Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations appear in many parts of the IGCSE Mathematics syllabus. You are expected to solve them algebraically, represent them graphically and use them to model real-life situations. This article consolidates all the essential methods: factorising, the quadratic formula, completing the square, and the discriminant, with clear worked examples and exam advice.

二次方程出现在 IGCSE 数学课程的许多部分。你需要会用代数方法求解、用图像表示，并用它们来建立实际问题的模型。本文整合了所有核心方法：因式分解法、求根公式、配方法以及判别式，并配有清晰例题与考试建议。

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

A quadratic equation in one variable is an equation that can be written in the standard form:

只含一个变量的二次方程是可以写成标准形式的方程：

ax² + bx + c = 0, 其中 a ≠ 0

Here, a is the coefficient of x², b is the coefficient of x, and c is the constant term. The highest power of x is 2, so a quadratic equation always has degree 2.

其中 a 是 x² 的系数，b 是 x 的系数，c 是常数项。x 的最高次数是 2，因此二次方程总是二次的。

Example: x² − 5x + 6 = 0. Here a = 1, b = −5, c = 6. Solving this equation means finding all values of x that make the statement true.

例如：x² − 5x + 6 = 0。这里 a = 1，b = −5，c = 6。解这个方程就是求出所有使等式成立的 x 值。

2. Factorising Quadratics | 因式分解法

If a quadratic equation can be factorised, it is often the quickest method. For a simple quadratic where a = 1, look for two numbers that multiply to give c and add to give b.

如果一个二次方程可以被因式分解，这通常是最快的方法。对于 a = 1 的简单二次式，寻找两个数使它们相乘等于 c 且相加等于 b。

For x² − 5x + 6 = 0, we need two numbers whose product is 6 and sum is −5. These are −2 and −3, so:

对于 x² − 5x + 6 = 0，我们需要找到两个数，乘积为 6，和为 −5。这两个数是 −2 和 −3，因此：

(x − 2)(x − 3) = 0

Then x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3. Always check by substituting back into the original equation.

于是 x − 2 = 0 或 x − 3 = 0，得到 x = 2 或 x = 3。记得代回原方程检验。

When a > 1, use the ‘ac method’. For 2x² + 5x − 3 = 0, multiply a by c: 2 × (−3) = −6. Find two numbers with product −6 and sum 5: these are 6 and −1. Rewrite the middle term:

当 a > 1 时，使用 ac 法。对于 2x² + 5x − 3 = 0，将 a 与 c 相乘：2 × (−3) = −6。找到两个数，乘积为 −6，和为 5：它们是 6 和 −1。重写中间项：

2x² + 6x − x − 3 = 0

Then factorise in pairs: 2x(x + 3) − 1(x + 3) = 0, so (2x − 1)(x + 3) = 0. Hence x = 1/2 or x = −3.

然后分组因式分解：2x(x + 3) − 1(x + 3) = 0，所以 (2x − 1)(x + 3) = 0。因此 x = 1/2 或 x = −3。

3. The Quadratic Formula | 求根公式

Not every quadratic can be factorised easily. The quadratic formula always works and should be memorised:

并非每个二次方程都能轻松分解。求根公式始终有效，必须牢记：

x = (−b ± √(b² − 4ac)) / (2a)

To use the formula, identify a, b and c from the equation ax² + bx + c = 0. Then substitute carefully.

使用公式时，先从 ax² + bx + c = 0 中确定 a、b 和 c，然后小心代入。

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

例如：解 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。

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

Since √40 = 2√10, x = 1 ± √10/2. The two solutions are x ≈ 2.58 and x ≈ −0.58 (to 2 decimal places).

