Quadratic Equation Solver
Solve quadratic equations of the form ax² + bx + c = 0. Get the discriminant, real or complex roots, vertex, and axis of symmetry with full step-by-step solutions.
ax² + bx + c = 0
How to Solve a Quadratic Equation
- Enter coefficients a, b, and c for the equation ax² + bx + c = 0.
- Click Solve — the calculator computes the discriminant D = b² − 4ac.
- Read the roots: two real roots if D > 0, one repeated root if D = 0, or complex roots if D < 0.
- Review the step-by-step solution to verify each stage of the calculation.
مرجع سريع
| من | إلى |
|---|---|
| x² + 5x + 6 = 0 | x = −2, −3 |
| x² − 4 = 0 | x = 2, −2 |
| 2x² − 3x − 2 = 0 | x = 2, −0.5 |
| x² − 1 = 0 | x = 1, −1 |
| x² + 2x + 1 = 0 | x = −1 |
| x² − 5x + 6 = 0 | x = 2, 3 |
حالات الاستخدام
- •Algebra class — solve x² + 5x + 6 = 0 to find roots for a homework assignment.
- •Physics — determine when a projectile hits the ground using a height equation.
- •Engineering — find equilibrium points in systems modeled by quadratic equations.
الصيغة
The quadratic formula is x = (−b ± √(b² − 4ac)) / 2a. The discriminant D = b² − 4ac determines the nature of the roots: D > 0 gives two real roots, D = 0 gives one repeated root, D < 0 gives two complex roots.
الأسئلة الشائعة
What is the discriminant and what does it tell us?
The discriminant D = b² − 4ac indicates the nature of the equation's roots. If D > 0, there are two distinct real roots. If D = 0, there is exactly one real root (a repeated root). If D < 0, there are two complex conjugate roots.
What is the vertex of a parabola?
The vertex is the highest or lowest point of the parabola y = ax² + bx + c. Its coordinates are (−b/(2a), f(−b/(2a))). If a > 0 the vertex is a minimum; if a < 0 it is a maximum.
Can a quadratic equation have no solutions?
Every quadratic equation has exactly two roots (counting multiplicity) in the complex numbers. However, when the discriminant is negative, there are no real solutions — only complex ones.