Q1. [1] § 2.3 Relationship between Zeroes and Coefficients of a Polynomial § 2.4 Summary
If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $p(x) = x^2 - ax - b$, then the value of $\alpha^2 + \beta^2$ is:
- (a) $a^2 - 2b$
- (b) $a^2 + 2b$
- (c) $b^2 - 2a$
- (d) $b^2 + 2a$
Previously asked in CBSE board exam
2023 30/2/1 Q16
Generated by claude-sonnet-4-6 · 2026-06-15 10:35 · grounding rag
Model Answer
(b) $a^2 + 2b$
For $p(x) = x^2 - ax - b$: $\alpha + \beta = a$ and $\alpha\beta = -b$.
$$\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta = a^2 - 2(-b) = a^2 + 2b$$
Source: Chapter 2, Section 2.3
Explanation
Use the identity $\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta$. From the polynomial $x^2-ax-b$ (comparing with $ax^2+bx+c$): sum of zeroes $=a$, product of zeroes $=-b$. Substituting gives $a^2+2b$. Watch the sign of the product carefully — it is a common error point.
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