Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations are one of the most frequently tested topics in IGCSE Mathematics. They appear in algebra, graphing, coordinate geometry, and problem-solving questions. A clear understanding of the standard form, solution methods, and the discriminant can help you secure full marks on this topic.

一元二次方程是 IGCSE 数学中最常考查的内容之一。它出现在代数、图象、坐标几何以及综合应用题中。扎实掌握标准形式、求解方法和判别式，可以帮助你在这一主题上拿到满分。

1. What Is a Quadratic Equation? | 什么是一元二次方程？

A quadratic equation is any equation that can be written in the standard form:

一元二次方程是指可以写成标准形式的方程：

ax² + bx + c = 0, where a ≠ 0

Here, x is the unknown variable, and a, b, and c are real numbers called coefficients. The condition a ≠ 0 is essential; if a = 0, the equation becomes linear rather than quadratic.

其中 x 是未知数，a、b、c 是称为系数的实数。条件 a ≠ 0 至关重要；如果 a = 0，方程就变成了一次方程而不是二次方程。

The word ‘quadratic’ comes from the Latin word ‘quadratus’, meaning square. The defining feature of a quadratic equation is the presence of the x² term.

‘quadratic’ 一词源自拉丁语 ‘quadratus’，意为”平方”。二次方程的核心特征是在 x² 项。

2. Identifying Coefficients | 识别系数

Before solving a quadratic equation, you must first identify the values of a, b, and c. Always rewrite the equation in the form ax² + bx + c = 0 before reading off the coefficients.

在求解二次方程之前，你必须先确定 a、b、c 的值。务必先将方程改写为 ax² + bx + c = 0 的形式，再读出系数。

For example, consider the equation 2x² − 3x + 5 = 0. Here, a = 2, b = −3, and c = 5.

例如，对于方程 2x² − 3x + 5 = 0，这里 a = 2，b = −3，c = 5。

Sometimes the equation is not given in standard form. For instance, 3x² + 7 = 5x must first be rearranged:

有时候方程并不是以标准形式给出的。例如，3x² + 7 = 5x 必须先整理为：

3x² − 5x + 7 = 0

Thus a = 3, b = −5, and c = 7. Rearranging is a common first step, and failing to do so is a frequent source of errors.

因此 a = 3，b = −5，c = 7。先整理方程是常见的第一步，漏掉这一步是导致出错的主要原因。

3. Solving by Factorisation | 因式分解法

If a quadratic expression can be factorised easily, solving by factorisation is the fastest method. The method relies on the zero product property: if p × q = 0, then p = 0 or q = 0.

如果二次式可以容易地因式分解，那么因式分解法就是最快的求解方法。该方法依赖零乘积性质：如果 p × q = 0，则 p = 0 或 q = 0。

Consider the equation x² − 5x + 6 = 0. We look for two numbers that multiply to 6 and add to −5. The numbers are −2 and −3:

考虑方程 x² − 5x + 6 = 0。我们需要找到两个数相乘得 6、相加得 −5。这两个数是 −2 和 −3：

(x − 2)(x − 3) = 0

By the zero product property, either x − 2 = 0 or x − 3 = 0. Hence x = 2 or x = 3.

根据零乘积性质，x − 2 = 0 或 x − 3 = 0。因此 x = 2 或 x = 3。

When the coefficient of x² is not 1, factorisation is slightly harder. For example, 2x² + 5x + 2 = 0 can be factored as:

当 x² 的系数不为 1 时，因式分解稍为复杂。例如，2x² + 5x + 2 = 0 可以分解为：

(2x + 1)(x + 2) = 0

Then x = −1/2 or x = −2. Always check your factorisation by expanding the brackets.

于是 x = −1/2 或 x = −2。解题后务必通过展开括号来检验你的分解是否正确。

4. Solving by Completing the Square | 配方法

Completing the square is a powerful technique that rewrites a quadratic in the form a(x + p)² + q. This form is especially useful for finding the turning point of a parabola.

配方法是一种强大的技巧，它可以把二次式改写为 a(x + p)² + q 的形式。这种形式在求抛物线顶点时特别有用。

Start with x² + 6x + 5 = 0. Take half of the coefficient of x, which is 3, and write:

从 x² + 6x + 5 = 0 开始。取 x 系数的一半，即 3，写成：

(x + 3)² − 9 + 5 = 0

Simplifying gives (x + 3)² − 4 = 0. Rearranging:

化简得 (x + 3)² − 4 = 0。整理后：

(x + 3)² = 4

Taking square roots on both sides, x + 3 = ±2. Therefore x = −1 or x = −5.

两边同时开平方根，x + 3 = ±2。因此 x = −1 或 x = −5。

For equations where the coefficient of x² is not 1, first divide the whole equation by that coefficient before completing the square.

当 x² 的系数不为 1 时，需要先将整个方程除以该系数，再进行配方。

5. The Quadratic Formula | 求根公式

The quadratic formula works for every quadratic equation, including those that cannot be solved by factorisation. The formula is:

求根公式适用于所有一元二次方程，包括那些无法用因式分解法求解的方程。公式为：

x = (−b ± √(b² − 4ac)) / 2a

This formula is derived by completing the square on ax² + bx + c = 0. You should memorise it, as it is not provided in the IGCSE formula booklet.

