Δ 判别式与一元二次方程求根Discriminant & Quadratic Roots Calculator
计算 ax² + bx + c 的判别式 Δ = b² − 4ac，并给出根（两个实根、重根或共轭复根 a ± bi）和顶点。系数按十进制精确计算，所以 0.1、0.2 这类小数不会因为浮点误差判错 Δ 的正负。本地处理。Discriminant Δ = b² − 4ac of ax² + bx + c, with the roots (two real roots, a double root or complex roots p ± qi) and the vertex. Coefficients are handled as exact decimals, so values such as 0.1 and 0.2 never get the sign of Δ wrong through floating-point error. Processed locally.
使用说明与常见问题How-to & FAQ
「判别式与一元二次方程求根」是什么？What is the Discriminant & Quadratic Roots Calculator tool?
计算 ax² + bx + c 的判别式 Δ = b² − 4ac，并给出根（两个实根、重根或共轭复根 a ± bi）和顶点。系数按十进制精确计算，所以 0.1、0.2 这类小数不会因为浮点误差判错 Δ 的正负。本地处理。Discriminant Δ = b² − 4ac of ax² + bx + c, with the roots (two real roots, a double root or complex roots p ± qi) and the vertex. Coefficients are handled as exact decimals, so values such as 0.1 and 0.2 never get the sign of Δ wrong through floating-point error. Processed locally.
如何使用「判别式与一元二次方程求根」？How do I use the Discriminant & Quadratic Roots Calculator tool?
在「判别式与一元二次方程求根」中，填写「a（x² 的系数）」、「b（x 的系数）」、「c（常数项）」等参数，结果会随输入实时计算并显示在下方。In the Discriminant & Quadratic Roots Calculator: Fill in "a (coefficient of x²)", "b (coefficient of x)", "c (constant term)"; results are recalculated instantly as you type.
怎么判断根的情形？How does Δ tell the roots apart?
Δ = b² − 4ac。Δ > 0 有两个不相等的实根，Δ = 0 有一个重根，Δ < 0 没有实根，而是一对共轭复根 p ± qi。例如 x² − 3x + 2：Δ = 1，根是 1 和 2；x² + 2x + 5：Δ = −16，根是 −1 ± 2i。Δ = b² − 4ac. Δ > 0 gives two distinct real roots, Δ = 0 one double root, and Δ < 0 no real roots but a pair of complex conjugate roots p ± qi. For x² − 3x + 2, Δ = 1 and the roots are 1 and 2; for x² + 2x + 5, Δ = −16 and the roots are −1 ± 2i.
小数系数会算错吗？Can decimal coefficients go wrong?
不会。系数按十进制精确处理，Δ 用大整数算出，所以 0.1x² + 0.2x + 0.1 的 Δ 正好是 0（重根 −1），不会因为浮点误差变成 −6.9×10⁻¹⁸ 而误判为复根。根和顶点按双精度计算并显示 10 位小数。No. Coefficients are handled as exact decimals and Δ is computed with big integers, so 0.1x² + 0.2x + 0.1 has Δ exactly 0 (double root −1) instead of a floating-point −6.9×10⁻¹⁸ that would wrongly report complex roots. Roots and the vertex use double precision and are shown to 10 decimals.
a = 0 会怎样？What if a = 0?
这时方程不是二次方程，没有判别式；页面会说明，并在 b ≠ 0 时给出一次方程 bx + c = 0 的根。Then the equation is not quadratic and has no discriminant; the page says so and, when b ≠ 0, gives the root of the linear equation bx + c = 0.
顶点怎么算？How is the vertex found?
顶点 = (−b ÷ 2a, −Δ ÷ 4a)，也就是抛物线 y = ax² + bx + c 的最高点或最低点。Vertex = (−b ÷ 2a, −Δ ÷ 4a), the highest or lowest point of the parabola y = ax² + bx + c.
能举个计算实例吗？Can you show a worked example?
例如在「判别式与一元二次方程求根」中填 a（x² 的系数） = 1，b（x 的系数） = -3，c（常数项） = 2，结果区显示：Δ = b² − 4ac：1；根的情形：Δ > 0：两个不相等的实根；x₁：1；x₂：2；顶点：(1.5, -0.25)For example, entering a (coefficient of x²) = 1, b (coefficient of x) = -3, c (constant term) = 2 in the Discriminant & Quadratic Roots Calculator gives: Δ = b² − 4ac: 1; Roots: Δ > 0: two distinct real roots; x₁: 1; x₂: 2; Vertex: (1.5, -0.25)
有哪些选项可以调整？What options can I adjust?
「判别式与一元二次方程求根」的设置：「a（x² 的系数）」默认 1；「b（x 的系数）」默认 -3；「c（常数项）」默认 2。每次生成结果都按当前设置处理。Settings in the Discriminant & Quadratic Roots Calculator — "a (coefficient of x²)" default 1; "b (coefficient of x)" default -3; "c (constant term)" default 2. The result always uses the current settings.
计算精度如何？How precise are the results?
「判别式与一元二次方程求根」使用 JavaScript 双精度浮点数（约 15–16 位有效数字），部分功能使用 BigInt 处理大整数。The Discriminant & Quadratic Roots Calculator uses double-precision floating point (about 15–16 significant digits); some features use BigInt for large integers.
「判别式与一元二次方程求根」免费吗？需要注册吗？Is the Discriminant & Quadratic Roots Calculator free? Do I need an account?
「判别式与一元二次方程求根」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Discriminant & Quadratic Roots Calculator is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.