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Quadratic Equation for the Given Roots

Find the valu...

Question

Find the values of ' $x$ ' which satisfies both the equations
$(a - b) x^{2} + (b - c) x + (c - a) = 0$ and $(c - a) x^{2} + (b - c) x + (a - b) = 0$

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Solution

$(a - b) x^{2} + (b - x) x + (c - a) = 0$
$(c - b) x^{2} + (b - c) x + (a - b) = 0$
has common root 'x'
Adding 2 equation we get a new equation which also has root 'X'.
$(c - b) x^{2} - 2 (c - b) x + (c - b) = 0$
(on adding 2 equation)
$\Rightarrow x^{2} - 2 x + 1 = 0$
$\Rightarrow (x - 1)^{2} = 0 \Rightarrow 1$

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