[JEE Advanced 1979] if alpha beta are roots of x 2 px q 0 and gamma delta
If $\alpha,\beta$ are roots of $x^2+px+q=0$ and $\gamma,\delta$ of $x^2+rx+s=0$, evaluate $(\alpha-\gamma)(\alpha-\delta)(\beta-\gamma)(\beta-\delta)$ and deduce the common-root condition.
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· 2d ago
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Answer: $(s-q)^2-p(r-p)(s-q)+q(r-p)^2$; common root $\iff$ this $=0$.
Since $\alpha^2=-p\alpha-q$, $f(\alpha)=(r-p)\alpha+(s-q)$ where $f(x)=x^2+rx+s$. Multiplying $f(\alpha)f(\beta)$ and using $\alpha+\beta=-p,\alpha\beta=q$ gives the stated expression; it vanishes exactly when the equations share a root.
JEE Advanced 1979 · Quadratic Equations and Inequations — verified solution by the Vidaara Team.
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