Form a quadratic equation whose roots are the numbers $\dfrac{1}{10 - \sqrt{72}}$ and $\dfrac{1}{10 + 6 \sqrt{2}}$
Roots of the required quadratic equation are
$\alpha = \dfrac{1}{10 - \sqrt{72}} = \dfrac{1}{10 - 6 \sqrt{2}}$ $\;$ and $\;$ $\beta = \dfrac{1}{10 + 6 \sqrt{2}}$
Sum of roots $= \alpha + \beta$
$\begin{aligned}
\alpha + \beta & = \dfrac{1}{10 - 6 \sqrt{2}} + \dfrac{1}{10 + 6 \sqrt{2}} \\\\
& = \dfrac{10 + 6 \sqrt{2} + 10 - 6 \sqrt{2}}{\left(10 + 6 \sqrt{2}\right) \left(10 - 6 \sqrt{2}\right)} \\\\
& = \dfrac{20}{100 - 72} \\\\
& = \dfrac{20}{28}
\end{aligned}$
Product of roots $= \alpha \cdot \beta$
$\begin{aligned}
\alpha \cdot \beta & = \left(\dfrac{1}{10 - 6 \sqrt{2}}\right) \times \left(\dfrac{1}{10 + 6 \sqrt{2}}\right) \\\\
& = \dfrac{1}{100 - 72} \\\\
& = \dfrac{1}{28}
\end{aligned}$
Quadratic equation in terms of its roots is
$x^2 - \left(\text{sum of roots}\right) x + \text{product of roots} = 0$
$\therefore \;$ The required quadratic equation is
$x^2 - \dfrac{20}{28}x + \dfrac{1}{28} = 0$
i.e. $\;$ $28 x^2 - 20x + 1 = 0$