Quadratic Equations

Question

If α and β are the roots of x² - px + q = 0, then the equation whose roots are α² and β² is

RankGuru · Accepted Answer

1. **Original Vieta**: $\alpha + \beta = p, \alpha\beta = q$.
2. **New Sum**: We need $S = \alpha^2 + \beta^2$.
   - $S = (\alpha + \beta)^2 - 2\alpha\beta = p^2 - 2q$.
3. **New Product**: We need $P = \alpha^2\beta^2 = (\alpha\beta)^2 = q^2$.
4. **Equation Construction**: Any quadratic equation is $x^2 - (	ext{Sum})x + (	ext{Product}) = 0$.
5. **Substitution**: $x^2 - (p^2 - 2q)x + q^2 = 0$.

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