Question

If \(\alpha\) and \(\beta\) are the roots of \(x^2-5x+6=0\), the value of \(\alpha^2+\beta^2\) is: 

• A. 13
• B. 25
• C. 10
• D. 15 Correct Answer: (A) 13 Solution:

Hint:

Remember the identity: \[ \alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta \] It is frequently used in questions involving roots of quadratic equations.

Answer 1

The Correct Option is A

Concept: For a quadratic equation \[ ax^2+bx+c=0 \] with roots \(\alpha\) and \(\beta\), \[ \alpha+\beta=-\frac{b}{a} \] and \[ \alpha\beta=\frac{c}{a} \] Also, \[ \alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta \]

Step 1: Find the sum and product of roots. Given: \[ x^2-5x+6=0 \] Thus, \[ \alpha+\beta=5 \] and \[ \alpha\beta=6 \]

Step 2: Apply the identity. \[ \alpha^2+\beta^2 =(\alpha+\beta)^2-2\alpha\beta \] Substituting values: \[ =(5)^2-2(6) \] \[ =25-12 \] \[ =13 \]

Step 3: Write the final answer. \[ \boxed{\alpha^2+\beta^2=13} \] Hence, the correct option is \(\boxed{(A)}\).