Explore the intricacies of solving quadratic and cubic equations in this 36-minute lecture from the MathFoundations series. Delve into the historical context of these formulas, tracing their origins back to 16th-century mathematicians. Examine the limitations of standard formulas in generating exact solutions and discover how Newton's method can be applied to find approximate solutions. Learn to create quadratic and cubic equations with known solutions, apply the quadratic formula, and understand the connection between algebraic equations and Newton's method. Gain insights into the logical challenges present in modern pure mathematics through this comprehensive exploration of fundamental mathematical concepts.