Question

If the roots of the equations $x^2-bx+c=0$ and $x^2-cx+b=0$ differ by the same quantity, then $b+c$ is equal to

• A. 4
• B. 0
• C. 1
• D. -4

Answer 1

The correct option is D

-4

Let the roots are $\alpha ,\beta$ of $x^2-bx+c=0$ and ${\alpha }^{\prime },{\beta }^{\prime }$ be roots of $x^2-cx+b=0$

Now, $\alpha -\beta =\sqrt{(\alpha +\beta {)}^2-4\alpha \beta }=\sqrt{b^2-4c}$ ..........(i)

$and{\alpha }^{\prime }-{\beta }^{\prime }=\sqrt{(\alpha +\beta {)}^2-4{\alpha }^{\prime }{\beta }^{\prime }}=\sqrt{c^2-4b}$ .........(ii)

$But\alpha -\beta ={\alpha }^{\prime }-{\beta }^{\prime }\Rightarrow \sqrt{b^2-4c}=\sqrt{c^2-4b}\Rightarrow b^2-4c=c^2-4b\Rightarrow b^2-c^2=4c-4b\Rightarrow (b+c)(b-c)=4(c-b)\Rightarrow b+c=-4$