Question

The sum of the values of '$m$' for which the equations $3x^2+4mx+2=0$ and $2x^2+3x-2=0$ may have a common root, is

• A. $\frac{3}{8}$
• B. $\frac{3}{4}$
• C. $\frac{5}{4}$
• D. $\frac{5}{8}$

Answer 1

The correct option is A $\frac{3}{8}$

$3x^2+4mx+2=0$
and $2x^2+3x-2=0$ have a common root.
$2x^2+3x-2=0$
$\Rightarrow (x+2)(2x-1)=0$
$\Rightarrow x=\frac{1}{2},-2$

If $x=\frac{1}{2}$ is the common root,
then $3\cdot {\left(\frac{1}{2}\right)}^2+4m\cdot \frac{1}{2}+2=0$
$\Rightarrow m=\frac{-11}{8}$

If $x=-2$ is the common root,
then $3\cdot {(-2)}^2+4m(-2)+2=0$
$\Rightarrow m=\frac{14}{8}$

Sum of the values of $m$ is,
$\frac{14}{8}+\frac{-11}{8}=\frac{3}{8}$


Alternate:
$(c_1{a}_2-c_2{a}_1{)}^2=(a_1{b}_2-a_2{b}_1\left)\right(b_1{c}_2-b_2{c}_1)$
$\Rightarrow (4+6{)}^2=(9-8m\left)\right(-8m-6)$
$\Rightarrow 100=(8m-9)(8m+6)$
$\Rightarrow 32m^2-12m-77=0$
$\therefore$ Sum of values of $m$ is $\frac{12}{32}=\frac{3}{8}$