因为 √40 = 2√10，所以 x = 1 ± √10/2。两个解约为 x ≈ 2.58 和 x ≈ −0.58（保留两位小数）。

4. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² + q. For a quadratic x² + bx + c, we use:

配方法将二次式改写为 (x + p)² + q 的形式。对于 x² + bx + c，我们使用：

x² + bx + c = (x + b/2)² − (b/2)² + c

To solve x² + 6x + 2 = 0, observe that (x + 3)² = x² + 6x + 9. Therefore:

要解 x² + 6x + 2 = 0，注意到 (x + 3)² = x² + 6x + 9。因此：

x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7

Set this equal to 0: (x + 3)² − 7 = 0 ⇒ (x + 3)² = 7. Taking square roots gives x + 3 = ±√7, so x = −3 ± √7.

令该式等于 0： (x + 3)² − 7 = 0 ⇒ (x + 3)² = 7。两边开平方得 x + 3 = ±√7，所以 x = −3 ± √7。

If a > 1, divide the whole equation by a first before completing the square.

如果 a > 1，先在整个方程两边除以 a，再进行配方。

5. The Graph of a Quadratic | 二次函数的图像

The graph of y = ax² + bx + c is a parabola. If a > 0, it opens upwards; if a < 0, it opens downwards.

y = ax² + bx + c 的图像是抛物线。若 a > 0，开口向上；若 a < 0，开口向下。

The solutions of ax² + bx + c = 0 correspond to the x-intercepts of the graph. The axis of symmetry and the x-coordinate of the vertex are given by:

方程 ax² + bx + c = 0 的解对应图像与 x 轴的交点。对称轴和顶点的 x 坐标为：

x = −b/(2a)

For example, y = x² − 4x + 3 has axis x = 2. Substituting x = 2 gives y = 4 − 8 + 3 = −1, so the vertex is (2, −1). The x-intercepts are x = 1 and x = 3, since x² − 4x + 3 = (x − 1)(x − 3).

例如，y = x² − 4x + 3 的对称轴为 x = 2。代入 x = 2 得 y = 4 − 8 + 3 = −1，因此顶点为 (2, −1)。x 轴交点为 x = 1 和 x = 3，因为 x² − 4x + 3 = (x − 1)(x − 3)。

Drawing a good sketch requires identifying the y-intercept, the x-intercepts, the vertex and the direction of opening.

绘制草图需要确定 y 轴截距、x 轴截距、顶点以及开口方向。

6. The Discriminant | 判别式

The discriminant is the expression inside the square root of the quadratic formula:

判别式是求根公式中根号内的表达式：

Δ = b² − 4ac

It tells us how many real roots a quadratic equation has:

它告诉我们二次方程有几个实数根：

Discriminant Δ	Number of real roots	Graph interpretation
Δ > 0	Two distinct real roots	Graph crosses x-axis twice
Δ = 0	One repeated root	Graph touches x-axis once
Δ < 0	No real roots	Graph does not cross x-axis

Example: x² + 4x + 5 = 0 has Δ = 16 − 20 = −4 < 0, so there are no real solutions.

例如：x² + 4x + 5 = 0 中 Δ = 16 − 20 = −4 < 0，因此没有实数解。

In IGCSE exams, if a question says “does not intersect the x-axis”, it often means Δ < 0.

在 IGCSE 考试中，如果题目说“不与 x 轴相交”，通常意味着 Δ < 0。

7. Solving Quadratics by Rearranging | 通过变形求解

Some equations are not written in standard quadratic form at first. Expand brackets, clear denominators and rearrange until you get ax² + bx + c = 0.

有些方程一开始并不是标准二次方程形式。先去括号、去分母，再移项整理为 ax² + bx + c = 0。

Example: solve (x + 1)(x − 2) = x + 3.

例如：解 (x + 1)(x − 2) = x + 3。

x² − x − 2 = x + 3 ⇒ x² − 2x − 5 = 0

This does not factorise simply, so use the formula with a = 1, b = −2, c = −5:

这个式子不易分解，因此用公式：a = 1，b = −2，c = −5：

x = (2 ± √(4 + 20)) / 2 = (2 ± √24) / 2 = 1 ± √6

So x = 1 + √6 or x = 1 − √6.

