First slide

Theory of equations

Question

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

Moderate

A

$- \sqrt{2}$
B

$- i \sqrt{3}$
C

$\sqrt{2}$
D

$\sqrt{3}$

Solution

$x^{2} + bx - 1 = 0$
$x^{2} + x + b = 0$ ….. (1)
Subtracting, we get (b - 1)x - 1 - b = 0
$\Rightarrow x = \frac{b + 1}{b - 1}$
This value of x satisfies equation (1)
$\begin{array}{l} \Rightarrow \frac{(b + 1)^{2}}{(b - 1)^{2}} + \frac{b + 1}{b - 1} + b = 0 \\ \Rightarrow b = \sqrt{3} i, - \sqrt{3} i, 0 \end{array}$

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