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Question: $x^2 + bx + ca = 0 $ and $ x^2 + cx + ab = 0 $ have only one non-zero common root , show that their other roots satisfy $ t^2 + at + bc =0 $ .

I have tried to solve it by first finding the common root which is equal to $-(b + c)$ . Then I found the other roots: $$ \frac{-ca}{b + c} \text{ and } \frac{-ab}{b + c}$$ If they are the roots of the given equation, then their sum is $-a$ and their product is $bc$.

Though I've found the sum is $-a$ but cannot prove their product. Help.

2 Answers2

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HINT: If $y$ is the common root, $$y^2+by+ca=0\ \ \ \ (1),y^2+cy+ab=0\ \ \ \ (2)$$

On subtraction, $(b-c)y=a(b-c)$

Safely assume $b\ne c$(why?) to find $\displaystyle y=a\implies a^2+ab+ca=0\iff a(a+b+c)=0$ (from $(1)$)

From $(2), a(a+c+b)=0$

Check if $a=0$ and if not

Can you take it home from here?

Alternatively solve $(1),(2)$ for $y; y^2$

and use the identity $y^2=(y)^2$

lab bhattacharjee
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  • Are my roots incorrect? I ve followed only cross multiplication and found the other roots by $\dfrac{c_1}{a_1}\dfrac{1}{alpha}$ and $\dfrac{c_2}{a_2}\dfrac{1}{alpha}$ where $alpha$ is the common root. –  Jul 02 '14 at 20:00
  • @ lab bhattacharjee :Thanks for answering; u ve again rescued me. –  Jul 03 '14 at 04:42
  • @user36790, My pleasure. I think you need to derive $a(a+b+c)=0$ – lab bhattacharjee Jul 03 '14 at 05:48
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HINT: The two roots you found should make both equations equal to $0$. Plug the values in and create equalities for the variables $a, b,$ and $c$, and then you will be able to prove the product.

EDIT: Your roots aren't correct, when you find the correct roots, follow the steps I described above.

Vishwa Iyer
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  • @ Vishwa Iyer : mine roots are correct since $ a + b + c = 0$ ,then common root is $a$. Just check it. –  Jul 03 '14 at 04:38
  • @user36790 I understand, but you didn't state that, so when you check for the if the sum of the roots are correct, you don't get an equality. – Vishwa Iyer Jul 03 '14 at 11:10