Question

If $\alpha$ and $\beta$ are the roots of the equation $ax^2 + bx + c = 0$, then what is the value of $(\alpha^2 + \beta^2)$?
I. $\alpha + \beta = -\left(\frac{b}{a}\right)$.
II. $2\alpha\beta = \left(\frac{c}{a}\right)$.

• A. The question cannot be answered even with the help of both the statements taken together.
• B. The question can be answered by any one of the statements
• C. Each statement alone is sufficient to answer the question, but not the other one.
• D. Both statements I and II together are needed to answer the question

Hint:

For symmetric expressions in roots, you usually need both sum and product of roots.

Answer 1

The Correct Option is D

We know: $\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta$.
I: Gives $\alpha+\beta$, but without $\alpha\beta$, cannot compute.
II: Gives $\alpha\beta$, but without $\alpha+\beta$, cannot compute.
Together: Both $\alpha+\beta$ and $\alpha\beta$ are known $\Rightarrow$ $\alpha^2+\beta^2$ can be found. %Quick tip