Quadratic Equation Solver
Our free quadratic equation solver helps you find exact solutions for any quadratic equation in the form ax² + bx + c = 0. Enter the coefficients below and get complete step-by-step solutions, discriminant analysis, and visualization of your equation.
Notes:
- A quadratic equation has the standard form ax² + bx + c = 0, where a ≠ 0.
- The discriminant (Δ = b² - 4ac) tells you the nature of the solutions: positive for two real solutions, zero for one real solution, negative for two complex solutions.
- The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) gives exact solutions.
- For complex roots, the answers are expressed in the form a + bi, where i is the imaginary unit.
- All decimals are rounded to 4 significant digits for display purposes.
- This calculator is designed for educational purposes to help students learn the quadratic formula.
Understanding Quadratic Equations
A quadratic equation is a second-degree polynomial equation in a single variable of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. These equations appear frequently in mathematics, science, engineering, and everyday applications such as calculating areas, analyzing projectile motion, or optimizing resources. Our quadratic equation solver provides complete step-by-step solutions, making it ideal for students learning algebra, teachers creating lesson plans, or anyone needing to solve quadratic equations accurately.
Frequently Asked Questions
What is a quadratic equation?
A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term in which the variable is squared (x²). The standard form is ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions to a quadratic equation are called roots or zeros.
How do you solve a quadratic equation?
There are several methods to solve quadratic equations:
- Factoring: If the quadratic expression can be factored as (px + q)(rx + s) = 0, then the solutions are x = -q/p and x = -s/r.
- Quadratic Formula: The solutions are given by x = (-b ± √(b² - 4ac)) / (2a).
- Completing the Square: Rewriting the equation to the form a(x + d)² = e and then solving for x.
- Graphical Method: Finding the x-intercepts of the parabola y = ax² + bx + c.
What does the discriminant tell me?
The discriminant is the expression under the square root in the quadratic formula: Δ = b² - 4ac. It determines the nature of the solutions:
- If Δ > 0, the equation has two distinct real solutions.
- If Δ = 0, the equation has exactly one real solution (a repeated root).
- If Δ < 0, the equation has two complex conjugate solutions.
Can this calculator show complex or imaginary roots?
Yes, our calculator can solve quadratic equations with complex roots (when the discriminant is negative). These solutions are displayed in the standard form a + bi, where i represents the imaginary unit (√-1). You can toggle between showing all solutions or only real solutions using the options provided.
Does this solver explain the steps or just give the answer?
Our quadratic equation solver provides a complete step-by-step solution, showing how to calculate the discriminant, apply the quadratic formula, and simplify the results. This makes it an excellent educational tool for learning how to solve quadratic equations, not just getting the final answer.
Real-World Applications of Quadratic Equations
Physics and Motion
Quadratic equations describe the path of projectiles under gravity, helping to calculate the height, distance, or time of flight of objects in motion.
Geometry and Area
When finding dimensions of rectangles with known area or perimeter, quadratic equations often arise in the calculation process.
Economics and Business
Quadratic equations help optimize profit functions, determine break-even points, and model supply-demand relationships in markets.
Engineering and Design
Engineers use quadratic equations when designing parabolic structures, calculating electrical circuits, or modeling fluid dynamics.