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.

[NUC Core] MTH 101: Elementary Mathematics I - Algebra and Trigonometry

Master the mathematics every scientist, engineer, and computer scientist must know. This track covers the NUC Core syllabus for MTH 101: Elementary Mathematics I. From set theory and sequences to trigonometry and complex numbers, you will gain the exact tools used in higher mathematics, algorithm design, data analysis, physics, and engineering. This programme is for first-year university students in mathematics, computer science, engineering, or physics, as well as serious learners preparing for undergraduate study. It also serves professionals who need a rigorous mathematical foundation for technical fields. By the end, you will handle formal set notation, progressions, quadratic models, combinatorics, induction proofs, binomial expansions, trigonometric analysis, and complex arithmetic. You will be equipped to solve real problems in science, engineering, computing, and finance, and ready for advanced study in calculus, statistics, and applied mathematics.

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.

Chapter lessons

1-1. Welcome

A direct statement of the course's purpose. This lesson defines the scope of quadratic theory and its essential role in algebra and applied sciences.

1-2. What is a quadratic equation?

Provides a formal definition of the quadratic equation. It establishes the standard form ax²+bx+c=0 and defines the terms coefficient, variable, and constant.

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.

Chapter lessons

2-1. Solving by factorisation

Covers the method of factorisation for solving quadratic equations where the expression can be broken down into linear factors. This is the most direct solution method when applicable.

2-2. Solving by completing the square

Details the process of completing the square to solve any quadratic equation. This technique is fundamental for understanding the derivation of the quadratic formula.

2-3. The quadratic formula

Introduces the quadratic formula as a universal tool for finding the roots of any quadratic equation. Its derivation from completing the square is outlined.

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.

Chapter lessons

3-1. Defining the discriminant

Formally defines the discriminant as the expression b²-4ac from within the quadratic formula. Its significance as an analytical tool is introduced.

3-2. The nature of roots

Explains how the value of the discriminant (positive, zero, or negative) determines whether a quadratic equation has two distinct real roots, one repeated real root, or no real roots.

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.

Chapter lessons

4-1. The sum and product of roots

Derives and explains the formulas for the sum (-b/a) and product (c/a) of the roots of a quadratic equation directly from its coefficients.

4-2. Constructing equations from roots

Details the method for constructing a quadratic equation given its roots, using the relationship x² - (sum of roots)x + (product of roots) = 0.

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.

Chapter lessons

5-1. Summary of quadratic theory

A concise review of the key methods and formulas covered in the course. This lesson ensures all analytical and procedural material has been consolidated.

5-2. Applications in science and engineering

Provides a brief overview of how quadratic equations are directly applied in modelling projectile motion in physics and designing parabolic structures in engineering.