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$2 x^{2}+3 x-\alpha-0$ has roots $-2$ and $\beta$ while the equation $x^{2}-3 m x+$ $2 \mathrm{~m}^{2}=0$ has both roots positive, where $\alpha>0$ and $\beta>0$.
If $\beta, 2,2 \mathrm{~m}$ are in GP, then what is the value of $\beta \sqrt{\mathrm{m}}$ ?
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If $\beta, 2,2 \mathrm{~m}$ are in GP, then what is the value of $\beta \sqrt{\mathrm{m}}$ ?
Solution:
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Verified Answer
The correct answer is:
1
$\beta=\frac{1}{2}$
$\Rightarrow \frac{2}{\beta}=\frac{2 m}{2}$
$\Rightarrow \mathrm{m}=\frac{2}{\beta}=2 \times \frac{2}{1}$
$\Rightarrow \mathrm{m}=4$
$\Rightarrow \beta \sqrt{m}=\frac{1}{2} \times \sqrt{4}=1$
$\Rightarrow \frac{2}{\beta}=\frac{2 m}{2}$
$\Rightarrow \mathrm{m}=\frac{2}{\beta}=2 \times \frac{2}{1}$
$\Rightarrow \mathrm{m}=4$
$\Rightarrow \beta \sqrt{m}=\frac{1}{2} \times \sqrt{4}=1$
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