Question

If one root of the equation $x^2+2x+3k=0$ and $2x^2+3x+5k=0$ is common then the value of $k$ is-

• A. $1,2$
• B. $0,-1$
• C. $1,3$
• D. $2$

Answer 1

Answer: (B)

$0,-1$

Only one root is common: let $\alpha$ be the common root

$\frac{{\alpha }^2}{b_1{c}_2-b_2{c}_1}=\frac{\alpha }{a_2{c}_1-a_1{c}_2}=\frac{1}{a_1{b}_2-a_2{b}_1}$

Above Formula is used to find common root between

$a_1{x}^2+b_{1}x+c_1=0$

$a_2{x}^2+b_{2}x+c_2=0$

Now comparing Given Equations with above Mentioned standard Equation

we get

$a_1=1,b_1=2,c_1=3k$ and $a_2=2,b_2=3,c_2=5k$

Now Putting above values in the given formula, we get

$\frac{{\alpha }^2}{k}=\frac{\alpha }{k}=\frac{1}{-1}$

now ${\alpha }^2=-k$ .....(1)

and $\alpha =-k$ .......(2)

Putting Value of $\alpha$ in (1)

we get

$\Rightarrow$ $k^2=-k$

$\Rightarrow$ $k^2+k=0$

$\Rightarrow$ $k(k+1)=0$ ......... (3)

$k=0,-1$ are the only values that are satisfying the equation$\left(3\right)$

Therefore answer is $\left(B\right)$