Given that (p + 1/2√3)(1 - √3)\(^2\) = 3- √3, find x the value of p.
Question 1 Report
Given that (p + 1/2√3)(1 - √3)\(^2\) = 3- √3,
find x the value of p.
Answer Details
Reading the expression as \(\left(p+\tfrac12\sqrt3\right)(1-\sqrt3)^{2}=3-\sqrt3\).
First expand the square:
\[(1-\sqrt3)^{2}=1-2\sqrt3+3=4-2\sqrt3\]
So:
\[\left(p+\tfrac{\sqrt3}{2}\right)(4-2\sqrt3)=3-\sqrt3\]
Note \(4-2\sqrt3=2(2-\sqrt3)\). Divide both sides:
\[p+\frac{\sqrt3}{2}=\frac{3-\sqrt3}{4-2\sqrt3}\]
Rationalise the right side by \(\times\dfrac{4+2\sqrt3}{4+2\sqrt3}\): denominator \(16-12=4\); numerator \((3-\sqrt3)(4+2\sqrt3)=12+2\sqrt3-6=6+2\sqrt3\). So the right side \(=\dfrac{6+2\sqrt3}{4}=\dfrac{3+\sqrt3}{2}\).
\[p=\frac{3+\sqrt3}{2}-\frac{\sqrt3}{2}=\frac{3}{2}\]
\[\boxed{p=\tfrac32}\]
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