Quadratic Equation
A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2, written in standard form as ax² + bx + c = 0 where a ≠ 0. Its graph is a parabola, and it can have two, one, or no real solutions depending on the value of the discriminant (b² − 4ac). Quadratic equations model projectile motion, area problems, and many optimisation scenarios in physics and engineering.
Key Formula
ax² + bx + c = 0
LaTeX: ax^2 + bx + c = 0
| Symbol | Meaning | Unit |
|---|---|---|
| a | Coefficient of x² (must be non-zero) | dimensionless |
| b | Coefficient of x | dimensionless |
| c | Constant term | dimensionless |
| x | Variable (unknown) | dimensionless |
Worked Example
Problem
Solve x² − 5x + 6 = 0 by factoring.
Solution
Step 1: Find two numbers that multiply to c = 6 and add to b = −5. (−2) × (−3) = 6 and (−2) + (−3) = −5 ✓ Step 2: Factor the quadratic. (x − 2)(x − 3) = 0 Step 3: Apply the zero-product property. x − 2 = 0 → x = 2 x − 3 = 0 → x = 3 Step 4: Verify x = 2: (4) − 10 + 6 = 0 ✓ Verify x = 3: (9) − 15 + 6 = 0 ✓
Answer
x = 2 and x = 3
Methods for Solving Quadratic Equations
| Method | Best Used When | Steps Required | Always Works? |
|---|---|---|---|
| Factoring | Integer roots exist | Low | No — only if factorable over integers |
| Square root method | No linear term (b = 0) | Low | Only when b = 0 |
| Completing the square | Deriving quadratic formula | Medium | Yes |
| Quadratic formula | Any quadratic | Medium | Yes — always works |
| Graphing | Approximate roots needed | Medium | Approximate only |
Interactive Tools
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Related Terms
Quadratic Formula
The quadratic formula is an algebraic formula that gives the solutions (roots) of any quadratic equation ax² + bx + c = 0 directly in terms of its coefficients a, b, and c. It is derived by completing the square on the general quadratic and is the most reliable method for solving quadratics, working even when factoring over integers is impossible. The formula also reveals the nature of the roots through the discriminant b² − 4ac.
Discriminant
The discriminant of a quadratic equation ax² + bx + c = 0 is the expression b² − 4ac, denoted by the Greek letter Δ (delta) or D. The sign of the discriminant determines the nature and number of roots: positive means two distinct real roots, zero means one repeated real root, and negative means two complex (non-real) conjugate roots. The discriminant is the part under the square root sign in the quadratic formula, making it the key indicator of solvability over the real numbers.
Factoring
Factoring (or factorisation) in algebra is the process of rewriting an algebraic expression as a product of simpler expressions called factors, reversing the process of expansion. For example, x² − 5x + 6 can be factored as (x − 2)(x − 3). Factoring is essential for solving polynomial equations, simplifying rational expressions, and finding roots, and it is a core skill that underpins much of higher mathematics.
From Latin "quadratus" meaning "square", referring to the x² (x-squared) term that defines these equations. The systematic study of quadratic equations dates to ancient Babylonia (circa 2000 BCE), with algebraic solutions developed by al-Khwarizmi in the 9th century.