Question

If equations $2x^2+kx-5=0$ and $x^2-3x-4=0$ have a common root, then the value of $k$ are

• A. $-3$ or $-\frac{27}{4}$
• B. $3$ or $\frac{27}{4}$
• C. $-3$ or $\frac{27}{4}$
• D. $3$ or $-\frac{27}{4}$

Answer 1

The correct option is A $-3$ or $-\frac{27}{4}$

$\Rightarrow 2{\alpha }^2+kx-5=0$

$\alpha +\beta =-\frac{k}{2}$, $\alpha \beta =-\frac{5}{2}$

$\to x^2-3x-4=0$

$⟹x^2-3x-4=0$

$⟹x^2-4x+x-4=0$

$⟹x(x-4)+1(x-4)=0$

$\therefore \alpha =-1,\beta =4$

If common root is $-1$,

$\therefore \beta =\frac{5}{2}$

$\therefore k=-3$

If common root is $4$

$\beta =-\frac{5}{8}$

$\therefore k=-\frac{27}{4}$