Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations appear in nearly every IGCSE Mathematics paper, whether as pure algebra questions, graph sketching tasks, or word problems. Mastering the three main solution methods — factorisation, the quadratic formula, and completing the square — is essential for top marks.

二次方程几乎出现在每一份 IGCSE 数学试卷中，无论是纯代数题、函数图像题还是应用题。掌握三种主要解法——因式分解法、求根公式法和配方法——是获得高分的关键。

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

A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2. The variable is often written as x, and the general form is ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

二次方程是次数为 2 的多项式方程，即变量的最高次数为 2。变量通常用 x 表示，一般形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。

If a = 0, the equation becomes linear, so the restriction a ≠ 0 is important. The values of the variable that satisfy the equation are called its roots or solutions.

如果 a = 0，方程就变成了一次方程，因此 a ≠ 0 这一限制条件十分重要。满足方程的变量的值称为方程的根或解。

2. The Standard Form ax² + bx + c = 0 | 标准形式 ax² + bx + c = 0

Before solving any quadratic, you should rearrange it into standard form. This means expanding brackets, collecting like terms, and moving every term to the left-hand side so that the right-hand side is 0.

在解任何二次方程之前，你都应该先将其整理成标准形式。也就是说，展开括号、合并同类项，并把所有项移到等号左边，使右边为 0。

For example, x(2x − 3) = 5 becomes 2x² − 3x = 5, and then 2x² − 3x − 5 = 0. Only in this standard form can you confidently apply factorisation or the quadratic formula.

例如，x(2x − 3) = 5 可化为 2x² − 3x = 5，再化为 2x² − 3x − 5 = 0。只有化为标准形式，才能放心地使用因式分解法或求根公式法。

3. Solving by Factorisation | 因式分解法

Factorisation is usually the quickest method when the quadratic can be decomposed into two linear brackets. The principle is the zero product property: if p × q = 0, then at least one of p or q must be 0.

当二次式可以分解成两个一次括号相乘时，因式分解法通常是最快的方法。其原理是零乘积性质：如果 p × q = 0，那么 p 和 q 中至少有一个为 0。

Consider x² + 5x + 6 = 0. We look for two numbers that multiply to 6 and add to 5. The numbers 2 and 3 work, so x² + 5x + 6 = (x + 2)(x + 3) = 0.

考虑 x² + 5x + 6 = 0。我们需要找两个数，它们相乘等于 6，相加等于 5。2 和 3 满足条件，因此 x² + 5x + 6 = (x + 2)(x + 3) = 0。

Setting each bracket to zero gives x + 2 = 0 or x + 3 = 0, so the solutions are x = −2 or x = −3. Always check your answer by substituting back into the original equation.

令每个括号等于零，得到 x + 2 = 0 或 x + 3 = 0，因此解为 x = −2 或 x = −3。记得把答案代回原方程验证。

4. Special Cases: Difference of Two Squares | 特殊情况：平方差公式

When a quadratic takes the form x² − a², it can always be factored as (x − a)(x + a). This is called the difference of two squares and requires no middle term.

当二次方程形如 x² − a² 时，它总可以分解为 (x − a)(x + a)。这称为平方差公式，此时没有一次项。

For example, x² − 25 = 0 becomes (x − 5)(x + 5) = 0, giving x = 5 or x = −5. The same idea works for expressions like 4x² − 9, which factors to (2x − 3)(2x + 3).

例如，x² − 25 = 0 可化为 (x − 5)(x + 5) = 0，得到 x = 5 或 x = −5。同样的思路适用于像 4x² − 9 这样的式子，它可分解为 (2x − 3)(2x + 3)。

Be careful with sums of squares: x² + 25 cannot be factored over real numbers, so do not attempt to split it into (x + 5)(x + 5).

注意平方和的情况：x² + 25 在实数范围内无法分解，所以千万不要将它拆成 (x + 5)(x + 5)。

5. Solving by the Quadratic Formula | 用求根公式求解

For any quadratic ax² + bx + c = 0, the roots can be found directly using the quadratic formula:

对于任意二次方程 ax² + bx + c = 0，可以直接使用求根公式求出根：

x = [−b ± √(b² − 4ac)] / (2a)

This formula works for every quadratic, even when factorisation is difficult or impossible. You do not need to memorise a derivation, but you must memorise the formula exactly.

这个公式对所有二次方程都适用，即使因式分解很困难或根本无法分解。你不需要背推导过程，但必须准确记住公式本身。

Example: solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, c = −3. Substituting into the formula gives x = [4 ± √(16 + 24)] / 4 = [4 ± √40] / 4. This simplifies to x = (4 ± 2√10) / 4, giving two exact surd answers.

例：解 2x² − 4x − 3 = 0。此处 a = 2，b = −4，c = −3。代入公式得 x = [4 ± √(16 + 24)] / 4 = [4 ± √40] / 4，化简为 x = (4 ± 2√10) / 4，得到两个精确的无理数答案。

6. The Discriminant and the Nature of Roots | 判别式与根的性质

The expression b² − 4ac under the square root is called the discriminant, usually denoted by Δ. Its value tells us the type of roots without fully solving the equation.

根号下的表达式 b² − 4ac 称为判别式，通常用 Δ 表示。它的值可以不求解方程就判断根的类型。

Discriminant Δ = b² − 4ac	Nature of Roots \| 根的性质
Δ > 0 (positive)	Two distinct real roots \| 两个不相等的实数根
Δ = 0 (zero)	One repeated real root \| 两个相等的实数根（重根）
Δ < 0 (negative)	No real roots \| 没有实数根

If Δ is a perfect square such as 9 or 16, the quadratic can be solved by factorisation; if not, the roots will involve surds or decimals.

