Multiple choice

The equation $\displaystyle x^{2}+Bx+C=0$ has 5 as the sum of its roots and 15 as the sum of the square of its roots. The value of C is

  1. $5$
  2. $7.5$
  3. $10$
  4. $12.5$
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A Correct answer
Explanation

Let roots be a, b. a+b = 5. a^2+b^2 = 15. (a+b)^2 = a^2+b^2+2ab. 25 = 15 + 2ab, so 2ab = 10, ab = 5. The product of roots is C/1 = ab = 5.

AI explanation

Using the identity for the sum of squares of roots, we write alpha squared plus beta squared as (alpha plus beta) squared minus 2 times alpha times beta. The sum of the roots alpha plus beta equals minus B, so this becomes B squared minus 2C, which we set equal to 15. Since the sum of the roots is 5, we have minus B equals 5, meaning B equals minus 5. Substituting this into the sum of squares equation gives 25 minus 2C equals 15, so C equals 5.

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