Multiple choice

For the quadratic equation ax2 + bx + c = 0, if the roots are in the ratio c : a, what can be inferred about their relationship?

  1. b2 = a(c + 1)2

  2. b2 = (c + a)2

  3. b = a2(c + 1)

  4. b = a(c + 1)2
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B Correct answer
Explanation

Let the roots of the quadratic equation be k*c and k*a. The sum of the roots is k(c + a) = -b/a, and the product of the roots is k^2 * a * c = c/a, which simplifies to k^2 = 1/a^2. Squaring the sum equation gives k^2 * (c + a)^2 = b^2 / a^2, and substituting k^2 = 1/a^2 yields b^2 = (c + a)^2.

AI explanation

Let the roots be ck and ak for some constant k, meaning their sum is k(a + c) = -b/a and their product is ack^2 = c/a. From the product equation, we get k^2 = 1/a^2, so k = 1/a or k = -1/a. Substituting k = 1/a into the sum equation gives (a + c)/a = -b/a, which leads to b = -(a + c) and squaring both sides results in b^2 = (a + c)^2.

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