Multiple choice

If the roots of the equation x2 - bx + c = 5 differ by 5, then which one of the following is correct?

  1. b2 = 4c + 5

  2. c2 = 4b - 5

  3. b2 + c2 = 5

  4. b2 - c2 = 5
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A Correct answer
Explanation

Let roots be r1 and r2. r1 - r2 = 5. Sum of roots = b, product = c-5. (r1 - r2)^2 = (r1 + r2)^2 - 4r1r2. 5^2 = b^2 - 4(c-5). 25 = b^2 - 4c + 20. b^2 - 4c = 5, or b^2 = 4c + 5.

AI explanation

Rewriting the equation gives x2 - bx + c - 5 = 0, so the product of the roots is c - 5 and the sum is b. Since the roots differ by 5, using the identity (root1 - root2)2 = (sum)2 - 4(product) gives 25 = b2 - 4(c - 5). Simplifying this relationship yields b2 = 4c + 5.

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