Use this free Quadratic Equation Solver to calculate real or complex roots, view detailed steps, check the vertex and axis of symmetry, and generate a parabola graph instantly.
Quadratic Equation Solver – Step-by-Step with Graph
Solve quadratic equations of the form ax² + bx + c = 0
Use our free Quadratic Equation Solver to calculate roots of equations in the form ax² + bx + c = 0. This interactive tool provides step-by-step solutions, discriminant analysis, vertex, axis of symmetry, table of values, and a dynamic parabola graph. Ideal for students, teachers, engineers, and exam prep.
What is a Quadratic Equation?
A quadratic equation is a second-degree polynomial of the form:
\(ax^2 + bx + c = 0\), where \(a \neq 0\).
- a: coefficient of \(x^2\)
- b: coefficient of \(x\)
- c: constant term
Quadratic Formula
The roots of a quadratic equation are calculated using:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
The discriminant is:
\[ \Delta = b^2 – 4ac \]
- \(\Delta > 0\): two distinct real roots
- \(\Delta = 0\): one repeated real root
- \(\Delta < 0\): two complex roots
Graph of a Quadratic
The graph of \(y = ax^2 + bx + c\) is a parabola:
- Opens upward if \(a > 0\)
- Opens downward if \(a < 0\)
Vertex coordinates:
\[ x_v = -\frac{b}{2a}, \quad y_v = \frac{4ac – b^2}{4a} \]
Axis of symmetry: \(x = -\frac{b}{2a}\)
How to Use the Solver
- Enter values of a, b, and c.
- Click Solve Equation.
- View step-by-step solution, discriminant, vertex, and range.
- See the interactive parabola graph and table of values.
- Reset or download the graph as PNG.
Features
- Step-by-step solution with discriminant
- Root classification with badges
- Interactive parabola graph
- Highlights real roots on the graph
- Displays vertex, axis, domain, and range
- Table of values near vertex
- Reset and download options
Examples
Example 1: Physics — Projectile Motion
A ball is thrown upward: \(h(t) = -4.9t^2 + 20t + 1\).
Equation: \(-4.9t^2 + 20t + 1 = 0\).
Solution gives:
- \(t \approx -0.049\) (ignore — negative time)
- \(t \approx 4.131\) s → time when the ball hits ground
Example 2: Business — Profit Maximization
Profit function: \(P(x) = -5x^2 + 100x\).
Vertex at \(x = -b/2a = 10\). So maximum profit occurs when 10 items are sold.
Maximum profit value: \(P(10) = 500\).
Who Can Use This?
- High-school & college students
- Teachers explaining quadratic concepts
- Physics & engineering learners
- Business users analyzing profit/cost models
- Exam prep (JEE, SAT, GRE, etc.)
Related Concepts
- Factoring quadratic expressions
- Completing the square
- Vertex form: \(y = a(x-h)^2 + k\)
Frequently Asked Questions
What if a = 0?
If \(a = 0\), the equation is linear, not quadratic. Use non-zero a.
Why does the graph not show complex roots?
Complex roots don’t lie on the real axis, so they can’t be plotted on the parabola. The solver still lists them algebraically.
Can I get exact symbolic roots?
The solver shows formula steps and decimals. For exact symbolic form, use a CAS tool.
Why is the graph hidden until I click Solve?
To keep the page fast and clean — the graph only loads after valid inputs.
Can I download the chart?
Yes, use the Download Graph button once the graph is displayed.
Developed by C4Calculators — trusted, free tools for students, teachers, and engineers.