如果 Δ 是像 9 或 16 这样的完全平方数，那么二次方程通常可以用因式分解法求解；如果不是，根将包含无理数或小数。

7. Solving by Completing the Square | 配方法

Completing the square rewrites x² + bx as (x + b/2)² − (b/2)². Even when factorisation fails, this method converts the equation into a perfect square plus a constant.

配方法将 x² + bx 改写为 (x + b/2)² − (b/2)²。即使因式分解失败，这种方法也能将方程转化为一个完全平方加上一个常数。

Example: solve x² + 6x + 2 = 0. First rewrite x² + 6x as (x + 3)² − 9. The equation becomes (x + 3)² − 9 + 2 = 0, so (x + 3)² = 7. Taking square roots gives x + 3 = ±√7, hence x = −3 ± √7.

例：解 x² + 6x + 2 = 0。先将 x² + 6x 改写为 (x + 3)² − 9。方程变为 (x + 3)² − 9 + 2 = 0，即 (x + 3)² = 7。两边开平方得 x + 3 = ±√7，因此 x = −3 ± √7。

This method is particularly useful for solving equations like (x + h)² = k directly and is the foundation for deriving the vertex form of a quadratic graph.

这种方法对于直接解形如 (x + h)² = k 的方程特别有用，也是推导二次函数图像顶点形式的基础。

8. Word Problems with Quadratics | 二次方程应用题

In exam word problems, you must translate a real-life situation into a quadratic equation. Look for key phrases: “product of two numbers”, “area of a rectangle”, or “the square of a number is increased by”.

在考试应用题中，你需将实际问题转化为二次方程。注意关键词：”两个数的乘积”、”矩形的面积”、”某数的平方再增加”等。

Example: the area of a rectangle is 40 cm² and its length is 3 cm more than its width. Let the width be x. Then the length is x + 3, so x(x + 3) = 40, which expands to x² + 3x − 40 = 0. Factorising gives (x + 8)(x − 5) = 0, so x = 5 (rejecting x = −8 as a length cannot be negative).

例：一个矩形的面积为 40 cm²，它的长比宽多 3 cm。设宽为 x，则长为 x + 3，所以 x(x + 3) = 40，展开得 x² + 3x − 40 = 0。因式分解得 (x + 8)(x − 5) = 0，所以 x = 5（舍去 x = −8，因为长度不能为负）。

Always interpret your answers with units and check that they make sense in the original context.

一定要带上单位并检验答案是否符合作应用题的实际情境。

9. Graphs of Quadratic Functions | 二次函数的图像

The graph of y = ax² + bx + c is a parabola. When a > 0, the parabola opens upward and has a minimum point; when a < 0, it opens downward and has a maximum point.

函数 y = ax² + bx + c 的图像是一条抛物线。当 a > 0 时，抛物线开口向上，有最小值点；当 a < 0 时，开口向下，有最大值点。

The solutions of ax² + bx + c = 0 correspond to the x-intercepts of the parabola. If the discriminant is negative, the graph never crosses the x-axis, confirming there are no real roots.

方程 ax² + bx + c = 0 的解对应于抛物线与 x 轴的交点。如果判别式为负，图像与 x 轴没有交点，这也证实了没有实数根。

For curve sketching, you should identify the roots, the y-intercept (0, c), and the axis of symmetry x = −b / (2a), which passes through the vertex.

画草图时，你需要确定根、y 轴截距 (0, c)，以及对称轴 x = −b / (2a)，对称轴经过顶点。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

One frequent mistake is forgetting to write the equation in the form ax² + bx + c = 0 before factorising. The zero product property only works when one side of the equation is exactly 0.

一个常见错误是忘记先把方程化为 ax² + bx + c = 0 的形式。零乘积性质只有在方程一边恰好为 0 时才成立。

When using the quadratic formula, check the sign of b: if b = −4, then −b = +4.
Remember that √(b² − 4ac) applies to the whole discriminant, not just b².
If the question asks for answers to 2 decimal places, give decimals; if it mentions surds, leave the answer in exact form.
Show all working steps — many IGCSE papers award full marks for method even if the final answer is slightly wrong.

使用求根公式时，注意 b 的正负号：如果 b = −4，那么 −b = +4。记住 √(b² − 4ac) 作用于整个判别式，而不只是 b²。如果题目要求保留两位小数，就给出小数；如果提到无理数，就保留精确形式。把每一步都写出来——许多 IGCSE 试卷即使最终答案有误，也会给方法分。

11. Quick Practice | 快速练习

Here are three quick questions to test yourself. Solve each equation and then check using substitution.

下面三道快速自测题可以帮助你检验学习效果。解每个方程，然后用代入法检验。

1. x² − 7x + 12 = 0
2. 4x² − 1 = 0
3. x² + 2x − 4 = 0 (give answers to 2 decimal places)

Answers: 1. x = 3 or x = 4. 2. x = 1/2 or x = −1/2. 3. x ≈ 1.24 or x ≈ −3.24.

答案：1. x = 3 或 x = 4。2. x = 1/2 或 x = −1/2。3. x ≈ 1.24 或 x ≈ −3.24。

Consistent practice with factorisation, the quadratic formula, and completing the square will make solving quadratics automatic. Remember to always check the discriminant, interpret word problems carefully, and present your method clearly.

坚持练习因式分解、求根公式和配方法，解二次方程就会变得得心应手。记住时刻检查判别式、仔细理解应用题，并清晰地呈现你的解题过程。

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Solving Quadratic Equations | 解二次方程