Representing $\frac{3 + \sqrt{5}}{2}$ as a square of a quadratic surd How can I represent $\frac{3 + \sqrt{5}}{2}$ as square of a quadratic surd? Actually, I was solving a question where $\frac{3 + \sqrt{5}}{2}$ was converted to $(\frac{1+\sqrt{5}}{2})^2$. How did the solution writer think of it?

We can set $(a+b\sqrt{5})^2=\frac{3}{2} + \frac{1}{2}\sqrt{5}$. Then, solve for rational $a$ and $b$.

Comparing the terms we obtain: $$a^2 + 5b^2 = \frac{3}{2},\ \ \ 2ab = \frac{1}{2}$$ Solving for $a$ and $b$, we get $a=b=\frac{1}{2}$ or $a=b=-\frac{1}{2}$, thus the two possible squares are: $$ \left(\frac{1+\sqrt{5}}{2}\right)^2,\ \ \ \left(\frac{-1-\sqrt{5}}{2}\right)^2$$