Question

Which of the following quadratic equations whose real roots \(x_1,x_2\) satisfy the conditions \[ x_1^2+x_2^2=5, \quad 3(x_1^5+x_2^5)=11(x_1^3+x_2^3)? \]

• A. \(x^2+3x+2=0\)
• B. \(x^2+3x+11=0\)
• C. \(x^2+5x+2=0\)
• D. \(x^2+5x+11=0\)

Hint:

For option-based quadratic root questions, it is often quickest to factorize the options and directly verify the given root conditions.

Answer 1

The Correct Option is A

Step 1: Test option (1) directly.
For \(x^2+3x+2=0\), roots are \(x_1=-1, x_2=-2\). Check: \(x_1^2+x_2^2=1+4=5\).

Step 2: Verify the second condition.
\(3(x_1^5+x_2^5)=3((-1)^5+(-2)^5)=3(-1-32)=-99\). \(11(x_1^3+x_2^3)=11(-1-8)=-99\). Both match. The answer is \(x^2+3x+2=0\). \[ \boxed{x^2+3x+2=0} \]