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

Step 1: Read off the sum and product.
For $x^2-5x+6=0$, the sum of roots is the negative of the middle term and the product is the last term.
\[ \alpha+\beta = 5, \qquad \alpha\beta = 6 \]

Step 2: Use a handy identity.
We can write the sum of squares using the sum and product.
\[ \alpha^2+\beta^2 = (\alpha+\beta)^2 - 2\alpha\beta \]

Step 3: Put in the numbers.
\[ = 5^2 - 2 \times 6 = 25 - 12 = 13 \]
\[ \boxed{13} \]