Solving Quadratic Equations
A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. The word 'quadratic' comes from the Latin 'quadratus', meaning square, reflecting the fact that the variable x appears raised to the second power. Quadratic equations arise naturally in problems involving areas, projectile motion, and optimisation, making them one of the most important equation types in all of mathematics. Unlike linear equations, which have exactly one solution, quadratic equations can have two distinct real solutions, one repeated real solution, or two complex (non-real) solutions. The number and nature of solutions is determined by the discriminant, b² − 4ac. There are three main algebraic methods for solving quadratic equations: factorisation, completing the square, and the quadratic formula. Each method has situations where it is most efficient, and a skilled mathematician chooses the approach that suits the equation at hand. Mastery of quadratic equations is a gateway to more advanced topics including polynomial functions, conic sections, and calculus. Understanding how to solve them — and why each method works — develops algebraic reasoning that underpins much of higher mathematics.
- Method 1: Factorisation
- Method 2: Completing the Square
- Method 3: The Quadratic Formula
- Choosing the Right Method
Method 1: Factorisation
Factorisation works when the quadratic expression ax² + bx + c can be written as a product of two linear factors. The key principle is the Zero Product Property: if P × Q = 0, then P = 0 or Q = 0. To factorise x² + bx + c, find two numbers p and q such that p + q = b and p × q = c; then x² + bx + c = (x + p)(x + q). For example, x² − 5x + 6 = 0 factors as (x − 2)(x − 3) = 0, giving x = 2 or x = 3. For ax² + bx + c with a ≠ 1, use the AC method: multiply a × c, find factors of this product that add to b, then split the middle term and factor by grouping. Factorisation is the fastest method when integer roots exist, but it cannot always be applied.
Method 2: Completing the Square
Completing the square transforms any quadratic into the form (x + h)² = k, from which x is isolated by taking square roots. The procedure for x² + bx + c = 0 is: (1) Move the constant: x² + bx = −c. (2) Add (b/2)² to both sides: x² + bx + (b/2)² = −c + (b/2)². (3) Write the left side as a perfect square: (x + b/2)² = (b² − 4c)/4. (4) Take square roots of both sides and solve for x. For example, to solve x² + 6x + 5 = 0: x² + 6x = −5; add 9 to both sides: (x + 3)² = 4; so x + 3 = ±2, giving x = −1 or x = −5. Completing the square always works and is the algebraic foundation from which the quadratic formula is derived.
Method 3: The Quadratic Formula
The quadratic formula gives the solutions of any equation ax² + bx + c = 0 directly: x = (−b ± √(b² − 4ac)) / (2a). This formula is derived by completing the square on the general equation. It is the most universal method — it works for every quadratic, including those with irrational or complex roots. The expression under the square root, Δ = b² − 4ac, is called the discriminant. If Δ > 0, there are two distinct real roots; if Δ = 0, there is exactly one real root (a repeated root); if Δ < 0, there are two complex conjugate roots with no real solutions. For example, 2x² − 4x − 6 = 0 has a = 2, b = −4, c = −6, so Δ = 16 + 48 = 64, and x = (4 ± 8)/4, giving x = 3 or x = −1.
Choosing the Right Method
A practical strategy: first check whether the quadratic factors easily (small integer coefficients, obvious factor pairs). If not, use the quadratic formula for speed and reliability. Use completing the square when you need to rewrite the quadratic in vertex form (x − h)² + k for graphing or when the equation has no linear term (bx missing), since then completing the square reduces to one step. Always simplify the equation to standard form before applying any method.
The Key Equation
- x — Solution (root) (dimensionless): The value(s) of the variable that satisfy the equation ax² + bx + c = 0
- a — Leading coefficient (dimensionless): The coefficient of x²; must be non-zero for the equation to be quadratic
- b — Linear coefficient (dimensionless): The coefficient of x in the standard form equation
- c — Constant term (dimensionless): The term with no x; the y-intercept of the corresponding parabola
- b^2 - 4ac — Discriminant (Δ) (dimensionless): Determines the number and type of roots: positive gives two real roots, zero gives one repeated root, negative gives two complex roots
Worked Example
Problem. Solve 3x² − 7x + 2 = 0 using all three methods.
Answer. x = 2 or x = 1/3
Origins · 1 minMuhammad ibn Musa al-Khwarizmi and how this was discovered
Al-Khwarizmi wrote 'Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wal-muqābala' (c. 820 CE), the foundational text of algebra. In it, he provided systematic methods for solving quadratic equations by completing the square, classifying six types of quadratic problems and providing geometric proofs for each solution. The word 'algebra' itself derives from 'al-jabr' in his title, and the word 'algorithm' comes from the Latin transliteration of his name. Al-Khwarizmi worked at the House of Wisdom in Baghdad, one of the great intellectual centres of the medieval world. He solved all quadratic equations geometrically using areas of squares and rectangles — centuries before symbolic algebra notation was invented.
Related Concepts
- Solving Linear Equations — Linear equations are the foundation of equation-solving; understanding them is essential before tackling the more complex quadratic case.
- The Discriminant — The discriminant b² − 4ac determines the number and type of roots of a quadratic equation, directly linked to the quadratic formula.
- Quadratic Functions and Parabolas — The roots of a quadratic equation are the x-intercepts of the corresponding parabola y = ax² + bx + c, connecting algebra to graphical analysis.
- Factorisation of Algebraic Expressions — Factorisation is one of the three core methods for solving quadratic equations and relies on manipulating algebraic expressions efficiently.
- Introduction to Complex Numbers — When the discriminant is negative, the quadratic formula produces complex number solutions, making this a natural bridge to complex numbers.