Question

Let \(\alpha\) and \(\beta\) be the roots of \[ x^2+bx+c=0. \] If \[ \alpha^2+\beta^2=14 \] and \[ \alpha\beta=3, \] then the value of \(b^2\) is:

• A. \(8\)
• B. \(16\)
• C. \(20\)
• D. \(28\)

Hint:

Whenever a question involves roots of a quadratic equation, immediately write \[ \alpha+\beta=-\frac{\text{coefficient of }x}{\text{coefficient of }x^2}, \qquad \alpha\beta=\frac{\text{constant term}}{\text{coefficient of }x^2}. \] Most root-based problems become straightforward after applying these relations.

Answer 1

The Correct Option is C

Concept: For a quadratic equation \[ x^2+bx+c=0, \] the roots satisfy \[ \alpha+\beta=-b, \] and \[ \alpha\beta=c. \] The identity \[ \alpha^2+\beta^2 = (\alpha+\beta)^2-2\alpha\beta \] is the key relation.

Step 1: Use the given information Given \[ \alpha^2+\beta^2=14 \] and \[ \alpha\beta=3. \] Substituting into the identity, \[ 14 = (\alpha+\beta)^2 -2(3). \] \[ 14 = (\alpha+\beta)^2-6. \] \[ (\alpha+\beta)^2 = 20. \]

Step 2: Relate to coefficient \(b\) Since \[ \alpha+\beta=-b, \] we have \[ b^2=(\alpha+\beta)^2. \] Therefore, \[ b^2=20. \] Hence, \[ \boxed{20} \]