不使用计算器，求整数 $a$ 和 $b$ 的值，使得方程 $xackslash√{18} - ackslash√{72} = xackslash√{3}$ 的解为 $\frac{a + 2\sqrt{b}}{5}$。

1. Rearrange the equation to group terms involving $x$ and simplify the surds.

$x\sqrt{18} - x\sqrt{3} = \sqrt{72}$
$x(3\sqrt{2} - \sqrt{3}) = 6\sqrt{2}$

2. Isolate $x$.

$x = \frac{6\sqrt{2}}{3\sqrt{2} - \sqrt{3}}$

3. Rationalise the denominator by multiplying the numerator and denominator by the conjugate.

$x = \frac{6\sqrt{2}(3\sqrt{2} + \sqrt{3})}{(3\sqrt{2} - \sqrt{3})(3\sqrt{2} + \sqrt{3})}$

4. Expand and simplify the expression.

$x = \frac{18(2) + 6\sqrt{6}}{(3\sqrt{2})^2 - (\sqrt{3})^2}$
$x = \frac{36 + 6\sqrt{6}}{18 - 3}$
$x = \frac{36 + 6\sqrt{6}}{15}$

5. Simplify the fraction to match the given form and identify $a$ and $b$.

Divide numerator and denominator by 3:
$x = \frac{12 + 2\sqrt{6}}{5}$
Comparing with $\frac{a + 2\sqrt{b}}{5}$, we have $a = 12$ and $b = 6$.