Theory of Quadratic Equations - Mathematics (Undergraduate Foundation)

Quadratic equations are the bedrock of advanced algebra and the starting point for mastering non-linear mathematical models. This course provides a rigorous analysis of these equations, covering standard forms, solving methods like factorisation and completing the square, and the critical use of the discriminant to determine root nature. You will study turning points for optimisation, solve complex quadratic inequalities, and master the relationship between roots and coefficients to build equations from scratch. Mastering these models is essential for careers in engineering, physics, and finance where trajectories and profit levels must be calculated precisely. You will use these principles to describe projectile motion, design parabolic structures, and identify boundary limits in economic models. This is the mathematics of optimisation, providing the necessary tools to find the highest or lowest possible values in any system governed by second-degree polynomials. By the end of this course, you will solve any quadratic equation using multiple methods; use the discriminant to identify real, identical, or imaginary roots; and determine the location and value of turning points. You will master solving quadratic inequalities; identifying valid ranges for rational expressions; and using symmetric identities to evaluate root properties without solving the original equation. These skills form the essential foundation for calculus and mechanics. This course is built for first-year university students in mathematics, physics, engineering, and economics who need a solid algebraic foundation. It also serves secondary school leavers preparing for advanced entrance exams or technical professionals looking to refresh their analytical skills. Even those in data science or basic programming will benefit from the logical structuring and problem-solving techniques required to handle non-linear variables.

9 hrs

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 establishes the foundation for quadratic analysis by defining the standard form and its real-world relevance. Mastering these basics is essential before attempting complex solving techniques or applications. You will master identifying quadratic components; rearranging equations into standard form; understanding the concept of roots; and applying the non-zero coefficient constraint.

Concept Overviews

3 Lessons

31:40

Problem Walkthroughs

2 Lessons

7:36

2. Solving Quadratic Equations

This chapter provides the procedural core of the course by teaching the three primary methods for solving quadratic equations. Mastering these techniques is mandatory for any further study in calculus, physics, or engineering. You will master solving equations by factorisation; using the method of completing the square; applying the universal quadratic formula; and reducing complex fractional equations into standard quadratic form.

Concept Overviews

3 Lessons

51:32

Problem Walkthroughs

4 Lessons

24:48

3. Nature of Roots

This chapter focuses on the discriminant to predict the properties of roots without solving the equation. It is a critical analytical tool for determining the existence and type of solutions in engineering and physics models. You will master calculating the discriminant; classifying roots as real, identical, or imaginary; identifying rational and irrational solutions; and applying tangency conditions to curves and lines.

Concept Overviews

4 Lessons

1:09:13

Problem Walkthroughs

8 Lessons

1:01:54

4. Maximum and Minimum Values

This chapter focuses on finding the highest or lowest points of quadratic graphs. It is vital for solving real-world optimisation problems in physics, engineering, and finance. You will master identifying maximum and minimum turning points; locating the vertex of a parabola; using the completing the square method for optimisation; and proving numerical boundary limits for quadratic expressions.

Concept Overviews

2 Lessons

25:17

Problem Walkthroughs

3 Lessons

16:59

5. Inequalities and Rational Expressions

This chapter covers solving quadratic inequalities and finding the ranges of rational functions. These skills are vital for defining valid domains and identifying excluded values in advanced physics and engineering models. You will master solving quadratic inequalities using graph shapes and critical values; applying the discriminant to find unknown coefficient ranges; rearranging rational expressions into quadratic form; and identifying the exact sets of values a function can or cannot take.

Concept Overviews

2 Lessons

26:50

Problem Walkthroughs

3 Lessons

45:27

6. Properties of Roots

This chapter shows how equation coefficients relate to their roots. Learning these shortcuts helps you study and build equations quickly without solving them first. You will master calculating the sum and product of roots; evaluating expressions like the sum of squares or cubes; constructing new equations from given roots; and identifying conditions for shared roots between equations.

Concept Overviews

4 Lessons

45:05

Problem Walkthroughs

8 Lessons

36:29

7. Conclusion

This chapter consolidates all theoretical and practical aspects of quadratic equations covered in the course. It ensures you have a firm command of every analytical tool and solution method before advancing to higher mathematics. You will master factorisation, completing the square, and the quadratic formula; interpreting the discriminant for the nature of roots; applying sum and product properties; and identifying turning points for optimisation.

Concept Overviews

1 Lesson

6:30