Question
Solve the following equations by using the method of completing the square:
$\sqrt3\text{x}^2+10\text{x}+7\sqrt3=0$
$\sqrt3\text{x}^2+10\text{x}+7\sqrt3=0$
✓
Answer
$\sqrt3\text{x}^2+10\text{x}+7\sqrt3=0$
$3\text{x}^2+10\sqrt3\text{x}+21=0$ $\big($Multiplying both sides by $\sqrt3\big)$
$\Rightarrow\text{3x}^2-10\sqrt3\text{x}=-21$
$\Rightarrow\big(\sqrt3\text{x}\big)^2-2\times\sqrt3\text{x}\times5+5^2\\=-21+5^2$ [Adding $5^2$ on both sides]
$\Rightarrow\big(\sqrt3\text{x}+5\big)^2=-21+25$
$=4=2^2$
$\Rightarrow\sqrt3\text{x}+5=\pm2$ (Taking square root on both sides)
$\Rightarrow\sqrt3\text{x}+5=2$ or $\sqrt3\text{x}+5=-2$
$\Rightarrow\sqrt3\text{x}=-3$ or $\sqrt3\text{x}=-7$
$\Rightarrow\text{x}=-\frac{3}{\sqrt3}=-\sqrt3$ or $\text{x}=-\frac{7}{\sqrt3}=-\frac{7\sqrt3}{3}$
Hence, $-\sqrt3$ and $-\frac{7\sqrt3}{3}$ are the roots of the given equation.
$3\text{x}^2+10\sqrt3\text{x}+21=0$ $\big($Multiplying both sides by $\sqrt3\big)$
$\Rightarrow\text{3x}^2-10\sqrt3\text{x}=-21$
$\Rightarrow\big(\sqrt3\text{x}\big)^2-2\times\sqrt3\text{x}\times5+5^2\\=-21+5^2$ [Adding $5^2$ on both sides]
$\Rightarrow\big(\sqrt3\text{x}+5\big)^2=-21+25$
$=4=2^2$
$\Rightarrow\sqrt3\text{x}+5=\pm2$ (Taking square root on both sides)
$\Rightarrow\sqrt3\text{x}+5=2$ or $\sqrt3\text{x}+5=-2$
$\Rightarrow\sqrt3\text{x}=-3$ or $\sqrt3\text{x}=-7$
$\Rightarrow\text{x}=-\frac{3}{\sqrt3}=-\sqrt3$ or $\text{x}=-\frac{7}{\sqrt3}=-\frac{7\sqrt3}{3}$
Hence, $-\sqrt3$ and $-\frac{7\sqrt3}{3}$ are the roots of the given equation.
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