Solving Quadratic Equations | 二次方程求解

📚 Solving Quadratic Equations | 二次方程求解

A quadratic equation is one of the most important algebraic skills in IGCSE Mathematics. It appears in topics ranging from algebra and graphs to area problems and sequences, so you need a reliable set of methods for solving it accurately.

二次方程是 IGCSE 数学中最重要的代数技能之一。它出现在代数、图像、面积问题和数列等多个主题中，因此你需要掌握一套可靠的方法来准确求解。

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

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of x is 2, which is why it is called a quadratic.

二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。x 的最高次数是 2，因此被称为二次方程。

If a = 0, the equation becomes bx + c = 0, which is a linear equation. So the condition a ≠ 0 is essential for the equation to remain quadratic.

如果 a = 0，方程将变为 bx + c = 0，这是一元一次方程。因此 a ≠ 0 这个条件是方程保持为二次方程的关键。

Degree 2: the highest exponent is 2.

次数为 2：最高指数是 2。
Standard form: ax² + bx + c = 0.

标准形式：ax² + bx + c = 0。
Solutions are called roots or x-intercepts.

解称为根或 x 轴截距。

2. Standard Form and Coefficients | 标准形式与系数

Before solving, rewrite the equation in the standard form. The coefficient a is attached to x², b is attached to x, and c is the constant term.

在求解之前，应先将方程写成标准形式。系数 a 对应 x² 项，b 对应 x 项，c 是常数项。

ax² + bx + c = 0

For example, in 3x² – 5x + 2 = 0, we have a = 3, b = -5 and c = 2. Sign errors often come from reading the coefficients incorrectly when the equation is not in order.

例如，在 3x² – 5x + 2 = 0 中，a = 3、b = -5、c = 2。如果方程没有按顺序排列，读数时很容易出现符号错误。

If the equation is given as 2 – 7x = 4x², you should rearrange it as 4x² + 7x – 2 = 0 by moving all terms to one side.

如果给出的方程是 2 – 7x = 4x²，你应将所有项移到一侧，重新整理为 4x² + 7x – 2 = 0。

3. Solving by Factorising | 因式分解法

Factorising is usually the fastest method when the quadratic has simple integer roots. You look for two numbers that multiply to give ac and add to give b, then split the middle term and factor by grouping.

当二次方程具有简单的整数根时，因式分解通常是最快的方法。你寻找两个数，使它们相乘等于 ac，相加等于 b，然后拆分中间项并进行分组分解。

For x² – 5x + 6 = 0, we need two numbers with product 6 and sum -5. The numbers are -2 and -3, so:

对于 x² – 5x + 6 = 0，我们需要两个数的乘积为 6、和为 -5。这两个数是 -2 和 -3，因此：

(x – 2)(x – 3) = 0

Setting each bracket to zero gives x – 2 = 0 or x – 3 = 0, so x = 2 or x = 3.

令每个括号等于零，得到 x – 2 = 0 或 x – 3 = 0，因此 x = 2 或 x = 3。

Always expand your brackets as a quick check: x² – 5x + 6. This confirms the factorisation is correct.

始终将括号展开作为快速检验：x² – 5x + 6。这样可以确认因式分解正确。

4. Solving by the Quadratic Formula | 公式法

The quadratic formula works for every quadratic equation, including those that cannot be factorised easily. It is given by:

求根公式适用于所有二次方程，包括不易因式分解的方程。公式如下：

x = [-b ± √(b² – 4ac)] / (2a)

Make sure the entire numerator is divided by 2a, not just part of it. Substitute a, b and c carefully, especially when they are negative.

要确保整个分子都除以 2a，而不仅仅是部分项。代入 a、b、c 时要仔细，尤其是当它们为负数时。

For 2x² + 3x – 2 = 0, a = 2, b = 3, c = -2, so:

对于 2x² + 3x – 2 = 0，a = 2、b = 3、c = -2，因此：

x = [-3 ± √(3² – 4 × 2 × -2)] / (2 × 2) = [-3 ± √25] / 4 = [-3 ± 5] / 4

This gives x = 1/2 or x = -2. The formula is especially useful when the roots are fractions or surds.

由此得到 x = 1/2 或 x = -2。当根是分数或根式时，公式法尤其有用。

5. Solving by Completing the Square | 配方法

Completing the square rewrites the quadratic in the form (x + p)² = q. This method is required for deriving the quadratic formula and for locating the vertex of a parabola.

配方法将二次方程改写为 (x + p)² = q 的形式。这种方法是推导求根公式以及确定抛物线顶点所必需的。

For x² + 6x + 2 = 0, first move the constant to the right: x² + 6x = -2. Half of 6 is 3, so add 3² = 9 to both sides:

对于 x² + 6x + 2 = 0，首先将常数移到右边：x² + 6x = -2。6 的一半是 3，因此两边加上 3² = 9：

x² + 6x + 9 = 7 → (x + 3)² = 7

Taking square roots gives x + 3 = ±√7, so x = -3 ± √7.

取平方根得到 x + 3 = ±√7，因此 x = -3 ±√7。

When the coefficient of x² is not 1, divide every term by a first. For 2x² + 8x + 6 = 0, divide by 2 to get x² + 4x + 3 = 0.

当 x² 的系数不是 1 时，首先将每一项除以 a。例如 2x² + 8x + 6 = 0，除以 2 得到 x² + 4x + 3 = 0。

6. The Discriminant | 判别式

The discriminant is the expression b² – 4ac under the square root in the quadratic formula. It tells you how many real roots a quadratic has before you solve it.

判别式是求根公式中平方根下的表达式 b² – 4ac。它可以在你求解之前告诉你二次方程有多少个实根。

If b² – 4ac > 0, there are two distinct real roots.

