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If $\alpha, \beta$ are the roots of $a x^2+b x+c=0$, then the quadratic equation whose roots are $\sqrt{5} \alpha, \sqrt{5} \beta$ is
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Verified Answer
The correct answer is:
$a x^2+\sqrt{5} b x+5 c=0$
$\alpha$ and $\beta$ are roots of $a x^2+b x+c=0$
Consider $\sqrt{5} \alpha=t \Rightarrow \alpha=\frac{t}{\sqrt{5}}$
$$
\begin{aligned}
& \therefore a\left(\frac{t}{\sqrt{5}}\right)^2+b\left(\frac{t}{\sqrt{5}}\right)+c=0 \\
& \Rightarrow \quad \frac{a t^2}{5}+\frac{b t}{\sqrt{5}}+c=0 \Rightarrow a t^2+b \sqrt{5} t+5 c=0
\end{aligned}
$$
$\therefore$ Equation whose roots are $\sqrt{5} \alpha, \sqrt{5} \beta$ is
$$
a x^2+b \sqrt{5} x+5 c=0
$$
Consider $\sqrt{5} \alpha=t \Rightarrow \alpha=\frac{t}{\sqrt{5}}$
$$
\begin{aligned}
& \therefore a\left(\frac{t}{\sqrt{5}}\right)^2+b\left(\frac{t}{\sqrt{5}}\right)+c=0 \\
& \Rightarrow \quad \frac{a t^2}{5}+\frac{b t}{\sqrt{5}}+c=0 \Rightarrow a t^2+b \sqrt{5} t+5 c=0
\end{aligned}
$$
$\therefore$ Equation whose roots are $\sqrt{5} \alpha, \sqrt{5} \beta$ is
$$
a x^2+b \sqrt{5} x+5 c=0
$$
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