Question

If the equations $x^2+px+q=0$ and $x^2+p^{\prime }x+q^{\prime }=0$ have a common root, then it must be equal to

• A. $\sqrt{\frac{p{q}^{\prime }-p^{\prime }q}{q-q^{\prime }}}$
• B. $\frac{q-q^{\prime }}{p^{\prime }-p}$
• C. $\frac{p^{\prime }-p}{q-q^{\prime }}$
• D. $\frac{p{q}^{\prime }-p^{\prime }q}{p-p^{\prime }}$

Answer 1

Answer: (A), (B)

The correct options are
A $\frac{q-q^{\prime }}{p^{\prime }-p}$
B $\sqrt{\frac{p{q}^{\prime }-p^{\prime }q}{q-q^{\prime }}}$

Given,
$x^2+px+q=0$ and $x^2+p^{\prime }x+q^{\prime }=0$
Let $\alpha$ be the common root.
$\Rightarrow {\alpha }^2+p\alpha +q=0$ and ${\alpha }^2+p^{\prime }\alpha +q^{\prime }=0$
Solving by Cramer's rule,
$\frac{{\alpha }^2}{p{q}^{\prime }-q{p}^{\prime }}=\frac{\alpha }{q-q^{\prime }}=\frac{1}{p^{\prime }-p}$
$\Rightarrow {\alpha }^2=\frac{p{q}^{\prime }-p^{\prime }q}{q-q^{\prime }}$ and $\alpha =\frac{q-q^{\prime }}{p^{\prime }-p}$
Hence option 'A' and 'B' are correct.