Some equations have no formula. There is no quadratic formula for $e^x+x-3=0$, and the formula for cubic equations is so laborious that hardly anyone uses it. In these cases the way forward is to approximate: guess a value, see how far off we are, and correct the guess. The question is how to correct it well. Newton’s example In De analysi, of 1669, Newton chose the equation $$y^3-2y-5=0.$$By trial he saw that the root is near $2$, since $2^3-2\cdot2-5=-1$. So he wrote $y=2+p$, where $p$ is the missing correction, and substituted: