Writing Quadratic Equations from Roots and Points

The Organic Chemistry Tutor

This math tutorial provides a comprehensive guide on how to reconstruct quadratic equations in standard form when given their roots (solutions) or specific points on the curve. The video begins by demonstrating the fundamental relationship between factors and zeros, showing how to convert integer and fractional solutions back into binomial factors. It progresses to more advanced concepts, explaining how irrational and imaginary roots always occur in conjugate pairs, and demonstrates techniques for expanding these complex factors into standard quadratic equations. The video explores several key topics in algebra, including the Factor Theorem, the properties of conjugate pairs for radicals and complex numbers, and the expansion of polynomials. It specifically covers how to handle fractional roots by clearing denominators to achieve integer coefficients, how to simplify expressions involving $i^2$, and utilizing shortcut methods for multiplying complex conjugates. The final segment shifts to a different approach: finding a quadratic equation given three coordinate points by setting up and solving a system of linear equations. This resource is highly valuable for high school Algebra II and Pre-Calculus classrooms. It reinforces the bidirectional nature of algebraic processes—teaching students not just how to solve equations, but how to build them. The clear, step-by-step walkthroughs allow teachers to flip the classroom or provide targeted remediation for students struggling with complex numbers or systems of equations. The inclusion of the "three points" problem also serves as an excellent introduction to curve fitting and mathematical modeling.