Multiple choice

Two numbers whose sum is $6$ and the absolute value of whose difference is $8$ are roots of the equation:

  1. $x^2 - 6x + 7 =0$
  2. $x^2 - 6x - 7 =0$
  3. $x^2 + 6x - 8 =0$
  4. $x^2 - 6x + 8 =0$
  5. $x^2 + 6x - 7 =0$
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B Correct answer
Explanation

Let roots be a and b. a + b = 6 and |a - b| = 8. Solving gives a = 7, b = -1. The equation is (x - 7)(x + 1) = x^2 - 6x - 7 = 0.

AI explanation

Let the two numbers be the roots alpha and beta. We are given the sum of the roots alpha + beta = 6 and the absolute difference gives the product. Solving the system using the identity (alpha - beta)^2 = (alpha + beta)^2 - 4(alpha * beta), we have 8^2 = 6^2 - 4(alpha * beta), so 64 = 36 - 4(alpha * beta) which means alpha * beta = -7. A quadratic equation is given by x^2 - (sum)x + product = 0, so the required equation is x^2 - 6x - 7 = 0.

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