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Solve quadratic equations of the form ax² + bx + c = 0. Find real and complex roots with step-by-step logic.
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.
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
计算示例
1. **Real**: x² - 5x + 6 = 0 roots are **2 and 3**.
2. **Complex**: x² + 1 = 0 roots are **±i**.
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该工具是学生和研究人员必不可少的生产力提升工具。通过为任何系数提供即时、无错误的根，它可以作为您作业的可靠验证器和现实世界建模的高速工具。 该工具是学生和研究人员必不可少的生产力提升工具。 通过为任何系数提供即时、无错误的根，它可以作为您作业的可靠验证器和现实世界建模的高速工具。
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