Theory of Quadratic Equations - Mathematics (Undergraduate Foundation)

This course provides a rigorous, in-depth analysis of quadratic equations. It covers all methods for solving quadratics, determining the nature of their roots using the discriminant, and constructing equations from given roots. A full command of this topic is a cornerstone of algebra. Quadratic models are essential in the physical and financial sciences. They are used in physics to describe projectile motion, in engineering to design parabolic structures like antennas, and in finance for profit optimisation. This is the mathematics of trajectories and optimisation. By the end of this course, you will be able to solve any quadratic equation by factorisation, completing the square, or the quadratic formula, use the discriminant to determine the nature of the roots, apply the sum and product of roots to solve problems, and construct a quadratic equation from a given set of roots. This course is critical for first-year university students in mathematics, physics, engineering, and economics. It provides the necessary algebraic foundation for calculus, mechanics, and any field involving the optimisation of non-linear models.

Enrolment valid for 12 months

This course is also part of the following learning track. You may join the track to gain comprehensive knowledge across related courses.

MTH 101: Elementary Mathematics I - Algebra and Trigonometry

Master the foundational mathematical structures essential for success in quantitative undergraduate degrees and professional technical roles. This comprehensive learning track delivers a rigorous treatment of algebra and trigonometry, moving rapidly from fundamental set theory and real number operations to advanced topics including matrix algebra, complex numbers, and analytical trigonometry. You will establish the critical problem-solving framework required for advanced study in calculus, engineering mechanics, and data science. This programme is primarily designed for first-year university students in STEM disciplines requiring strong analytical bases, particularly engineering, physics, computer science, and economics. It also serves as an intensive, high-level refresher for professionals returning to academia or shifting into data-driven roles demanding precise numerical literacy and logical structuring. Prior competence in standard secondary school mathematics is assumed; focus is placed strictly on mastery and application of core definitions. Upon completion, you will possess the skills to construct rigorous logical arguments using set theory and mathematical induction, model complex relationships with functions and matrices, and analyze periodic systems using advanced trigonometry. You will demonstrate competence in solving diverse equation types, from quadratics to linear systems, and manipulating complex numbers in engineering applications. This track prepares you directly for the mathematical demands of second-year university studies and technical professional certification exams.

Course Chapters

1. Introduction

This chapter introduces the quadratic equation and its importance in algebra. It establishes the standard form of a quadratic equation and outlines the core theoretical questions that this course will address, such as finding roots and analysing their properties. Key objectives include recognising a quadratic equation, understanding the concept of a 'root' or 'solution', and appreciating the structure of the course.

Concept Overviews

2 Lessons

2. Solving Quadratic Equations

This chapter covers the three standard algebraic methods for solving quadratic equations. Mastery of these techniques is non-negotiable, as they provide the procedural foundation for all quadratic analysis and its applications. Key methods covered are solving by factorisation, by completing the square, and by using the quadratic formula. Worked examples demonstrate each technique with standard problems.

Concept Overviews

3 Lessons

Problem Walkthroughs

4 Lessons

3. Nature of Roots

This chapter focuses on the discriminant of a quadratic equation. It explains how this single value can be used to determine the nature of the roots without having to solve the equation itself, an essential skill for efficient analysis. Key topics include the definition of the discriminant, its use in identifying real, equal, or non-real roots, and solving for unknown coefficients based on root conditions.

Concept Overviews

2 Lessons

Problem Walkthroughs

4 Lessons

4. Properties of Roots

This chapter covers the relationship between the coefficients of a quadratic equation and its roots. It provides powerful shortcuts for finding the sum and product of roots and for constructing equations from given information. Key topics include the formulas for the sum and product of roots, constructing quadratic equations from given roots, and solving problems involving symmetric expressions of roots.

Concept Overviews

2 Lessons

Problem Walkthroughs

4 Lessons

5. Conclusion

This chapter consolidates the theory of quadratic equations. It provides a structured summary of the methods for solving equations and analysing their roots, reinforcing the core principles of this foundational topic. The conclusion summarises the key solution methods and analytical tools. It also provides a forward look to how these concepts are applied in physics, engineering, and higher mathematics.

Concept Overviews

2 Lessons