Quadratic Equations: Solving and Applications | 二次方程：求解与应用

📚 Quadratic Equations: Solving and Applications | 二次方程：求解与应用

≠ 0. In this article, you will learn how to solve quadratics by factorising, completing the square, and using the quadratic formula. You will also study the discriminant, graph interpretation, and common exam techniques.

二次方程是 IGCSE 数学的核心内容。它出现在代数、坐标几何和现实建模问题中。二次方程可以写成一般形式 ax² + bx + c = 0，其中 a ≠ 0。本文将介绍如何通过因式分解、配方法和求根公式求解二次方程，并学习判别式、图像解释和常见考试技巧。

1. What is a Quadratic Equation? | 什么是二次方程？

s, and constant terms. Examples include x² + 5x + 6 = 0, 2x² − 3x + 1 = 0, and x² − 9 = 0.

如果未知变量的最高次数是 2，则该方程是二次方程。它可以包含 x² 项、x 项和常数项。例如 x² + 5x + 6 = 0、2x² − 3x + 1 = 0 和 x² − 9 = 0。

In IGCSE, you are usually asked to solve a quadratic equation, which means finding the values of x that make the equation true. A quadratic equation can have two real solutions, one repeated solution, or no real solutions.

在 IGCSE 考试中，通常会要求解二次方程，即找出使方程成立的 x 值。二次方程可能有两个实数解、一个重根解，或者没有实数解。

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2. Standard Form and Key Terms | 标准形式与关键术语

e constants and a ≠ 0. For example, in 3x² − 7x + 2 = 0, we have a = 3, b = −7, and c = 2.

二次方程的标准形式是 ax² + bx + c = 0，其中 a、b、c 是常数且 a ≠ 0。例如在 3x² − 7x + 2 = 0 中，a = 3，b = −7，c = 2。

Always rearrange an equation into standard form before using factorising, completing the square, or the quadratic formula. Collect all terms on one side so that the other side equals 0.

在使用因式分解、配方法或求根公式之前，一定要将方程整理成标准形式。将所有项移到一边，使另一边等于 0。

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3. Solving by Factorising | 因式分解法

two linear brackets. For example, x² + 5x + 6 = 0 can be factorised as (x + 2)(x + 3) = 0.

当二次式可以写成两个线性括号的乘积时，因式分解通常是最快的方法。例如 x² + 5x + 6 = 0 可以因式分解为 (x + 2)(x + 3) = 0。

then at least one factor must be zero. So x + 2 = 0 or x + 3 = 0, giving x = −2 or x = −3.

使用零积性质：如果两个因式的乘积为零，则至少有一个因式为零。因此 x + 2 = 0 或 x + 3 = 0，得到 x = −2 或 x = −3。

+ 7x + 3 = 0, factorise as (2x + 1)(x + 3) = 0. Then the solutions are x = −1/2 or x = −3.

对于首项系数大于 1 的二次式，例如 2x² + 7x + 3 = 0，可以因式分解为 (2x + 1)(x + 3) = 0。于是解为 x = −1/2 或 x = −3。

• Write the equation in standard form. 将方程写成标准形式。
• Factorise the quadratic expression fully. 将二次式完全因式分解。
• Set each bracket equal to zero. 令每个括号等于零。
• Solve each linear equation. 解每个线性方程。

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4. Solving by Completing the Square | 配方法

goal is to rewrite ax² + bx + c = 0 in the form (x + p)² = q, and then take square roots.

当二次式不容易因式分解时，配方法很有用。其目标是将 ax² + bx + c = 0 改写成 (x + p)² = q 的形式，然后开平方求解。

3, and square it to get 9. Add 9 to both sides: x² + 6x + 9 = 7. This gives (x + 3)² = 7.

例如解 x² + 6x + 2 = 0。先将常数移到右边：x² + 6x = −2。取 x 系数的一半，即 3，将其平方得到 9。两边同时加 9：x² + 6x + 9 = 7。于是 (x + 3)² = 7。

Take the square root of both sides: x + 3 = ±√7. Therefore x = −3 ± √7.

两边开平方：x + 3 = ±√7。因此 x = −3 ± √7。

If the coefficient of x² is not 1, divide the whole equation by that coefficient first. Completing the square also helps find the coordinates of the vertex of a parabola.

如果 x² 的系数不是 1，请先在整个方程中除以该系数。配方法还有助于求抛物线顶点的坐标。

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5. Solving by the Quadratic Formula | 求根公式法

The quadratic formula gives the solutions of ax² + bx + c = 0 directly. It is:

求根公式可直接给出 ax² + bx + c = 0 的解：

x = [−b ± √b² − 4ac] ÷ (2a)

wo solutions: one using the plus sign and one using the minus sign. Always substitute the values of a, b, and c carefully, including negative signs.

符号 ± 表示通常有两个解：一个取

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