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

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- Sep 6th 2007, 06:10 AMieatfoodfinding common roots
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 - Sep 6th 2007, 07:26 AMred_dog
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 $\displaystyle \displaystyle x_0=-\frac{1}{2}$.