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Solving a Quadratic Equation by Completion of Squares Method

Simplify: 3...

Question

Simplify:
$3 \sqrt{45} - \frac{5}{2} \sqrt{\frac{1}{3}} + 4 \sqrt{3}$

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Solution

$3 \sqrt{45} - \frac{5}{2} \sqrt{\frac{1}{3}} + 4 \sqrt{3}$
$= 3 \sqrt{9 \times 5} - \frac{5}{2} \sqrt{\frac{1}{3}} + 4 \sqrt{3}$
$= 3 \sqrt{3^{2} \times 5} - \frac{5}{2} \sqrt{\frac{1}{3}} + 4 \sqrt{3}$
$= 3 \times 3 \sqrt{5} - \frac{5}{2} \sqrt{\frac{1}{3}} + 4 \sqrt{3}$
$= 9 \sqrt{5} - (\frac{5}{2} \sqrt{\frac{1}{3}} \times \sqrt{\frac{3}{3}}) + 4 \sqrt{3}$
$= 9 \sqrt{5} - \frac{5}{6} \sqrt{3} + 4 \sqrt{3}$
$= 9 \sqrt{5} + \sqrt{3} (4 - \frac{5}{6})$
Hence, $3 \sqrt{45} - \frac{5}{2} \sqrt{\frac{1}{3}} + 4 \sqrt{3} = 9 \sqrt{5} + \sqrt{3} (4 - \frac{5}{6})$

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