Solving Quadratic Equations – A Level Further Mathematics CCEA Revision

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.
The solutions of the quadratic equation are given by the quadratic formula, x = [-b ± sqrt(b² - 4ac)] / (2a).
If the discriminant (b² - 4ac) is positive, the equation has two distinct real roots. If it is zero, the equation has one real root, also known as a repeated root. If it is negative, the equation has two complex roots.

Quadratic equations have a wide range of applications in physics, economics, and engineering, among others.
In the context of complex numbers, quadratic equations allow us to find complex solutions for equations that don’t have real solutions.
In many scenarios, it’s more convenient to solve quadratic equations in the complex plane because this allows us to use the De Moivre theorem, the geometry of the complex plane, or other techniques related to complex numbers.

Solving quadratic equations involves understanding how changes to the coefficients a, b, and c affect the discriminant and, consequently, the nature of the roots.
In quadratic equations with complex roots, the term sqrt(b² - 4ac), or the square root of the discriminant, involves the square root of a negative number, which is defined in terms of an imaginary unit, i.
Therefore, the solutions have an imaginary part and can be represented in the complex plane.

Quadratic equations are the foundation for the further study of higher-degree polynomial equations, where complex roots are common.
They also prove vital when delving into complex dynamics, a subfield of mathematics that studies the behaviour of iterates of holomorphic functions.
Being able to solve quadratic equations, particularly with complex roots, is a critical skill, not only in pure mathematics, but in many areas of applied mathematics and physics as well.