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.
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.