If $\alpha$ and $\beta$ are roots of the quadratic equation $x^2 - 7x + 10 = 0$, find the quadratic equation whose roots are $\alpha^2$ and $\beta^2$.
Generated by claude-sonnet-4-6 · 2026-06-15 10:33 · grounding rag
Model Answer
For $x^2 - 7x + 10 = 0$, by Vieta's formulas:
$$\alpha + \beta = 7, \quad \alpha\beta = 10$$
For the new equation with roots $\alpha^2$ and $\beta^2$:
Sum of new roots:
$$\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = (7)^2 - 2(10) = 49 - 20 = 29$$
Product of new roots:
$$\alpha^2 \cdot \beta^2 = (\alpha\beta)^2 = (10)^2 = 100$$
The required quadratic equation is:
$$x^2 - (\alpha^2 + \beta^2)x + \alpha^2\beta^2 = 0$$
$$\boxed{x^2 - 29x + 100 = 0}$$
Source: Chapter 4, Quadratic Equations
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Explanation
- Examiners award marks for: (1) correctly finding sum and product from original equation, (2) computing sum and product of new roots, (3) writing the final equation.
- The key identity needed is $\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta$ — write it explicitly to earn step marks.
- The standard form of a quadratic with known sum (S) and product (P) of roots is $x^2 - Sx + P = 0$. Always quote this form.