Question

The quadratic equation whose sum of roots is 11 and sum of squares of roots is 61 is:

• A. \(x^2 + 11x - 30 = 0\)
• B. \(x^2 + 11x + 30 = 0\)
• C. \(x^2 - 11x - 30 = 0\)
• D. \(x^2 - 11x + 30 = 0\)

Hint:

For quadratic with given sum and sum of squares of roots, compute product using \(\alpha^2+\beta^2 = (\alpha+\beta)^2 - 2\alpha\beta\) and then form \(x^2 - (\text{sum})x + (\text{product}) = 0\).

Answer 1

The Correct Option is D

Step 1: Let roots be \(\alpha\) and \(\beta\).
\[ \alpha + \beta = 11, \quad \alpha^2 + \beta^2 = 61 \]

Step 2: Express product of roots using sum of squares.
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta \] \[ 61 = 11^2 - 2\alpha \beta \] \[ 61 = 121 - 2\alpha \beta \implies 2\alpha \beta = 60 \implies \alpha \beta = 30 \]

Step 3: Form the quadratic equation.
\[ x^2 - (\alpha+\beta)x + (\alpha\beta) = 0 \] \[ x^2 - 11x + 30 = 0 \]

Step 4: Final conclusion.
\[ \boxed{x^2 - 11x + 30 = 0} \]