Question

If $x^2 - 11x + a$ and $x^2 - 14x + 2a$ will have a common factor, then a =

• A. 24
• B. 0, 24
• C. 3, 24
• D. 0, 3

Answer 1

Answer: (B)

0, 24

Answer 2

To find the values of 'a' for which the two quadratic expressions $x^2 - 11x + a$ and $x^2 - 14x + 2a$ have a common factor, we can assume they share a common root, say $\alpha$. If $\alpha$ is a common root, it must satisfy both equations:

1. $\alpha^2 - 11\alpha + a = 0$
2. $\alpha^2 - 14\alpha + 2a = 0$

Subtract equation (1) from equation (2): $(\alpha^2 - 14\alpha + 2a) - (\alpha^2 - 11\alpha + a) = 0$ $\alpha^2 - 14\alpha + 2a - \alpha^2 + 11\alpha - a = 0$ $-3\alpha + a = 0$

This gives us a relationship between $a$ and $\alpha$: $a = 3\alpha$

Now substitute $a = 3\alpha$ into equation (1): $\alpha^2 - 11\alpha + (3\alpha) = 0$ $\alpha^2 - 8\alpha = 0$

Factor out $\alpha$: $\alpha(\alpha - 8) = 0$

This equation yields two possible values for $\alpha$:

Case 1: $\alpha = 0$ Case 2: $\alpha = 8$

Now, we find the corresponding values of 'a' using $a = 3\alpha$:

Case 1: If $\alpha = 0$ $a = 3 \times 0$ $a = 0$

Case 2: If $\alpha = 8$ $a = 3 \times 8$ $a = 24$

Thus, the values of 'a' for which the two expressions have a common factor are 0 and 24.