Question

If the equations \(4x^2 -x-1 = 0\) and \(3x^2 +(\lambda + \mu)x + \lambda - \mu = 0\) have a common root, then the rational values of \(\lambda\) and \(\mu\) are

Answer 1

\(4x^2 -x-1=0\)
\(\Rightarrow x= \dfrac{1 \pm \sqrt{17}}{2}\)
Roots of \(4x^2 -x-1=0\) are irrational and exist as conjugate pairs.

$\therefore$ One root common implies both roots are common as coefficients of $2$nd equation are also rational.
\(\therefore \dfrac{4}{3} = \dfrac{-1}{\lambda + \mu} = \dfrac{-1}{\lambda - \mu} \)
On solving, we get
\(\lambda = \dfrac{-3}{4}, \mu = 0\)