Question

Let $f\left(x\right)=x^3-3x+4b$ and $g\left(x\right)=x^2+bx-3$ where, b is a real number. If the equations $f\left(x\right)=0$ and $g\left(x\right)=0$ have a common root, then

• A. number of possible values of $b$ is $3$
• B. number of possible values of $b$ is $2$
• C. sum of all possible values of $b$ is $0$
• D. sum of all possible values of $b$ is $1$

Answer 1

Answer: (A), (C)

The correct options are
A number of possible values of $b$ is $3$
C sum of all possible values of $b$ is $0$

If $x$ is the common root of both equations, then
$x^3-3x+4b=0$.......$\left(1\right)$
$x^2+bx-3=0$..........$\left(2\right)$
$\left(2\right)\times x-\left(1\right)\Rightarrow$
$b{x}^2-4b=0$
$\Rightarrow b=0$ or $x=\pm 2$
Now, $x=2,\Rightarrow b=-\frac{1}{2}$
and $x=-2,\Rightarrow b=\frac{1}{2}$