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Simon's Favorite Factoring Trick practice

An equation like xy + 3x + 4y = 24 looks stuck, until you add one number to both sides and it factors.

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How it works

The trick rests on one expansion:

(x + b)(y + a) = xy + ax + by + ab

So if you see xy + ax + by, add ab to both sides and the left side factors. For xy + 3x + 4y = 24, add 3 × 4 = 12 to get (x + 4)(y + 3) = 36.

The payoff comes with integer problems. Once the equation is a product equal to a number, list the factor pairs of that number. Each pair gives one solution. If the coefficient of xy isn't 1, multiply through first so it factors cleanly.

Simon's Favorite Factoring Trick practice problems with solutions

These problems come from Mathyessy's question bank. Try each one before opening the solution.

  1. If xy+3x+4y = 24, what is (x+4)(y+3)?

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    Answer: 36

    (x + 4)(y + 3) = xy + 3x + 4y + 12 = 24 + 12 = 36.

  2. If xy+6x+y = 12, what is (x+1)(y+6)?

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    Answer: 18

    (x + 1)(y + 6) = xy + 6x + y + 6 = 12 + 6 = 18.

  3. If (x+3)(y+1) = 10, and xy+x+3y is an integer, what is xy+x+3y?

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    Answer: 7

    Expand: (x + 3)(y + 1) = xy + x + 3y + 3. So xy + x + 3y = 10 − 3 = 7.

  4. How many positive integer pairs (x,y) satisfy (x+1)(y+1) = 30? (Ordered pairs count separately.)

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    Answer: 6

    Both factors must be at least 2. The ordered factorizations of 30 with both parts ≥ 2 are 2×15, 3×10, 5×6, 6×5, 10×3 and 15×2: 6 pairs. (1×30 is excluded because it would make x = 0.)

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