I answered a nice question on WyzAnt about approximating roots to a quadratic equation that had real but irrational roots. Calculus enables us to find rational approximations to these irrational roots. I will re-present the solution to the problem, and continue to find a general expression to do this for an arbitrary polynomial. "Problem: approximate roots of -9x^2+8x+5. There is a nice approach using calculus to estimate/approximate a function without a square root and calculator. We can use the concept of moments to get an approximation to a function. For this example, we have a quadratic function in (x) with coefficients, a=-9, b=8, and c=5, as indicated in a previous solution. Thus f(x)=ax^2+bx+c. First we need to find a general idea of where a root (an x where f(x)=0) is located. To do this we can check the value of the function for some easy numbers, x=0 and x=1 are always... read more