Multiple choice

If f(x) = x2 + bx + c be a quadratic function with real coefficients, and the roots of the function are α and β such that (α + β = 9) and α2 + β2 = 41, find the quadratic function f(x).

  1. x2 - 9x + 20

  2. x2 + 9x - 20

  3. x2 - 9x - 20

  4. x2 + 9x + 20

  5. None of these
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A Correct answer
Explanation

Given the sum of roots alpha + beta = 9 and the sum of squares alpha^2 + beta^2 = 41, we use the identity (alpha + beta)^2 = alpha^2 + beta^2 + 2*alpha*beta. Substituting the values, 81 = 41 + 2*alpha*beta, which gives alpha*beta = 20. The quadratic equation is x^2 - (sum of roots)x + (product of roots) = 0, resulting in x^2 - 9x + 20.

AI explanation

Using the algebraic identity for the sum of squares, alpha squared plus beta squared equals the square of the sum of the roots minus twice their product. Substituting the given values gives 41 equals 9 squared minus 2 alpha beta, so alpha beta equals 20. Since the sum of the roots is negative b and the product is c, the function is f of x equals x squared minus 9 x plus 20.

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