Multiple choice

The equation ax2 + bx + c = 0 has two roots, δ and θ. If δ2 + θ2 = 10 and the sum of the roots is 6, which of the following could represent (a, b, c)? A. (1, -6, 9) B. (2, -12, 26) C. (1, 4, -10)

  1. Only A

  2. Only B

  3. Only C

  4. Only A and B

  5. All A, B, and C
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given roots delta and theta, delta + theta = -b/a = 6 and delta * theta = c/a. Since delta^2 + theta^2 = (delta + theta)^2 - 2(delta * theta) = 10, we have 6^2 - 2(c/a) = 10, so 36 - 2(c/a) = 10, which means 2(c/a) = 26, or c/a = 13. Option A: c/a = 9/1 = 9 (No). Option B: c/a = 26/2 = 13 (Yes). Option C: c/a = -10/1 = -10 (No).

AI explanation

Using the relationship between roots and coefficients, the sum of the roots gives the equation b divided by a equals negative 6. The sum of the squares of the roots is found using the identity delta squared plus theta squared equals the sum squared minus twice the product, giving 10 equals 36 minus twice the product, so the product is 13 and c divided by a equals 13. Testing the options, only the triplet (2, -12, 26) satisfies both ratios, since -12 divided by 2 is negative 6 and 26 divided by 2 is 13. Therefore, only B is correct.

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