因此 x = 1 + √6 或 x = 1 − √6。

Equations with fractions such as 2/x + x = 5 can also be solved by multiplying every term by x to obtain a quadratic.

含分数的方程如 2/x + x = 5，也可以通过两边同乘 x 转化为二次方程来求解。

8. Quadratic Equations from Word Problems | 应用题列二次方程

Many IGCSE questions present a real-world context and require you to form and solve a quadratic equation. Always define the unknown and check that your answer makes sense in the original context.

许多 IGCSE 题目提供实际情境，要求你建立并解二次方程。务必定义未知数，并检查答案是否符合原情境。

Example: A rectangle has length (x + 2) cm and width x cm. Its area is 48 cm². Find x.

例如：一个长方形的长为 (x + 2) cm，宽为 x cm，面积为 48 cm²。求 x。

x(x + 2) = 48 ⇒ x² + 2x − 48 = 0

Factorise: (x + 8)(x − 6) = 0, so x = −8 or x = 6. Since length cannot be negative, x = 6. The width is 6 cm and the length is 8 cm.

因式分解：(x + 8)(x − 6) = 0，所以 x = −8 或 x = 6。长度不能为负，因此 x = 6。宽为 6 cm，长为 8 cm。

Remember to reject negative or nonsense answers in word problems, even if they satisfy the equation.

记住在应用题中要舍去负数或无意义的答案，即使它们满足方程。

9. The Sum and Product of Roots | 根的和与积

If a quadratic equation ax² + bx + c = 0 has roots α and β, then the following relationships hold:

若二次方程 ax² + bx + c = 0 的根为 α 和 β，则有以下关系：

α + β = −b/a, αβ = c/a

For example, the roots of 2x² + 3x − 5 = 0 have sum −3/2 and product −5/2. This is useful for checking answers or writing a quadratic from its roots.

例如，2x² + 3x − 5 = 0 的根的和为 −3/2，积为 −5/2。这可用于检验答案，或已知根求二次方程。

To form a quadratic with given roots α and β, use:

要由已知根 α 和 β 构造二次方程，可使用：

x² − (α + β)x + αβ = 0

If the roots are 2 and −7, then the equation is x² − (−5)x + (−14) = 0, that is x² + 5x − 14 = 0.

若根为 2 和 −7，则方程为 x² − (−5)x + (−14) = 0，即 x² + 5x − 14 = 0。

10. Exam Tips | 考试技巧

To maximise marks on quadratic equation questions, follow a consistent process and always show working.

为了在二次方程题目中拿到满分，应该遵循一致的步骤，并写出完整的计算过程。

Try factorising first; if it works, it is usually fastest and avoids calculation errors.

先尝试因式分解；如果能分解，通常最快且不易出错。
If the discriminant is not a perfect square, use the quadratic formula and leave answers in exact surd form when asked.

如果判别式不是完全平方数，使用求根公式，并在要求时以根式形式保留精确答案。
Read whether the question asks for answers to 1 decimal place, 2 decimal places, or 3 significant figures.

仔细阅读题目要求精确到小数点后一位、两位还是三位有效数字。
Check both solutions by substitution into the original equation.

将两个解代入原方程进行检验。
When sketching a graph, label the x-intercepts, y-intercept and vertex clearly.

画图时，清楚标出 x 轴截距、y 轴截距和顶点。

Also look for hidden quadratics, such as x⁴ − 5x² + 6 = 0. Let u = x², solve u² − 5u + 6 = 0, then u = x² and solve for x.

同时要注意隐藏的二次形式，如 x⁴ − 5x² + 6 = 0。设 u = x²，解 u² − 5u + 6 = 0，再解 x² = u 得到 x。

With these strategies, quadratic equation questions become a reliable source of marks in your IGCSE Mathematics exam.

掌握这些策略后，二次方程题将成为你 IGCSE 数学考试中一个稳定的得分点。

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Solving Quadratic Equations | 解二次方程