Quadratic equations are a cornerstone of algebra and appear frequently in physics, engineering, and economics to model parabolic motion, optimize costs, or calculate trajectories. Any equation that can be rearranged into the standard form of ax² + bx + c = 0 is a quadratic equation. Solving these by hand using factoring or completing the square can be time-consuming and prone to errors, especially when dealing with non-integer coefficients. Our Quadratic Equation Solver uses the universal quadratic formula to find the exact roots (solutions) for any set of coefficients, and it even handles complex numbers when the discriminant is negative.
结果
x₁ = -2.0000, x₂ = -3.0000
Roots for 1x² + 5x + 6 = 0 are -2.0000 and -3.0000
x = [-b ± √(b² - 4ac)] / 2a
a1
b5
c6
判别式1
工作原理
This solver takes your inputs for a, b, and c and first calculates the discriminant (D = b² - 4ac). Based on the value of the discriminant, it then applies the quadratic formula. If the discriminant is non-negative, it provides one or two real roots. If it is negative, the tool calculates the real and imaginary parts separately to provide the two complex conjugate roots in the form of 'x + yi'.
计算公式
x = [-b ± √(b² - 4ac)] / 2a
Calculation Examples
1. **Two Real Roots**: Solve x² - 5x + 6 = 0. Here a=1, b=-5, c=6. The solver finds roots **x = 2** and **x = 3**.
2. **One Real Root**: Solve x² - 4x + 4 = 0. Here a=1, b=-4, c=4. The discriminant is 0, so there is a single repeated root at **x = 2**.
3. **Complex Roots**: Solve x² + x + 1 = 0. The discriminant is -3, so the roots are complex: **-0.5 ± 0.866i**.
Why use this tool in your daily life?
This tool is an essential productivity booster for students and researchers. By providing instant, error-free roots for any coefficient, it serves as a reliable validator for your homework and a high-speed tool for real-world modeling.