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.
Key Formula
x = (−b ± √(b² − 4ac)) / (2a)
LaTeX: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
| Symbol | Meaning | Unit |
|---|---|---|
| x | Solution(s) of the quadratic equation | dimensionless |
| a | Coefficient of x² (must be non-zero) | dimensionless |
| b | Coefficient of x | dimensionless |
| c | Constant term | dimensionless |
| ± | Gives two solutions: one with + and one with − | dimensionless |
Worked Example
Problem
Solve 2x² − 7x + 3 = 0 using the quadratic formula.
Solution
Step 1: Identify a = 2, b = −7, c = 3. Step 2: Calculate the discriminant. b² − 4ac = (−7)² − 4(2)(3) = 49 − 24 = 25 Step 3: Apply the formula. x = (−(−7) ± √25) / (2×2) x = (7 ± 5) / 4 Step 4: Find both solutions. x₁ = (7 + 5) / 4 = 12/4 = 3 x₂ = (7 − 5) / 4 = 2/4 = 1/2 Step 5: Verify x = 3: 2(9) − 21 + 3 = 18 − 21 + 3 = 0 ✓
Answer
x = 3 or x = 1/2
Quadratic Formula — Solution Cases Based on Discriminant
| Discriminant (Δ = b² − 4ac) | Sign | Nature of Roots | Example |
|---|---|---|---|
| Δ > 0 | Positive | Two distinct real roots | x² − 5x + 4 = 0 → x = 4 or x = 1 |
| Δ = 0 | Zero | One repeated real root | x² − 4x + 4 = 0 → x = 2 (double) |
| Δ < 0 | Negative | Two complex conjugate roots | x² + x + 1 = 0 → x = (−1 ± i√3)/2 |
| Δ is perfect square | Positive | Two rational roots | x² − 5x + 6 = 0 → x = 2 or x = 3 |
Interactive Tools
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Related Terms
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.
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.
The word "quadratic" comes from Latin "quadratus" (square). The formula itself was known in various forms to Babylonian, Greek, and Indian mathematicians. Brahmagupta gave a clear version in 628 CE in his "Brahmasphutasiddhanta". The symbolic form using coefficients a, b, c emerged with René Descartes and later Leonhard Euler in the 17th–18th centuries.