Question

If equations $a{x}^2+bx+c=0$, ($a,b,cϵR,a\ne 0$) and $2x^2+3x+4=0$ have a common root, then $a:b:c$ equals to

• A. $1:1:3$
• B. $1:2:3$
• C. $2:3:4$
• D. $3:4:5$

Answer 1

The correct option is C $2:3:4$

$\Rightarrow$ The given quadratic equation is $2x^2+3x+4=0$

Now, we are going to find roots of the given equation.

$x=\frac{-3\pm \sqrt{(3{)}^2-4(2\left)\right(4)}}{\left(2\right)\left(2\right)}$

$x=\frac{-3\pm \sqrt{9-32}}{4}$

$x=\frac{-3\pm \sqrt{-23}}{4}$

$\therefore$ $x=\frac{-3\pm \sqrt{23}i}{4}$

So, here $2x^2+3x+4=0$ will give complex roots.

It is given that $a{x}^2+bx+c=0$ and $2x^2+3x+4=0$ have a common root.

So, $a{x}^2+bx+c=0$ also ahve these complex pairs.

This means least value will be obtained at $a=2,b=3$ and $c=4$

$\therefore$ $a:b:c=2:3:4$