By definition, to find $x$-intercept of a graph, let $y$ be zero and solve the equation for $x$.

$$\begin{align*} y&=4\cdot x^{2}+3\\ 0&=4\cdot x^{2}+3\\ -3&=4\cdot x^{2} \\ \dfrac{-3}{4}&=x^{2} \\ \end{align*}$$