First slide

Theory of equations

Question

A value of $b$ for which the equations $x^{2} + b x - 1 = 0$ and $x^{2} + x + b = 0$ have one root in common, is

Moderate

A

$\sqrt{2}$
B

$- i \sqrt{3}$
C

$i \sqrt{5}$
D

$- \sqrt{2}$

Solution

Let $α$ be a common root of the given equations.
Then,
$α^{2} + b α - 1 = 0$ …(i)
$α^{2} + α + b = 0$ …(ii)
On subtracting, we get
$(b - 1) a = b + 1 > a = \frac{b + 1}{b - 1}$
Substituting the value of $α$ in (i), we get
$(b + 1)^{2} + b (b + 1) (b - 1) - (b - 1)^{2} = 0$
$\begin{array}{l} \Rightarrow b (b^{2} - 1) + 4 h = 0 \\ \Rightarrow b (b^{2} + 3) = 0 \Rightarrow b = 0, \pm i \sqrt{3} \end{array}$

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