Multiple choice

Two engineers, Riya and Kabir, were modelling the shape of a parabolic bridge arch using a quadratic equation. Riya made a mistake copying the constant term of the equation and, after solving her (incorrect) equation, obtained roots 3 and 7. Kabir made a mistake copying the coefficient of x and, after solving his (incorrect) equation, obtained roots 1 and 16. The correct roots of the quadratic equation representing the arch are:

  1. 4, 6

  2. 2, 8

  3. 1, 16

  4. −2, −8
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B Correct answer
Explanation

Riya's roots 3 and 7 imply the equation is x^2 - 10x + c = 0. Kabir's roots 1 and 16 imply the equation is x^2 + bx + 16 = 0. Combining the correct coefficient of x (-10) and the correct constant (16), the equation is x^2 - 10x + 16 = 0. Factoring gives (x-2)(x-8) = 0, so the roots are 2 and 8.

AI explanation

Let the correct quadratic equation be x2 - Sx + P = 0, where S is the correct sum of the roots and P is the correct product. Riya made a mistake in the constant term P, but her sum of the roots is correct, so the correct sum S is 3 + 7 = 10. Kabir made a mistake in the coefficient of x S, but his product of the roots is correct, so the correct product P is 1 times 16 = 16. The correct quadratic equation is x2 - 10x + 16 = 0. Factoring this equation gives (x - 2)(x - 8) = 0, which means the correct roots are 2 and 8.

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