Q.Using Sridharacharya's formula, solve $4x^2 - 4x + 1 = 0$ and comment on the nature of the roots.

Compute the discriminant first to know what kind of roots to expect.

Here $a=4, b=-4, c=1$. Compute the discriminant: $D=(-4)^2-4(4)(1)=16-16=0$.

Since $D=0$, the equation has exactly one real (repeated) root.

Apply Sridharacharya's formula: $x=\dfrac{-b\pm\sqrt D}{2a}=\dfrac{4\pm0}{8}=\dfrac12$. …

Alternative — recognize $4x^2-4x+1=(2x)^2-2(2x)(1)+1^2=(2x-1)^2$ directly as a perfect square, giving $x …

Students sometimes report two DIFFERENT roots close to $\tfrac12$ by mis-simplifying $\sqrt0$ as a small nonzero value due to a rounding habit carried over from calculator-based arithmetic — $\ …