该公式通过对 ax² + bx + c = 0 配方推导而来。你应该牢记此公式，因为它不出现在 IGCSE 的公式表中。

Let us solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, c = −3. Substituting:

我们用该公式求解 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。代入得：

x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4

Simplifying, x = (4 ± 2√10) / 4, so x = 1 ± √10/2. These are two irrational roots.

化简得 x = (4 ± 2√10) / 4，因此 x = 1 ± √10/2。这是两个无理数根。

When using the quadratic formula, be careful with negative signs. A common mistake is to write −b incorrectly when b itself is negative; always check your substitution.

使用求根公式时，要特别注意负号。一个常见错误是在 b 本身为负时写错 −b；请务必检查你的代入过程。

6. The Discriminant | 判别式

The expression b² − 4ac inside the quadratic formula is called the discriminant. It tells us the nature of the roots without solving the full equation:

求根公式中根号内的代数式 b² − 4ac 称为判别式。无需完整求解方程，它就能告诉我们根的性质：

If b² − 4ac > 0, there are two distinct real roots.
若 b² − 4ac > 0，方程有两个不相等的实数根。

If b² − 4ac = 0, there is exactly one repeated real root.
若 b² − 4ac = 0，方程有一个重根（两个相等的实数根）。

If b² − 4ac < 0, there are no real roots.
若 b² − 4ac < 0，方程没有实数根。

For example, for x² + 2x + 5 = 0, the discriminant is 2² − 4(1)(5) = 4 − 20 = −16 < 0. Therefore this equation has no real roots.

例如，对于 x² + 2x + 5 = 0，判别式为 2² − 4(1)(5) = 4 − 20 = −16 < 0。因此该方程没有实数根。

Exam questions often ask you to find the value of a parameter for which an equation has equal roots. Set the discriminant equal to 0 and solve for the parameter.

考试题经常要求你求出使方程有两个相等根的参数值。此时令判别式等于 0，然后解出该参数即可。

7. Graphs of Quadratic Functions | 二次函数的图象

The graph of a quadratic function y = ax² + bx + c is a parabola. If a > 0, the parabola opens upward, like a ‘U’ shape; if a < 0, it opens downward.

二次函数 y = ax² + bx + c 的图象是一条抛物线。当 a > 0 时，抛物线开口向上，形如字母”U”；当 a < 0 时，开口向下。

The solutions of the equation ax² + bx + c = 0 correspond to the x-intercepts of the graph. A quadratic equation with no real roots corresponds to a parabola that never crosses the x-axis.

方程 ax² + bx + c = 0 的解对应图象与 x 轴的交点。无实数根的二次方程所对应的抛物线与 x 轴没有交点。

The turning point of the parabola can be found by completing the square. For y = (x − 2)² − 3, the turning point is at (2, −3). The x-coordinate of the turning point is also given by x = −b / 2a.

抛物线的顶点可以通过配方法求得。对于 y = (x − 2)² − 3，顶点在 (2, −3)。顶点的 x 坐标也可由 x = −b / 2a 得到。

Sketching the graph of a quadratic requires identifying three features: the y-intercept (c), the x-intercepts (real roots), and the turning point.

画二次函数草图需要确定三个特征：y 轴截距 (c)、x 轴截距（实数根）和顶点。

8. Word Problems | 应用题

Quadratic equations frequently arise from geometry and other contextual problems. A rectangular garden has an area of 40 m². Its length is 3 m longer than its width. Let the width be x metres.

二次方程经常出现在几何和其他实际情境中。例如，一块长方形花园的面积为 40 m²，其长比宽长 3 m。设宽为 x 米。

Then the length is x + 3, and the area condition gives x(x + 3) = 40. Expanding and rearranging:

则长为 x + 3，根据面积条件可得 x(x + 3) = 40。展开并整理：

x² + 3x − 40 = 0

Factorising gives (x + 8)(x − 5) = 0, so x = −8 or x = 5. Since a width cannot be negative, x = 5. The rectangle is 5 m by 8 m.

分解因式得 (x + 8)(x − 5) = 0，因此 x = −8 或 x = 5。因为宽度不能为负，所以 x = 5。该矩形为 5 m × 8 m。

Many real-world problems lead to quadratics where one root is physically impossible. Always interpret your solutions in the context of the question.

许多实际问题求得的两个根中有一个在物理上无意义。务必根据题目情境对解进行合理取舍。

9. Summary and Exam Tips | 总结与考试技巧

To solve a quadratic equation successfully, follow this checklist:

为顺利求解一元二次方程，请遵循以下检查清单：

Always rearrange the equation into standard form ax² + bx + c = 0 first.
先整理方程为标准形式 ax² + bx + c = 0。

Try factorisation first; if factorisation is difficult, use the quadratic formula.
优先尝试因式分解；若分解困难，则使用求根公式。

Check the discriminant to understand the nature of the roots before solving.
求解前先计算判别式，了解根的性质。

Substitute your answers back into the original equation to verify them.
将答案代回原方程进行检验。

Common mistakes include forgetting to rearrange the equation, dropping a negative sign, and discarding an answer without justification in word problems.

常见错误包括忘记整理方程、漏掉负号，以及在应用题中不说明理由就舍去某个解。

By practising all four methods — factorisation, completing the square, the quadratic formula, and graphing — you will develop flexibility and speed for the exam.

通过练习四种方法——因式分解、配方、求根公式和图象法——你将在考试中更加灵活和迅速。

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Solving Quadratic Equations | 解一元二次方程