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 $2 x^{2}+3 x+5=0$ 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 $
$\Rightarrow 2 x^{2}+9 x+45=0$