How to Solve Quadratic Equations
Our quadratic equation solver finds the roots of ax² + bx + c = 0 using the quadratic formula. Includes real and complex solutions, discriminant analysis, and vertex form conversion.
Understanding Quadratic Equations
Quadratic equations form parabolas when graphed. They have the form ax² + bx + c = 0 where a ≠ 0. Solutions (roots) are where the parabola crosses the x-axis. The discriminant determines how many real solutions exist.
Quadratic Formula
x = (-b ± √(b² - 4ac)) / 2a. The discriminant D = b² - 4ac determines solutions: D > 0 → two real roots, D = 0 → one real root, D < 0 → two complex roots.
Example:
2x² + 5x - 3 = 0: a=2, b=5, c=-3. D = 25 + 24 = 49. x = (-5 ± 7)/4. Solutions: x = 0.5 and x = -3.
Common Use Cases
Real-world applications for this calculator
Physics
Projectile motion, falling objects, and trajectory problems.
Engineering
Optimization problems and system modeling.
Finance
Profit maximization and break-even analysis.
Tips
- Always check solutions by substituting back into original equation.
- Factor first if coefficients are small integers.
- The sum of roots = -b/a; product of roots = c/a.
- Completing the square gives vertex form directly.
Frequently Asked Questions
What is the quadratic formula?
x = (-b ± √(b² - 4ac)) / 2a. It solves any equation ax² + bx + c = 0. The ± gives two solutions (or one if discriminant is 0).
What is the discriminant?
The discriminant is b² - 4ac (the part under the square root). If positive: 2 real solutions. If zero: 1 real solution. If negative: 2 complex solutions.
Can I solve quadratics without the formula?
Yes! Try factoring (reverse FOIL), completing the square, or graphing. The quadratic formula always works, but other methods can be faster for simple equations.
What is the vertex of a parabola?
The vertex is the minimum or maximum point. Its x-coordinate is -b/(2a). Substitute back to find y. The vertex form is a(x - h)² + k where (h, k) is the vertex.
What are complex roots?
When the discriminant is negative, roots involve imaginary numbers (i = √-1). For example, x² + 1 = 0 has roots x = ±i. These represent parabolas that don't cross the x-axis.