Question

The quadratic equation whose roots are three times the roots of the equation $2x^2+3x+5=0,$ is

• A. $2x^2+9x+45=0$
• B. $2x^2+9x-45=0$
• C. $5x^2+9x+45=0$
• D. $2x^2-9x+45=0$
• E. $2x^2+9x+49=0$

Answer 1

Answer: (A)

Let the roots of the given equation $2x^2+3x+5=0$ be $\alpha$ and $\beta$.
Then, $\alpha +\beta =-\frac{3}{2}$ and $\alpha \beta =\frac{5}{2}$
Let the roots of required equation be ${\alpha }^{\prime }$ and ${\beta }^{\prime }$. It is given that,
${\alpha }^{\prime }=3\alpha$ and${\beta }^{\prime }=3\beta$
Now,${\alpha }^{\prime }+{\beta }^{\prime }=3\alpha +3\beta$
$=3(\alpha +\beta )$
$=3\left(-\frac{3}{2}\right)=-\frac{9}{2}$
Also${\alpha }^{\prime }\cdot {\beta }^{\prime }=(3\alpha )(3\beta )=9\alpha \beta$
$=9\left(\frac{5}{2}\right)=\frac{45}{2}$
Hence, required equation is
$x^2-\left(-\frac{9}{2}\right)x+\frac{45}{2}=0$
$\Rightarrow 2x^2+9x+45=0$