如果 b² – 4ac > 0，则有两个不同的实根。
If b² – 4ac = 0, there is exactly one repeated real root.

如果 b² – 4ac = 0，则恰好有一个重复的实根。
If b² – 4ac < 0, there are no real roots.

如果 b² – 4ac < 0，则没有实根。

For x² + 2x + 5 = 0, the discriminant is 2² – 4 × 1 × 5 = 4 – 20 = -16, so the equation has no real roots.

对于 x² + 2x + 5 = 0，判别式为 2² – 4 × 1 × 5 = 4 – 20 = -16，因此该方程没有实根。

This concept is often tested through questions such as ‘Find the value of k for which the equation has two equal roots’. Set the discriminant to zero and solve for k.

这个概念经常以这样的问题考查：’求使方程有两个相等实根的 k 值’。令判别式为零并求解 k 即可。

7. Graphical Interpretation | 图像解释

The graph of y = ax² + bx + c is a parabola. The solutions of ax² + bx + c = 0 are the x-coordinates where the graph crosses the x-axis.

y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的解就是图像与 x 轴相交处的 x 坐标。

Two roots: the parabola crosses the x-axis at two points.

两个根：抛物线与 x 轴有两个交点。
One repeated root: the parabola touches the x-axis at exactly one point.

一个重根：抛物线恰好与 x 轴相切于一点。
No real roots: the parabola stays fully above or fully below the x-axis.

没有实根：抛物线完全位于 x 轴上方或完全位于下方。

If a > 0, the parabola opens upwards, giving a minimum point. If a < 0, it opens downwards, giving a maximum point.

如果 a > 0，抛物线开口向上，存在最小值点。如果 a < 0，抛物线开口向下，存在最大值点。

8. Solving Word Problems | 求解应用题

Many IGCSE quadratic questions are hidden in word problems involving areas, products of consecutive numbers, or projectile paths. You must define the variable, form an equation, solve it, and then reject invalid answers.

许多 IGCSE 二次方程题都隐藏在涉及面积、相邻数乘积或抛射路径的应用题中。你必须设出变量、建立方程、求解方程，然后舍去不合理的答案。

Example: a rectangle has length 2 cm more than its width, and its area is 48 cm². If the width is x cm, then length is x + 2, so:

例如：一个矩形的长比宽多 2 cm，面积为 48 cm²。如果宽为 x cm，则长为 x + 2，因此：

x(x + 2) = 48 → x² + 2x – 48 = 0 → (x + 8)(x – 6) = 0

The solutions are x = -8 or x = 6. Since width cannot be negative, the width is 6 cm and the length is 8 cm.

解为 x = -8 或 x = 6。由于宽度不能为负数，因此宽为 6 cm，长为 8 cm。

9. Common Mistakes | 常见错误

Forgetting that a must not be zero before using the quadratic formula.

在使用求根公式之前忘记 a 不能为零。
Dividing only part of the numerator by 2a in the formula.

在公式中只将分子的部分项除以 2a。
Losing a negative sign when substituting b or c into the formula.

将 b 或 c 代入公式时丢失负号。
Factoring incorrectly and not checking by expanding the brackets.

因式分解错误，且没有通过展开括号进行检查。
Giving negative lengths or times as valid answers in context questions.

在应用题中将负数长度或时间作为合理答案。

Write out every step, especially the substitution step, and double-check signs. A small arithmetic error can change both roots.

写下每一步，尤其是代入步骤，并仔细检查符号。一个小的算术错误可能会改变两个根。

10. Choosing the Best Method | 选择最佳方法

Method	Use when
Factorising	Roots are simple integers and you can spot the pair quickly
Quadratic formula	Roots are fractions, surds, or the equation looks messy
Completing the square	You need the vertex form or exact surd answers

In an exam, if factorising does not work within a few seconds, switch to the quadratic formula. It always works and reduces the risk of guessing incorrectly.

在考试中，如果因式分解在几秒钟内无法完成，请改用求根公式。它总是有效，并能降低猜测错误的风险。

11. Worked Example | 完整例题

Solve 3x² – 7x – 6 = 0. Try factorising: we need two numbers with product 3 × -6 = -18 and sum -7. The numbers are -9 and 2. Split the middle term:

求解 3x² – 7x – 6 = 0。尝试因式分解：我们需要两个数，乘积为 3 × -6 = -18，和为 -7。这两个数是 -9 和 2。拆分中间项：

3x² – 9x + 2x – 6 = 0 → 3x(x – 3) + 2(x – 3) = 0 → (x – 3)(3x + 2) = 0

Therefore x – 3 = 0 or 3x + 2 = 0, giving x = 3 or x = -2/3.

因此 x – 3 = 0 或 3x + 2 = 0，得到 x = 3 或 x = -2/3。

Check: substitute x = 3 into the original equation: 3(9) – 21 – 6 = 0. Substitute x = -2/3: 3(4/9) + 14/3 – 6 = 0. Both work.

检验：将 x = 3 代入原方程：3(9) – 21 – 6 = 0。将 x = -2/3 代入：3(4/9) + 14/3 – 6 = 0。两者均成立。

12. Key Points Summary | 重点总结

Always write the equation in standard form ax² + bx + c = 0 first.

始终先写成标准形式 ax² + bx + c = 0。
Factorising is fast for simple integer roots; the formula is the universal method.

因式分解法适用于简单整数根；公式法是通用方法。
Completing the square gives exact surd answers and helps find the vertex.

配方法得到精确根式解，并有助于确定顶点。
Use b² – 4ac to predict the number of real roots.

使用 b² – 4ac 判断实根的个数。
Check your answers by substitution and reject non-real or impossible context values.

通过代入检验答案，并舍去不符合实际意义的值。

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Solving Quadratic Equations | 二次方程求解