Quadratic EquationHard
Question
Let $f(x) = x^{3} - 3x + b$ and $g(x) = x^{2} + bx - 3$, where b is a real number. If the equations $f(x) = 0$ and $g(x) = 0$ have a common root, then
Options
A.Number of possible values of $b$ is 3 .
B.Number of possible value of $b$ is 2 .
C.Sum of all possible values of $b$ is 0 .
D.Sum of all possible values of $b$ is 2 .
Solution
Let x be the common root of equation.
$$\begin{array}{r} x^{3} - 3x + b = 0\#(1) \\ x^{2} + bx - 3 = 0\#(2) \end{array}$$
(2) $\times x -$ (1) $\Rightarrow {bx}^{2} - b = 0 \Rightarrow \text{ }b = 0$ or $x = \pm 1$.
$$\begin{matrix} \text{~Put~} & x = 1, \Rightarrow b = 2 \\ & x = - 1, \Rightarrow b = - 2 \\ \therefore & b = 0,2, - 2 \end{matrix}$$
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