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: (C)

$5x^2+9x+45=0$

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, ${\alpha }^{{}^{\prime }}+{\beta }^{{}^{\prime }}=3\alpha +3\beta$

$=3(\alpha +\beta )$

$=3(-\frac{3}{2})=-\frac{9}{2}$

Also, ${\alpha }^{{}^{\prime }}\cdot {\beta }^{{}^{\prime }}=\left(3\alpha \right)\left(3\beta \right)=9\alpha \beta$

$=9\left(\frac{5}{2}\right)=\frac{45}{2}$

Hence, required equation is

$x^2-(-\frac{9}{2})x+\frac{45}{2}=0$

$\Rightarrow 2x^2+9x+45=0$