Question

If the quadratic equations $x^2+abx+c=0$ and $x^2+acx+b=0$ have a common root, the equation containing their other roots is/are:

• A. $x^2+a(b+c)x-a^{2}bc=0$
• B. $x^2-a(b+c)x+a^{2}bc=0$
• C. $a(b+c)x^2-(b+c)x+abc=0$
• D. $a(b+c)x^2+(b+c)x-abc=0$

Answer 1

The correct option is B $x^2-a(b+c)x+a^{2}bc=0$

Let, $\alpha$ and $\beta$ be the roots of the equation $x^2+abx+c=0$ then

$\alpha +\beta =-ab$......(1)

$\alpha .\beta =c$.......(2).

And also let $\beta$ and $\gamma$ be the roots of the equation $x^2+acx+b=0$, then

$\beta +\gamma =-ac$......(3)

$\beta .\gamma =b$......(4).

Now, subtracting (2) from (1) we get,

$\alpha -\gamma =a(c-b)$.......(4).

Now, dividing (1) by (2) we get,

$\frac{\alpha }{\gamma }=\frac{c}{b}$

or, $\alpha =\gamma .\frac{c}{b}$.

Putting this in equation (4) we get

$\gamma \left(\frac{c-b}{b}\right)=a(c-b)$

or, $\gamma =ab$.

Using this in equation (4) we get,

$\alpha =ac$.

Now, the quadratic equation formed by $\alpha$ and $\gamma$ is

$x^2-(\alpha +\gamma )x+\alpha .\gamma =0$

or, $x^2-a(b+c)x+a^{2}bc=0$