How do I use the Quadratic Solver?

Enter the coefficients a, b, and c from your quadratic equation ax² + bx + c = 0. The solver will compute the roots using the quadratic formula and display the solutions, including complex roots if the discriminant is negative.

What if the discriminant is negative?

If the discriminant (b² - 4ac) is negative, the equation has no real roots. The solver will show the complex roots in the form of a + bi, where i is the imaginary unit.

Can the solver handle a = 0?

If a = 0, the equation is not quadratic but linear. The solver will indicate that the equation is not quadratic and may not provide a solution. Ensure a ≠ 0 for a valid quadratic equation.

数学

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二次方程求解器.

求解 ax² + bx + c = 0 二次方程

设备端计算无需注册

输入数值

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02 / 答案

求解：ax² + bx + c = 0

系数 a

系数 b

系数 c

1x² -5x + 6 = 0

判别式分析

判别式 (Δ)1.0000

根的性质✓ 两个不等实根

解 x₁

3.0000

解 x₂

2.0000

抛物线性质

顶点(2.5000, -0.2500)

对称轴x = 2.5000

Y 轴截距(0, 6)

开口方向↑ 向上（最小值）

理解二次方程

求根公式：x = (-b ± √(b² - 4ac)) / 2a。此公式可求解任意二次方程 ax² + bx + c = 0。

判别式 (Δ)：Δ = b² - 4ac 的值决定根的性质：

• Δ > 0：两个不等实根（抛物线与 x 轴相交于两点）
• Δ = 0：一个重根（抛物线与 x 轴相切）
• Δ < 0：两个共轭复根（抛物线不与 x 轴相交）

顶点式：顶点代表抛物线的最小值点（a > 0）或最大值点（a < 0）。顶点位于 x = -b/(2a) 处。

Cite this calculator

Canonical URL: https://mathify.one/zh/math/quadratic

Cite as: Mathify. (n.d.). 二次方程求解器. https://mathify.one/zh/math/quadratic

Use Cases

Homework and Exam Preparation

Quickly verify your manual solutions to quadratic equations, check your work, and understand the step-by-step process.

Example: Solve x² - 5x + 6 = 0 to get roots 2 and 3.

Engineering and Physics Calculations

Find roots of quadratic equations that arise in projectile motion, circuit analysis, and optimization problems.

Example: Determine the time when a projectile hits the ground using h = -4.9t² + 20t + 1.

Frequently Asked Questions

How do I use the Quadratic Solver?: Enter the coefficients a, b, and c from your quadratic equation ax² + bx + c = 0. The solver will compute the roots using the quadratic formula and display the solutions, including complex roots if the discriminant is negative.
What if the discriminant is negative?: If the discriminant (b² - 4ac) is negative, the equation has no real roots. The solver will show the complex roots in the form of a + bi, where i is the imaginary unit.
Can the solver handle a = 0?: If a = 0, the equation is not quadratic but linear. The solver will indicate that the equation is not quadratic and may not provide a solution. Ensure a ≠ 0 for a valid quadratic equation.

Tips & Common Mistakes

Tips

Ensure the equation is in standard form ax² + bx + c = 0 before entering coefficients.
Double-check the signs of b and c; a common error is entering a negative b as positive.
Use the discriminant (b² - 4ac) to predict the nature of roots: positive for two real, zero for one real, negative for complex.
If a = 0, the equation is linear, not quadratic, so the solver may not work correctly.

Common Mistakes to Avoid

Entering coefficients in the wrong order (e.g., swapping a and c).
Forgetting to include the sign of the coefficient (e.g., entering 5 instead of -5 for b).
Assuming the solver can handle a = 0; it's designed for quadratic equations only.

Last updated: August 13, 2026