Question

If the equation $a{x}^2+2bx+c=0$ and $a{x}^2+2cx+b=0,a\ne 0,b\ne c$, have a common root, then their other roots are the roots of the quadratic equation

• A. $a^{2}x(x+1)+4bc=0$
• B. $a^{2}x(x+1)+8bc=0$
  
• C. $a^{2}x(x+2)+8bc=0$
  
• D. $a^{2}x(1+2x)+8bc=0$

Answer 1

The correct option is D $a^{2}x(1+2x)+8bc=0$

By condition of common roots

$\Rightarrow$ Common root $\alpha =\frac{1}{2}$ and $\frac{a}{4}+b+c=0$

$\Rightarrow \beta =\frac{2c}{a}$ and $\gamma =\frac{2b}{a}$ (using product of roots)

Equation whose roots are $\beta$ and $\gamma$ is

$x^2-(\frac{2c}{a}+\frac{2b}{a})x+\frac{4bc}{a^2}=0$

$\Rightarrow 2a^2{x}^2+a^{2}x+8bc=0$