Solve the following equations by using formula:

256x² - 32x + 1 = 0

The given equation is 256x² - 32x + 1 = 0.

Comparing it with ax² + bx + c = 0, we get
a = 256 , b = -32 , c = 1

By using formula,

$x = \dfrac{-b ± \sqrt{b^2 - 4ac}}{2a}$

we obtain:

$\Rightarrow x = \dfrac{-(-32) ± \sqrt{(-32)^2 - 4\times 256 \times 1}}{2 \times 256} \\[1em] \Rightarrow x = \dfrac{32 ± \sqrt{1024 - 1024}}{512} \\[1em] \Rightarrow x = \dfrac{32 ± \sqrt{0}}{512} \\[1em] \Rightarrow x = \dfrac{32 + 0}{512} \text{ or } \dfrac{32 - 0}{512} \\[1em] \Rightarrow x = \dfrac{32}{512} \text{ or } \dfrac{32 - 0}{512} \\[1em] x = \dfrac{1}{16} \text{ or } \dfrac{1}{16}$

Hence roots of the given equation are $\dfrac{1}{16} , \dfrac{1}{16}$.