Precision Utility
Quadratic Equation
Calculator
Standard Form
ax² + bx + c = 0
Solution
Quadratic Formula
Solve any quadratic equation in seconds. Enter the coefficients a, b and c and the calculator instantly finds both roots, the discriminant, the vertex and the axis of symmetry. Handles real and complex roots.
Equation Coefficients
Your Equation
1x² - 5x + 6 = 0
Solutions
x = 2, x = 3
Root Type
2 Real Roots
Discriminant
1
Vertex X
2.5
Vertex Y
-0.25
Root 1
x = 3
Root 2
x = 2
Discriminant (b²-4ac)
1
Vertex (-b/2a, f(-b/2a))
(2.5, -0.25)
How the quadratic calculator works
Enter the three coefficients of your quadratic equation: a (the x² coefficient), b (the x coefficient) and c (the constant term). The equation must be in the standard form ax² + bx + c = 0.
The calculator applies the quadratic formula x = (-b ± sqrt(b² - 4ac)) / 2a to find both solutions instantly. It first computes the discriminant (b² - 4ac) to determine whether the roots are real or complex.
You will also see the vertex of the parabola, calculated as (-b/2a, f(-b/2a)). The vertex is the maximum or minimum point of the curve y = ax² + bx + c. If a is positive the parabola opens upward and the vertex is a minimum; if a is negative it opens downward and the vertex is a maximum.
Results update automatically as you type — no need to press a button.
What you need to know about quadratic equations
A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b and c are real numbers and a is not zero. The solutions (also called roots or zeros) are the x-values where the parabola crosses the x-axis.
Key concepts:
- The quadratic formula — x = (-b ± sqrt(b² - 4ac)) / 2a solves every quadratic equation. It works whether the roots are rational, irrational or complex.
- The discriminant — the value b² - 4ac determines the nature of the roots. Positive means two distinct real roots, zero means one repeated real root, negative means two complex conjugate roots.
- Real vs complex roots — real roots are ordinary numbers on the number line. Complex roots involve the imaginary unit i (where i² = -1) and always come in conjugate pairs: a + bi and a - bi.
- The vertex — the turning point of the parabola is at x = -b/(2a). Substituting back gives the y-coordinate. This is the minimum value when a > 0 or the maximum when a < 0.
- Axis of symmetry — the vertical line x = -b/(2a) divides the parabola into two equal halves.
Quadratic equations appear throughout maths, physics and engineering — from projectile motion and area problems to optimisation and circuit analysis.
Quadratic equations in UK education and engineering
Quadratic equations are one of the most heavily tested topics in UK secondary mathematics. At GCSE level, all three major exam boards — AQA, Edexcel and OCR — require students to solve quadratics by factorising, completing the square and using the quadratic formula. On the higher tier, questions often combine quadratics with simultaneous equations, inequalities or graph sketching, and are typically worth 3–5 marks per question.
At A-Level Mathematics, quadratics become foundational to almost every pure maths topic. The discriminant is used to determine the number of intersections between a line and a curve, completing the square is essential for coordinate geometry (finding the centre and radius of a circle), and the quadratic formula underpins work on polynomial roots and factor theorems. Both Edexcel and OCR A-Level specifications list quadratic functions as a core Year 12 topic.
Beyond the classroom, quadratic equations are used extensively in UK engineering and physics. Structural engineers apply them when calculating beam deflections and load distributions under British Standards (BS 5950 for steel, BS 8110 for concrete). In physics, projectile motion — studied in every A-Level mechanics module — is modelled with quadratic equations, using metric units (metres and seconds) as standard in the UK. Civil engineers working on UK road design use quadratic curves to calculate vertical alignment and sight distances on highways built to DMRB (Design Manual for Roads and Bridges) standards.
Worked examples for UK students
Example 1: GCSE factorisation question
Solve x² − 7x + 12 = 0. (Typical AQA/Edexcel GCSE question, 2 marks.)
Solution: Find two numbers that multiply to give 12 and add to give −7. Those numbers are −3 and −4. So x² − 7x + 12 = (x − 3)(x − 4) = 0. Therefore x = 3 or x = 4. You can verify with this calculator by entering a = 1, b = −7, c = 12.
Example 2: A-Level projectile motion
A cricket ball is hit vertically upward from a height of 1.2 metres at 15 m/s. Taking g = 9.8 m/s², when does it hit the ground?
Solution: Height h = 1.2 + 15t − 4.9t². Setting h = 0 gives −4.9t² + 15t + 1.2 = 0, or equivalently 4.9t² − 15t − 1.2 = 0. Using the quadratic formula with a = 4.9, b = −15, c = −1.2: discriminant = 225 + 23.52 = 248.52, so t = (15 ± √248.52) / 9.8. The positive root gives t ≈ 3.14 seconds. The ball hits the ground after approximately 3.14 seconds — a standard A-Level mechanics problem using SI units.
Example 3: Garden fencing optimisation
A UK homeowner has 20 metres of fencing to enclose a rectangular vegetable patch against an existing garden wall. What dimensions give the maximum area?
Solution: Let the width perpendicular to the wall be x metres. The length parallel to the wall is (20 − 2x) metres. Area A = x(20 − 2x) = 20x − 2x². To find the maximum, note this is a downward-opening parabola with vertex at x = −20 / (2 × −2) = 5 metres. So the optimal dimensions are 5 m × 10 m, giving a maximum area of 50 m² — a practical application of quadratic optimisation.
Frequently asked questions
What is the quadratic formula?
The quadratic formula is x = (-b ± sqrt(b² - 4ac)) / 2a. It finds the values of x that satisfy any equation in the form ax² + bx + c = 0, where a is not zero.
What is the discriminant and what does it tell you?
The discriminant is b² - 4ac. If it is positive the equation has two distinct real roots, if it equals zero there is one repeated root, and if it is negative the equation has two complex (imaginary) roots.
Can a quadratic equation have no real solutions?
Yes. When the discriminant (b² - 4ac) is negative, there are no real solutions. Instead the equation has two complex roots expressed in the form a + bi and a - bi, where i is the imaginary unit.
What is the vertex of a parabola?
The vertex is the turning point of the parabola — its highest or lowest point. For y = ax² + bx + c the vertex x-coordinate is -b/(2a) and the y-coordinate is found by substituting that x back into the equation.
What happens when a = 0?
When a = 0 the equation is no longer quadratic — it becomes the linear equation bx + c = 0. The single solution is x = -c/b (assuming b is also not zero).
How do I find the axis of symmetry?
The axis of symmetry of a parabola y = ax² + bx + c is the vertical line x = -b/(2a). It passes through the vertex and divides the parabola into two mirror-image halves.