Quadratic Formula Calculator
Solve quadratic equations ax² + bx + c = 0 using the quadratic formula. Find roots, discriminant, vertex, and view the parabola graph with step-by-step solution.
Formula:x = (-b ± √(b²-4ac)) / 2a
Solutions
Roots
3, 2
Two real roots
Discriminant1
Vertex(2.5, -0.25)
Y-Intercept6
Try These Examples
Common quadratic equations
Enter Coefficients
For the equation ax² + bx + c = 0
Your equation:
x² - 5x + 6 = 0
Parabola Graph
Opens upward (∪)Axis: x = 2.5
Solutions (Roots)
First Root (x₁)
3
Second Root (x₂)
2
Factored Form
(x -3)(x -2) = 0
Parabola Properties
Discriminant
1
b² - 4ac
Vertex
(2.5, -0.25)
Minimum point
Axis of Symmetry
x = 2.5
-b / (2a)
Y-Intercept
(0, 6)
Where x = 0
Step-by-Step Solution
1. Identify coefficients:
a = 1, b = -5, c = 6
2. Calculate the discriminant (b² - 4ac):
Δ = (-5)² - 4(1)(6)
Δ = 25 - 24
Δ = 1
Since Δ > 0, there are two distinct real roots
3. Apply the quadratic formula:
x = (-b ± √Δ) / (2a)
x = (-(-5) ± √1) / (2 × 1)
x = (5 ± 1) / 2
4. Solutions:
x = 3, 2
Quadratic Formula
Interactive formula breakdown
a-var="x">x = (-b ± √(b²-4ac)) / 2a
x
Solution
The values that satisfy the equation
Current value: x = 3, 2
a
Quadratic coefficient
Coefficient of x² term
Current value: 1
b
Linear coefficient
Coefficient of x term
Current value: -5
c
Constant term
The constant value
Current value: 6
Δ
Discriminant
b² - 4ac, determines root type
Current value: 1
Practice Quadratic Problems
Test your understanding of quadratic equations
Practice with 6 problems to test your understanding.
Solutions
Roots
3, 2
Two real roots
Discriminant1
Vertex(2.5, -0.25)
Y-Intercept6
?How to Use the Quadratic Formula
The quadratic formula x = (-b +/- sqrt(b^2-4ac)) / 2a solves any quadratic equation ax^2 + bx + c = 0. The discriminant (b^2-4ac) determines the nature of roots: if positive, two real roots exist; if zero, one repeated root; if negative, two complex conjugate roots. The vertex is at x = -b/(2a), and the parabola opens up when a > 0.
What is a Quadratic Equation?
A quadratic equation is a polynomial equation of degree 2 in the form ax^2 + bx + c = 0, where a is not equal to 0. The quadratic formula provides a method to find the values of x (called roots or solutions) that make the equation true. These roots represent the x-intercepts of the corresponding parabola on a graph.
Key Facts About Quadratic Equations
- Quadratic formula: x = (-b +/- sqrt(b^2-4ac)) / 2a
- Discriminant (b^2-4ac) determines root type: positive = 2 real, zero = 1 real, negative = 2 complex
- Vertex of parabola is at x = -b/(2a), which is the axis of symmetry
- Parabola opens upward when a > 0 (minimum at vertex), downward when a < 0 (maximum)
- Y-intercept is always the constant term c (where x = 0)
- Complex roots always come in conjugate pairs: a + bi and a - bi
- Factored form exists when roots are rational: (x - r1)(x - r2) = 0
- Sum of roots = -b/a and product of roots = c/a (Vieta's formulas)
Quick Answer
The quadratic formula x = (-b +/- sqrt(b^2-4ac)) / 2a solves any quadratic equation ax^2 + bx + c = 0. The discriminant (b^2-4ac) determines the nature of roots: if positive, two real roots exist; if zero, one repeated root; if negative, two complex conjugate roots. The vertex is at x = -b/(2a), and the parabola opens up when a > 0.
Frequently Asked Questions
The quadratic formula x = (-b ± √(b²-4ac)) / 2a solves any quadratic equation ax² + bx + c = 0. It gives the x-values where the parabola crosses the x-axis (the roots or solutions).
The discriminant (b² - 4ac) determines the nature of the roots. If positive, there are two distinct real roots. If zero, there is one repeated real root. If negative, there are two complex conjugate roots.
The vertex is the turning point of the parabola - its highest or lowest point. For y = ax² + bx + c, the vertex is at x = -b/(2a), and the y-coordinate is found by substituting this x value.
The axis of symmetry is the vertical line that divides the parabola into two mirror images. It passes through the vertex and has the equation x = -b/(2a).
A parabola opens upward (∪ shape) when a > 0, with a minimum at the vertex. It opens downward (∩ shape) when a < 0, with a maximum at the vertex.
Complex roots occur when the discriminant is negative. They involve the imaginary unit i (where i² = -1) and always come in conjugate pairs like a + bi and a - bi. The parabola doesn't cross the x-axis.
Last updated: 2025-01-15
How this works
Formulas follow standard definitions from the NIST Digital Library of Mathematical Functions and classical textbook derivations. Calculations run entirely in your browser. Where a closed-form solution exists, it is used; where an iterative or numerical method is required, the implementation is named on the page.
Sources
- [1]NIST Digital Library of Mathematical FunctionsAcademicdlmf.nist.govAccessed Apr 21, 2026
Joseph OrdunaFounder & Software Engineer
Full-stack software engineer specializing in embedded systems, web architecture, and AI/ML. Founder of Practical Web Tools. Built the gesture-controlled drone IP acquired by KD Interactive (Aura Drone, sold on Amazon).
Solutions
Roots
3, 2
Two real roots
Discriminant1
Vertex(2.5, -0.25)
Y-Intercept6