Prove the identity $\sqrt{17+2\sqrt{30}}-\sqrt{17-2\sqrt{30}}=2\sqrt{2}$

Question

Prove the identity $$\sqrt{17+2\sqrt{30}}-\sqrt{17-2\sqrt{30}}=2\sqrt{2}.$$ We have $$\left(\sqrt{17+2\sqrt{30}}-\sqrt{17-2\sqrt{30}}\right)^2=17+2\sqrt{30}-2\sqrt{17+2\sqrt{30}}\cdot\sqrt{17-2\sqrt{30}}+17-2\sqrt{30}=34-2\sqrt{(17)^2-(2\sqrt{30})^2}=34-2\sqrt{289-120}=34-2\sqrt{169}=34-2.13=8=(2\sqrt2)^2, $$ so $$\sqrt{17+2\sqrt{30}}-\sqrt{17-2\sqrt{30}}=2\sqrt{2}.$$ In the hints the authors have written that I should use the fact that the LHS is positive and square it. What would be the problem if it wasn't positive? The identity obviously won't hold because LHS<0, RHS>0...

Answer 1

both of your original real numbers are roots of $$ x^4 - 34 x^2 + 169 $$

Standard bit for quartic with no cubic term and no linear, $$ (x^2 - 13)^2 - 8 x^2 = x^4 - 34 x^2 + 169 $$ $$ (x^2 - 13)^2 - (x \sqrt 8)^2 = x^4 - 34 x^2 + 169 $$

This becomes ( because a difference of squares) $$ (x^2 - x \sqrt 8 - 13)(x^2 + x \sqrt 8 - 13) $$ so that your numbersare two out of four values $$ \frac{\pm \sqrt 8 \pm \sqrt {60}}{2} $$ or $$ \pm \sqrt 2 \pm \sqrt {15} $$

This leads to checking $$ (\sqrt 2 + \sqrt {15} )^2 = 17 + 2 \sqrt{30} $$ along with $$ (-\sqrt 2 + \sqrt {15} )^2 = 17 - 2 \sqrt{30} $$

Your original expression is equal to $$ (\sqrt 2 + \sqrt {15} ) - (-\sqrt 2 + \sqrt {15} ) $$

Answer 2

Since the roots on the LHS get "annihilated", it makes very much sense to look for natural $m,n$ such that

\begin{eqnarray*} 17+2\sqrt{30} & = & \left(\sqrt n + \sqrt m\right)^2 & = & n + m + 2\sqrt{nm}\\ 17-2\sqrt{30} & = & \left(\sqrt n - \sqrt m\right)^2 & = & n + m - 2\sqrt{nm} \end{eqnarray*}

From this approach you get \begin{eqnarray*} n+m & = & 17\\ \sqrt{nm} & = & \sqrt{30} \end{eqnarray*}

Obviously $n = 15, m=2$ fit, hence

$$\sqrt{17+2\sqrt{30}}-\sqrt{17-2\sqrt{30}}= \sqrt{15}+\sqrt 2 - (\sqrt{15}-\sqrt 2)=2\sqrt{2}$$