Quadratic Equation Solver - Calculate Roots with Steps
Solve quadratic equations (ax² + bx + c = 0) instantly. Find real and complex roots, the discriminant, and the vertex with step-by-step explanations.
Quadratic Equation Inputs
Results
What is a Quadratic Equation Solver?
A Quadratic Equation Solver is a mathematical tool that finds the roots (solutions) of a quadratic equation in the form ax² + bx + c = 0. It determines the values of x where the parabola crosses the x-axis.
This calculator works for:
- Finding Roots - Calculates the two values of x that satisfy the equation.
- Vertex Calculation - Identifies the peak or valley of the parabola.
- Discriminant Analysis - Determines the nature of the roots (real or complex).
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How the Quadratic Solver Works
The calculation uses the Quadratic Formula:
Where:
- a, b, c = Coefficients of the equation
- ± = Indicates two solutions (one plus, one minus)
- b² - 4ac = Discriminant (Δ)
Key Concepts Explained
Discriminant (Δ)
The value b² - 4ac determines the number and type of roots.
Vertex
The turning point of the parabola, given by (-b/2a, f(-b/2a)).
How to Use This Calculator
Enter Coefficients
Input values for a, b, and c.
Review Equation
Check the displayed equation format.
Get Results
See roots, vertex, and steps.
Benefits of Using This Calculator
- • Step-by-Step Logic: See exactly how the solution is derived using the formula.
- • Complex Roots Support: Handles equations with no real solutions correctly.
- • Instant Analysis: Get discriminant and vertex information immediately.
Factors Affecting Results
1. Value of a
If a = 0, the equation is linear, not quadratic. This calculator handles that check.
2. Sign of Discriminant
Positive = 2 real roots, Zero = 1 real root, Negative = 2 complex roots.
Frequently Asked Questions (FAQ)
Q: What is a quadratic equation?
A: A quadratic equation is a polynomial equation of the second degree, typically written in the form ax² + bx + c = 0, where a, b, and c are constants and a is not equal to zero.
Q: What is the discriminant?
A: The discriminant is the part of the quadratic formula underneath the square root symbol: b² - 4ac. It tells you how many roots the equation has and whether they are real or complex.
Q: How many roots can a quadratic equation have?
A: A quadratic equation always has two roots. If the discriminant is positive, there are two distinct real roots. If it's zero, there is one real root (repeated). If it's negative, there are two complex roots.
Q: Can this calculator solve for complex roots?
A: Yes, this calculator can solve for complex roots when the discriminant is negative. It will display the roots in the form x = real ± imaginary i.
Q: What is the vertex of a parabola?
A: The vertex is the highest or lowest point on the parabola represented by the quadratic equation. Its x-coordinate is given by -b/(2a).