初中数学公式之一元二次方程公式

初中数学公式 — 一元二次方程

一、定义

只含有一个未知数，且未知数的最高次数是 $2$ 的整式方程叫做一元二次方程。

一般形式：$ax^2 + bx + c = 0$（$a \neq 0$）文章源自公式库网-https://www.gongshiku.com/html/201808/chuzhongshuxuegongshizhiyiyuanercifangchenggongshi.html

二、解法

$x^2 = p$ → $x = \pm\sqrt{p}$（$p \geq 0$）文章源自公式库网-https://www.gongshiku.com/html/201808/chuzhongshuxuegongshizhiyiyuanercifangchenggongshi.html

$ax^2 + bx + c = 0$ → $a\left(x + \dfrac{b}{2a}\right)^2 + \dfrac{4ac-b^2}{4a} = 0$文章源自公式库网-https://www.gongshiku.com/html/201808/chuzhongshuxuegongshizhiyiyuanercifangchenggongshi.html

求根公式：$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$（$b^2 - 4ac \geq 0$）文章源自公式库网-https://www.gongshiku.com/html/201808/chuzhongshuxuegongshizhiyiyuanercifangchenggongshi.html

判别式 $\Delta = b^2 - 4ac$：文章源自公式库网-https://www.gongshiku.com/html/201808/chuzhongshuxuegongshizhiyiyuanercifangchenggongshi.html

• $\Delta \gt 0$ → 两个不相等实根
• $\Delta = 0$ → 两个相等实根
• $\Delta \lt 0$ → 无实根

化为 $(x - x_1)(x - x_2) = 0$ 的形式。文章源自公式库网-https://www.gongshiku.com/html/201808/chuzhongshuxuegongshizhiyiyuanercifangchenggongshi.html

三、根与系数的关系（韦达定理）

$x_1 + x_2 = -\dfrac{b}{a}$，$x_1 \cdot x_2 = \dfrac{c}{a}$文章源自公式库网-https://www.gongshiku.com/html/201808/chuzhongshuxuegongshizhiyiyuanercifangchenggongshi.html

文章源自公式库网-https://www.gongshiku.com/html/201808/chuzhongshuxuegongshizhiyiyuanercifangchenggongshi.html 文章源自公式库网-https://www.gongshiku.com/html/201808/chuzhongshuxuegongshizhiyiyuanercifangchenggongshi.html文章源自公式库网-https://www.gongshiku.com/html/201808/chuzhongshuxuegongshizhiyiyuanercifangchenggongshi.html