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: Find the product of the roots.
Let the roots be \(\alpha,\beta\) with \(\alpha+\beta=11\) and \(\alpha^2+\beta^2=61\).
\(\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta \Rightarrow 61=121-2\alpha\beta \Rightarrow \alpha\beta=30\)

Step 2: Verify by finding the actual roots.
We need two numbers with sum 11 and product 30. Testing 5 and 6: \(5+6=11\) and \(5\times 6=30\), both check out, and \(5^2+6^2=25+36=61\) confirms them.

Step 3: Build the quadratic.
\(x^2-(\alpha+\beta)x+\alpha\beta=0 \Rightarrow x^2-11x+30=0\), which indeed factors as \((x-5)(x-6)=0\).

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