please help me with this question
for which real values of b do the equations
x^3 + bx^2 + 2bx -1 =0 and x^2 + (b-1)x + b = 0
have a common root
thank you
Let $\displaystyle x_0$ be the common root.
$\displaystyle \displaystyle\left\{\begin{array}{cc}x_0^3+bx_0^2+ 2bx_0-1=0 & (1)\\
x_0^2+(b-1)x_0+b=0 & (2)\end{array}\right.$
We observe that $\displaystyle x_0\neq 0$.
Multiplying (2) by $\displaystyle x_0$ and then substracting from (1), we have
$\displaystyle x_0^2+bx_0-1=0\Rightarrow x_0^2+bx_0=1$ (3)
The equality (2) can be written as $\displaystyle x_0^2+bx_0-x_0+b=0$
Using (3) we have $\displaystyle 1-x_0+b=0\Rightarrow x_0=b+1$.
Plug $\displaystyle x_0$ in (2) and we have $\displaystyle 2b^2+3b=0\Rightarrow b_1=0, \ b_2=\displaystyle-\frac{3}{2}$.
If $\displaystyle b=0$, the common root is $\displaystyle x_0=1$.
If $\displaystyle \displaystyle b=-\frac{3}{2}$, the common root is $\displaystyle \displaystyle x_0=-\frac{1}{2}$.