A value of b for which the equations x 2 + b x - 1 = 0 x 2 + x + b = 0 have one root in common is [2011]

Question

A value of $b$ for which the equations

$x^{2} + b x - 1 = 0$

$x^{2} + x + b = 0$

have one root in common is [2011]

Smart Achievers · Accepted Answer

(2)

$Let α be the common root of given equations, then$

$α^{2} + b α - 1 = 0 . . . (i)$

$and α^{2} + α + b = 0 . . . (i i)$

$On subtracting (ii) from (i), we get$

$(b - 1) α - (b + 1) = 0$

$\Rightarrow α = \frac{b + 1}{b - 1}$

$Substituting this value of α in equation (i), we get$

${(\frac{b + 1}{b - 1})}^{2} + b (\frac{b + 1}{b - 1}) - 1 = 0$

$\Rightarrow b^{3} + 3 b = 0$

$\Rightarrow b = 0, i \sqrt{3}, - i \sqrt{3}$

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