positive and one is negative, and the bigger of the two takes the sign of 8.

  3. 03

    Now list the pairs that multiply to 345. There are only 4, so this is a short search rather than a guess: 1 and 345, 3 and 115, 5 and 69, 15 and 23.

  4. 04

    Which of them adds to 8? 23 × −15 = −345 and 23 + −15 = 8. So the two numbers are 23 and −15.

  5. 05

    Put them in the brackets: x² + 8x − 345 = (x + 23)(x − 15). Multiply it back out if you want to check — the x terms should add up to 8x again.

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