Question

If every pair among the equations $x^2+ax+bc=0,x^2+bx+ca=0$ and $x^2+cx+ab=0$ have a common root, then which of the following is/are correct?

• A. Product of the common roots is $abc$
• B. Product of the common roots is $-abc$
• C. Sum of the common roots is $\frac{1}{2}(a+b+c)$
• D. Sum of the common roots is $-\frac{1}{2}(a+b+c)$

Answer 1

Answer: (A), (B), (D)

Sum of the common roots is $-\frac{1}{2}(a+b+c)$

Assuming $x^2+ax+bc=0\cdots \left(1\right)$ has roots $\alpha ,\beta$
$x^2+bx+ca=0\cdots \left(2\right)$ has roots $\beta ,\gamma$
$x^2+cx+ab=0\cdots \left(3\right)$ has roots $\alpha ,\gamma$

Therefore,
$\alpha +\beta =-a,\alpha \beta =bc$
$\beta +\gamma =-b,\beta \gamma =ca$
$\gamma +\alpha =-c,\gamma \alpha =ab$

Adding, we get
$2(\alpha +\beta +\gamma )=-(a+b+c)\Rightarrow \alpha +\beta +\gamma =-\frac{1}{2}(a+b+c)$

Also, by multiplying, we get
${\alpha }^2{\beta }^2{\gamma }^2=a^2{b}^2{c}^2\Rightarrow \alpha \beta \gamma =\pm abc$