Question

solve the problem
Q. If the equations $a{x}^2+bx+c=0andc{x}^2+bx+a=0$ may have a common root then prove that a + b + c or a - b + c = 0.

Answer 1

Dear Student

$$\begin{gathered} Let\alpha bethecommonrootofa{x}^2+bx+c=0andc{x}^2+bx+a=0 \\ \Rightarrow a{\alpha }^2+b\alpha +c=0 \\ c{\alpha }^2+b\alpha +a=0 \\ \Rightarrow \frac{{\alpha }^2}{\left(ab-bc\right)}=\frac{\alpha }{c^2-a^2}=\frac{1}{ab-bc} \\ Fromlasttwoweget \\ \alpha =\frac{c^2-a^2}{ab-bc}=\frac{-\left(a+c\right)}{b} \\ so,forfirstandlastweget \\ {\left(c^2-a^2\right)}^2=\left(ab-bc\right)\left(ab-bc\right) \\ \Rightarrow {\left((c-a)(c+a)\right)}^2=b^2{\left(c-a\right)}^2 \\ \Rightarrow {\left(c+a\right)}^2=b^2 \\ \Rightarrow c+a=\pm b \\ \Rightarrow a+b+c=0 \\ anda-b+c=0 \end{gathered}$$

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