Quadratic Formula Calculator
Solve quadratic equations with step-by-step solutions
Quadratic Formula Calculator
Use our quadratic formula calculator to solve quadratic equations of the form ax² + bx + c = 0. Enter the coefficients a, b, and c to find the roots.
📊 Equation Coefficients
✅ Solution
Solution Steps
📚 Understanding Quadratic Equations
The Quadratic Formula
The quadratic formula x = (-b ± √(b² - 4ac)) / 2a provides the exact solutions to any quadratic equation ax² + bx + c = 0.
The Discriminant
The discriminant (b² - 4ac) determines the nature of the roots: positive means two real roots, zero means one repeated root, negative means complex roots.
Real-World Applications
Quadratic equations model projectile motion, optimize profit functions, calculate areas, and solve many physics and engineering problems.
What is a Quadratic Equation?
A quadratic equation is a polynomial equation of the second degree with the general form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants and 'a' ≠ 0. The term 'quadratic' comes from the Latin word 'quadratus', meaning square, because the variable is squared.
The solutions to a quadratic equation are called roots or zeros, and they represent the x-values where the parabola (the graph of the quadratic function) intersects the x-axis.
Types of Solutions
- Two Real Roots: When discriminant > 0, the parabola crosses the x-axis at two points
- One Real Root: When discriminant = 0, the parabola touches the x-axis at one point (vertex)
- Complex Roots: When discriminant < 0, the parabola doesn't intersect the x-axis
- Perfect Square: When the discriminant is a perfect square, roots are rational numbers