Common root of quadratic.

Question

If the quadratic equations, $x^2+bx+c=0$ & $bx^2+cx+1=0$ have a common root. Prove that either $b+c+1=0$ or $b^2+c^2+1=bc+b+c$.

Please also explain What should be the logic / approach we should use to solve these kind of problems?

Answer 1

If $y$ is the common root

$$y^2+by+c=0, by^2+cy+1=0$$

Solve for $\displaystyle y,y^2$ to find $\displaystyle y=\frac{bc-1}{c-b^2},y^2=\frac{b-c^2}{c-b^2}$

Using $\displaystyle y^2=(y)^2, \frac{b-c^2}{c-b^2}=\left(\frac{bc-1}{c-b^2}\right)^2$

Assuming $\displaystyle c-b^2\ne0,$

$\displaystyle\implies (b-c^2)(c-b^2)=(bc-1)^2\iff b^3+c^3+1^3-3b\cdot c\cdot1=0$

Now use from If $a,b,c \in R$ are distinct, then $-a^3-b^3-c^3+3abc \neq 0$.,

$$A^3+B^3+C^3=(\sum A)(\sum A^2-\sum BC)$$