Multiple choice

If the roots of the equation (x+1)(x+9) + 8 = 0 are a and b, then the roots of the equation (x + a) (x + b) - 8 = 0 are

  1. 1 and 9

  2. -4 and -6

  3. 4 and 6

  4. Cannot be determined
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A Correct answer
Explanation

The first equation is x^2 + 10x + 17 = 0, roots a and b. Sum a+b = -10, product ab = 17. The second equation is x^2 + (a+b)x + ab - 8 = 0. Substituting values: x^2 - 10x + 17 - 8 = 0, so x^2 - 10x + 9 = 0. Factors are (x-1)(x-9) = 0. Roots are 1 and 9.

AI explanation

Expanding the first equation (x+1)(x+9) + 8 = 0 gives x2 + 10x + 17 = 0, so by Vieta's formulas the sum of its roots a + b equals -10 and the product ab equals 17. Expanding the second equation (x + a)(x + b) - 8 = 0 gives x2 + (a+b)x + ab - 8 = 0. Substituting the known sum and product results in x2 - 10x + 17 - 8 = 0, which simplifies to x2 - 10x + 9 = 0. Factoring this quadratic equation gives (x - 1)(x - 9) = 0, so the roots are 1 and 9.

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