Quadratic Equation Solver
Solve quadratic equations ax² + bx + c = 0 with step-by-step solutions and graphical visualization
ax² + bx + c = 0
x² coefficient
x coefficient
Constant term
Coefficient 'a' must not be zero for a quadratic equation.
First Root (x₁)
Second Root (x₂)
Discriminant (Δ)
Δ = b² - 4ac
Vertex X
Vertex Y
Sum of Roots
Product of Roots
Step-by-Step Solution
The Quadratic Formula
x = (-b ± √(b² - 4ac)) / 2a
Δ > 0
Two distinct real roots. The parabola crosses the x-axis twice.
Δ = 0
One repeated root. The parabola touches the x-axis at its vertex.
Δ < 0
Two complex roots. The parabola never crosses the x-axis.
Vieta's Formulas
For a quadratic equation ax² + bx + c = 0 with roots r₁ and r₂:
Sum of Roots
r₁ + r₂ = -b/a
Product of Roots
r₁ × r₂ = c/a
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About Quadratic Equation Solver
What is a Quadratic Equation Solver?
A Quadratic Equation Solver is a tool that finds the solutions (roots) to quadratic equations in the standard form ax² + bx + c = 0, where a, b, and c are coefficients and a ≠ 0. The solutions represent the x-values where the parabola crosses the x-axis.
How to Use This Calculator
- Enter Coefficient a: The coefficient of x² (must not be zero)
- Enter Coefficient b: The coefficient of x
- Enter Coefficient c: The constant term
- View Results: See both roots, discriminant analysis, and step-by-step solution
The Quadratic Formula
The quadratic formula provides the solutions for any quadratic equation:
x = (-b ± √(b² - 4ac)) / 2a
This formula gives two solutions:
- x₁ = (-b + √(b² - 4ac)) / 2a
- x₂ = (-b - √(b² - 4ac)) / 2a
Understanding the Discriminant
The discriminant (Δ = b² - 4ac) determines the nature of the roots:
Positive Discriminant (Δ > 0)
Two distinct real roots. The parabola crosses the x-axis at two different points.
Zero Discriminant (Δ = 0)
One repeated real root (double root). The parabola touches the x-axis at exactly one point (vertex).
Negative Discriminant (Δ < 0)
Two complex conjugate roots. The parabola does not cross the x-axis.
Alternative Solution Methods
Factoring
When the equation can be written as a(x - r₁)(x - r₂) = 0, where r₁ and r₂ are the roots.
Completing the Square
Rewriting the equation in the form a(x - h)² = k, then solving for x.
Graphical Method
Finding where the parabola y = ax² + bx + c intersects the x-axis.
Vertex Form and Properties
The vertex of the parabola is at:
- x-coordinate (h): -b / 2a
- y-coordinate (k): c - b² / 4a
The axis of symmetry is the vertical line x = -b / 2a.
Frequently Asked Questions
What if coefficient 'a' equals zero?
If a = 0, the equation becomes linear (bx + c = 0), not quadratic. The solution is simply x = -c/b.
Can quadratic equations have no solution?
In the real number system, if the discriminant is negative, there are no real solutions. However, there are always two solutions in the complex number system.
What is the relationship between roots and coefficients?
For ax² + bx + c = 0 with roots r₁ and r₂:
- Sum of roots: r₁ + r₂ = -b/a
- Product of roots: r₁ × r₂ = c/a
How do I verify my solutions?
Substitute each root back into the original equation. If ax² + bx + c = 0, the solution is correct.
Note: This calculator handles both real and complex solutions, showing the complete step-by-step working for educational purposes.