Arithmetic and geometric mean condition

If the arithmetic mean of two positive numbers $a$ and $b$ is twice their geometric mean, show that $\frac{a}{b}=\frac{2+\sqrt{3}}{2-\sqrt{3}}$.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. V. Geometrical Progression, Examples V.b, No. 18