Multiple choice

The quadratic equation 2x2 + dx + e = 0 has two roots 5a and 2a, where a is an integer. Which of the following is a possible value of d2 - e?

  1. 702

  2. 1586

  3. 2114

  4. 2816
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D Correct answer
Explanation

Roots are 5a and 2a. Sum of roots = -d/2 = 7a => d = -14a. Product of roots = e/2 = 10a^2 => e = 20a^2. d^2 - e = (-14a)^2 - 20a^2 = 196a^2 - 20a^2 = 176a^2. For a=4, 176*16 = 2816.

AI explanation

For the equation 2x2 + dx + e = 0 with roots 5a and 2a, the sum of the roots is 7a and the product is 10a2. Using the relationships between roots and coefficients, we have -d/2 = 7a and e/2 = 10a2, which means d = -14a and e = 20a2. We need to find a possible value for d2 - e, so substitute the expressions for d and e to get (-14a)2 - 20a2. This simplifies to 196a2 - 20a2, which equals 176a2. Testing the options, only 2816 yields a perfect integer when divided by 176, since 2816 divided by 176 gives a2 = 16 and a = 4. The possible value is 2816.

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