Motion2020 was so close!

This involves a strategy with algebra that entails manipulating both sides of the equation in order to match the like terms, then solving from there.

We are told that:

\((63 - 36\sqrt{3})^{\frac{1}{2}}\)

can be expressed as:

\(x + y\sqrt{3}\)

This is equivalent to saying:

\((63 - 36\sqrt{3})^{\frac{1}{2}}\) \(=\) \(x + y\sqrt{3}\)

Let's square both sides:

\((63 - 36\sqrt{3})\) = \((x + y\sqrt{3})\)\(^2\)

Expanding the right side:

\((63 - 36\sqrt{3})\) \(=\) \(x^2 + 2xy\sqrt{3} + 3y^2\)

Let's group the \(3y^2\) with the \(x^2\) and rewrite a bit to elucidate the strategy:

\((63) + (-36\sqrt{3})\) \(=\) \((x^2 + 3y^2) + (2xy\sqrt{3})\)

I did this because, in order for this equation to be true, we must have that:

\(63 = x^2 + 3y^2\)

and

\(-36\sqrt{3}\) = \(2xy\sqrt{3}\)

The \(\sqrt{3}\) terms go together, leaving the other terms to equal 63.

Now we could solve this for \(x\) and \(y\), but look closely at the second equation:

\(-36\sqrt{3}\) = \(2xy\sqrt{3}\)

Let's simplify this:

\(-36 = 2xy\)

\(-18 = xy\)

Therefore the answer is C