If each pair of three equations $x^{2} + p x + q r = 0 m, x^{2} + q x + p r = 0$ and $x^{2} + r x + p q = 0$ have a common root, find the sum and the product of the common roots.

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Solution

Let $α, β, γ$ be the common root
$∴ α^{2} + p α + q r = 0; β^{2} + q β + p r = 0$
$α^{2} + q α + p r = 0; β^{2} + r β + p q = 0$
$γ^{2} + γ p + q r = 0; γ^{2} + r γ + p q = 0$
$∴ - p = α + γ; q r = α γ$
$∴ - q = α + β; p r = α β$
$∴ - r = β + γ; p q = β γ$
$∴ - (p + q + r) = 2 (α + β + γ)$
$∴ α + β + γ = - \frac{(p + q + r)}{2}$
$∴ (q r) (p r) (p q) = α^{2} β^{2} γ^{2}$
$∴ p q r = α β γ$

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If each pair of three equations x2+px+qr=0m,x2+qx+pr=0 and x2+rx+pq=0 have a common root, find the sum and the product of the common roots.

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If each pair of three equations $x^{2} + p x + q r = 0 m, x^{2} + q x + p r = 0$ and $x^{2} + r x + p q = 0$ have a common root, find the sum and the product of the common roots.