Quadratic Equation Solver
Solve any quadratic equation ax² + bx + c = 0 instantly. Get the discriminant, roots, step-by-step working and a parabola graph. Pro Mode adds vertex form, factored form, completing the square, Vieta's formulas and full curve analysis.
Quadratic Equation Solver — Step-by-Step Working, Graph and Full Analysis
A quadratic equation has the form ax² + bx + c = 0 where a ≠ 0. It is one of the most fundamental equation types in mathematics — appearing in physics (projectile motion), engineering (structural loads), economics (profit maximisation), and throughout GCSE, A-Level and university maths. Our free quadratic equation solver finds the roots instantly using the quadratic formula, shows the full step-by-step working including verification, and classifies the roots using the discriminant. Switch to Pro Mode for the parabola graph, three equation forms, completing the square, Vieta's formulas and complete curve analysis.
The quadratic formula works for every quadratic equation without exception. Unlike factoring — which only works cleanly when roots are rational and easy to spot — the formula handles irrational roots, repeated roots and complex roots with equal reliability. Learning to apply it systematically, interpret the discriminant and convert between equation forms is a core requirement from GCSE through to university mathematics.
Why the discriminant matters before you solve: Checking b² − 4ac before applying the formula tells you immediately what type of roots to expect — without any further calculation. A negative discriminant means no real solutions exist regardless of how the equation looks. Knowing this in advance saves time, avoids errors and is often asked directly in exam questions: "Without solving, determine the nature of the roots."
The Three Cases — What the Discriminant Tells You
Worked Example — Solving x² − 5x + 6 = 0 Step by Step
The Four Equation Forms — Standard, Vertex, Factored and Completed Square
| Form | Expression | What It Shows | When to Use |
|---|---|---|---|
| Standard | ax² + bx + c = 0 | Coefficients a, b, c directly visible | Starting point — apply quadratic formula directly |
| Vertex | a(x − h)² + k = 0 | Vertex (h, k) immediately visible | Graphing, finding max/min, identifying axis of symmetry |
| Factored | a(x − x₁)(x − x₂) = 0 | Roots x₁ and x₂ directly visible | Verifying roots, expanding back to check working |
| Completing the square | (x + b/2a)² = b²/4a² − c/a | Derivation of quadratic formula | Deriving vertex form, proving the formula, exam technique |
Vieta's Formulas — Root Relationships Without Solving
Vieta's formulas reveal a remarkable fact: you can find the sum and product of the roots of any quadratic without solving the equation at all. For ax² + bx + c = 0 with roots x₁ and x₂, the sum of roots = −b/a and the product of roots = c/a. This means for x² − 5x + 6 = 0, the sum of roots is 5 (= −(−5)/1) and the product is 6 (= 6/1) — which we can verify: 3 + 2 = 5 ✓ and 3 × 2 = 6 ✓. Vieta's formulas are frequently tested in A-Level and university entrance exams.
Pro Mode — Seven Advanced Features
How to Use the Quadratic Equation Solver
For related mathematical tools, use our Simultaneous Equation Solver for systems of two or three linear equations, our Scientific Calculator for general arithmetic and trigonometry, and our Triangle Calculator for geometry problems involving area, angles and side lengths.
Frequently Asked Questions
Quadratic equations, the formula and how this solver works
The quadratic formula is x = (−b ± √(b² − 4ac)) / 2a. It solves any quadratic equation ax² + bx + c = 0 where a ≠ 0. The ± sign means there are two solutions — one using + (giving x₁) and one using − (giving x₂). The expression under the square root, b² − 4ac, is the discriminant and determines whether the roots are real or complex without needing to complete the full calculation.
The discriminant is Δ = b² − 4ac — the expression inside the square root of the quadratic formula. Δ > 0: two distinct real roots. Δ = 0: one repeated real root (the vertex touches the x-axis). Δ < 0: two complex conjugate roots (no real solutions, parabola does not cross the x-axis). The discriminant is frequently tested directly in exams: "determine the nature of the roots without solving."
1) Identify a, b, c from ax² + bx + c = 0. 2) Calculate Δ = b² − 4ac. 3) Apply x = (−b ± √Δ) / 2a. 4) Calculate x₁ (+ branch) and x₂ (− branch). 5) Verify by substituting back — the result should equal 0. If Δ < 0, the roots are complex: x = (−b ± i√|Δ|) / 2a. This solver shows all seven steps with full working.
Vertex form is y = a(x − h)² + k where (h, k) is the vertex of the parabola. The vertex is found using h = −b/2a and k = c − b²/4a. Vertex form makes the maximum or minimum value of the function immediately visible — k is the minimum when a > 0 and the maximum when a < 0. Pro Mode shows vertex form alongside standard and factored form for every equation you solve.
Completing the square rewrites ax² + bx + c into a(x + p)² + q by adding and subtracting (b/2a)². It is used to solve quadratic equations, derive the quadratic formula, find the vertex and convert between standard and vertex form. A-Level maths frequently asks for completing the square method specifically — Pro Mode shows the full 6-step algebraic working for any equation you enter.
Vieta's formulas give the relationships between roots and coefficients without solving: for ax² + bx + c = 0, x₁ + x₂ = −b/a (sum of roots) and x₁ × x₂ = c/a (product of roots). These are used to check solutions, construct a quadratic from given roots, and solve advanced problems. Pro Mode shows Vieta's formulas with exact fraction values and a verification check for every equation.
Complex roots occur when Δ < 0. Since √(negative) is undefined in the reals, roots involve i = √(−1) and take the form a ± bi. They always appear as conjugate pairs. For x² + 2x + 5 = 0: Δ = 4 − 20 = −16, so x = −1 ± 2i. The parabola for such an equation sits entirely above or below the x-axis and does not intersect it at any point.
Simple mode gives discriminant analysis, real or complex roots, and a 7-step quadratic formula working with verification. Pro mode (free) adds: parabola graph with roots, vertex and y-intercept; three equation forms (standard, vertex, factored); completing the square steps; Vieta's formulas; full curve analysis (domain, range, concavity, intervals); decimal precision 2–10 places; and exact fraction mode for rational roots.