If $30 x^2-15 x+1=0$, then what is the value of $25 x^2+\left(36 x^2\right)^{-1}$ ?

$\frac{55}{12}$

If $30 x^2-15 x+1=0$,

then what is the value of $25 x^2+\left(36 x^2\right)^{-1}$ ?

Divide both the sides of If $30 x^2-15 x+1=0$,  by 6x to get the desired format of the equation,

5x + \(\frac{1}{6x}\) = \(\frac{5}{2}\)

If $K+\frac{1}{K}=n$

then, $K^2+\frac{1}{K^2}$ = n² – 2 × k × \(\frac{1}{k}\)

$25 x^2+\left(36 x^2\right)^{-1}$ = (\(\frac{5}{2}\))² – 2× 5x × \(\frac{1}{6x}\)

$25 x^2+\left(36 x^2\right)^{-1}$ = \(\frac{25}{4}\) - \(\frac{5}{3}\)

$25 x^2+\left(36 x^2\right)^{-1}$ = \(\frac{75 - 20}{12}\) = $\frac{55}{12}$