Use A quadratic function to find two real numbers with a sum of 11 and a product of -42

9514 1404 393

  +14, -3

Step-by-step explanation:

If x represents one of the numbers, then the equation can be written as ...

  x(11 -x) = -42

  x^2 -11x -42 = 0 . . . . quadratic equation whose solutions are the numbers

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If we are to solve this by factoring, we look for factors of -42 that have a sum of 11 (the original problem). The factorization in integers can be any of ...

  -42 = 42(-1) = 21(-2) = 14(-3) = 7(-6)

The sums of these factor pairs are 41, 19, 11, 1. Clearly the numbers we're interested in are 14 and -3.

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There are ways to solve quadratics that don't involve searching for integer factors by trial and error. A graphical solution is attached. The quadratic formula can also be used.

The solution to ax^2+bx+c = 0 is ...

  x = (-b±√(b^2-4ac))/(2a)

Then the solution to our quadratic is ...

  x = (-(-11)±√((-11)^2-4(1)(-42)))/(2(1)) = (11±√289)/2 = (11±17)/2

  x ∈ {-3, 14}

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Additional comment

When the solutions are integers, sometimes the easiest way to find them is to look for the factors of the product that have the correct sum. If the numbers are such that finding factor pairs is difficult or tedious, the more direct approach of graphing or using the quadratic formula may be preferred.

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