Question

If the quadratic equations $3x^2+ax+1=0$ and $2x^2+bx+1=0$ have a common root, then the value of the expression $5ab-2a^2-3b^2$ is

• A. $0$
• B. $1$
• C. $-1$
• D. None of these

Answer 1

Answer: (B)

$1$

Quadratic equations $3x^2+ax+1=0$ and $2x^2+bx+1=0$ have a common root.

Let $\alpha$ be the common root.
$\Rightarrow 3{\alpha }^2+a\alpha +1=0\dots \left(1\right)$

and $2{\alpha }^2+b\alpha +1=0\dots \left(2\right)$

Multiplying $\left(1\right)$ by $2$ and $\left(2\right)$ by $3$.

Further solving those two equations for $\alpha$ gives

$\alpha =b-a\dots \left(3\right)$

Substituting $\left(3\right)$ in$\left(1\right)$

$\Rightarrow 2(b-a{)}^2+b(b-a)+1=0$

$\Rightarrow 3b^2+2a^2-5ab+1=0$

$\therefore 5ab-3b^2-2a^2=1$

Hence, option B.