Question

The value of ${}^{\prime }{a}^{\prime }$ for which the equations $x^2-3x+a=0$ and $x^2+ax-3=0$ have a common root is

• A. $3$
• B. $1$
• C. $-2$
• D. $2$

Answer 1

The correct option is D $2$

Given

$x^2-3x+a=0$

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

lly $x^2+ax-3=0$

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

we get

$\alpha +\beta =3--\left(1\right),\alpha \beta =a--\left(2\right)$

$\beta +\gamma =-a,\beta \gamma =-3$

we get

$\alpha -\beta =3+a$

$\beta (\alpha -\gamma )=a+3$

$(\alpha -\gamma )=\beta (\alpha -\gamma )$

$\because \alpha -\gamma \ne 0$

$\beta =1$

put $\beta =1$ in eq $\left(1\right)$

$\alpha =2$

$\alpha =2,\beta =1$ in eq $\left(2\right)$

$a=2$