Solving Quadratic Equations | 求解二次方程

📚 Solving Quadratic Equations | 求解二次方程

Quadratic equations appear in every IGCSE Mathematics exam. They are the bridge between simple linear algebra and the more advanced functions you will meet at A-Level. Mastering the three main solution methods — factorising, completing the square and the quadratic formula — is essential for top marks.

二次方程是 IGCSE 数学考试中的必考内容，也是从简单线性代数走向 A-Level 高阶函数的桥梁。掌握因式分解、配方法和二次公式三大核心解法，是冲击高分的关键。

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

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of the variable x is 2. If a = 0, the equation becomes linear.

二次方程是指可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数，且 a ≠ 0。变量的最高次幂为 2。若 a = 0，方程就退化为一次方程。

Examples of quadratic equations include:

二次方程的典型例子包括：

x² − 5x + 6 = 0
2x² + 3x − 2 = 0
x² = 9 (which can be written as x² − 9 = 0)

x² − 5x + 6 = 0
2x² + 3x − 2 = 0
x² = 9（可写成 x² − 9 = 0）

Notice that a quadratic equation may have missing b or c terms, but the x² term must always be present.

注意，二次方程可能缺少 b 或 c 项，但 x² 项必须存在。

2. The Standard Form and Roots | 标准形式与根

The standard form of a quadratic equation is ax² + bx + c = 0. The values of x that satisfy the equation are called roots or solutions. A quadratic equation can have two distinct roots, one repeated root, or no real roots.

二次方程的标准形式为 ax² + bx + c = 0。满足方程的 x 值称为根或解。二次方程可以有两个不同的根、一个重根，或没有实数根。

Example: For x² − 5x + 6 = 0, the roots are x = 2 and x = 3, because (2)² − 5(2) + 6 = 4 − 10 + 6 = 0, and similarly for x = 3.

例如：对于 x² − 5x + 6 = 0，根为 x = 2 和 x = 3，因为 (2)² − 5(2) + 6 = 4 − 10 + 6 = 0，x = 3 同理。

3. Solving by Factorising | 因式分解法

Factorising is usually the quickest method when the quadratic has simple integer roots. The idea is to rewrite ax² + bx + c as a product of two binomial brackets.

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。基本思路是将 ax² + bx + c 改写为两个二项式括号的乘积。

Step-by-step example: Solve x² + 7x + 12 = 0

分步示例：解 x² + 7x + 12 = 0

Find two numbers that multiply to 12 and add to 7. The numbers are 3 and 4. So:

找两个数，它们相乘等于 12，相加等于 7。这两个数是 3 和 4。因此：

(x + 3)(x + 4) = 0

Then set each bracket equal to zero:

然后令每个括号等于零：

x + 3 = 0 ⇒ x = −3

x + 4 = 0 ⇒ x = −4

So the roots are x = −3 and x = −4.

所以根为 x = −3 和 x = −4。

Always check if a common factor can be taken out first. For example, 2x² + 8x = 0 becomes 2x(x + 4) = 0, giving x = 0 or x = −4.

注意先检查是否能提出公因式。例如，2x² + 8x = 0 可化为 2x(x + 4) = 0，得到 x = 0 或 x = −4。

4. Solving by Completing the Square | 配方法

Completing the square is a powerful technique that works for any quadratic, even when factorising is difficult. It also helps when sketching graphs or solving inequalities.

配方法是一种适用于任何二次方程的强大的技巧，即使因式分解很困难也能使用。它还有助于绘制图像或解不等式。

To complete the square for x² + bx, add and subtract (b/2)². Then rewrite as (x + b/2)² − (b/2)².

对 x² + bx 配方，需要加上并减去 (b/2)²，然后重写为 (x + b/2)² − (b/2)²。

Example: Solve x² + 6x − 7 = 0 by completing the square.

示例：用配方法解 x² + 6x − 7 = 0。

First move the constant to the other side:

先把常数项移到另一边：

x² + 6x = 7

Add (6/2)² = 9 to both sides:

两边加上 (6/2)² = 9：

x² + 6x + 9 = 16

Now write the left side as a perfect square:

现在将左边写成完全平方：

(x + 3)² = 16

Take the square root of both sides:

两边开平方：

x + 3 = ±4

So x = 1 or x = −7. You can check these roots by substituting back into the original equation.

因此 x = 1 或 x = −7。你可以将根代回原方程验证。

5. Solving by the Quadratic Formula | 二次公式法

The quadratic formula works for all quadratics, including those with irrational or complex roots. It is derived from completing the square, but you can use it directly.

二次公式适用于所有二次方程，包括带有无理数根或复数根的情况。它由配方法推导而来，你可以直接使用。

For ax² + bx + c = 0, the formula is:

对于 ax² + bx + c = 0，公式为：

x = (−b ± √(b² − 4ac)) / (2a)

Example: Solve 2x² − 4x − 3 = 0 using the formula.

示例：用公式法解 2x² − 4x − 3 = 0。

Here a = 2, b = −4, c = −3. Substitute:

这里 a = 2，b = −4，c = −3。代入：

x = (−(−4) ± √((−4)² − 4×2×(−3))) / (2×2)

x = (4 ± √(16 + 24)) / 4

x = (4 ± √40) / 4

Since √40 = 2√10, the exact solutions are:

由于 √40 = 2√10，精确解为：

x = (4 ± 2√10) / 4 = 1 ± √10/2

It is important to write the formula carefully and to substitute the coefficients without sign errors.

书写公式时要仔细，代入系数时注意不要弄错符号。

6. The Discriminant | 判别式

In the quadratic formula, the expression b² − 4ac is called the discriminant. It tells us how many real roots the quadratic equation has without solving it.

在二次公式中，表达式 b² − 4ac 称为判别式。它告诉我们二次方程有多少个实数根，而无需求解。

The three cases are:

三种情况如下：

Discriminant	Number of real roots	Graph shape
b² − 4ac > 0	Two distinct roots	Cuts the x-axis twice
b² − 4ac = 0	One repeated root	Touches the x-axis once
b² − 4ac < 0	No real roots	Does not meet the x-axis

判别式大于零，有两个不同实数根；等于零，有一个重根；小于零，没有实数根。

For example, for x² − 4x + 4 = 0, the discriminant is (−4)² − 4×1×4 = 0, so there is exactly one root, x = 2.

例如，x² − 4x + 4 = 0 的判别式为 (−4)² − 4×1×4 = 0，因此只有一个根 x = 2。

7. Solving Quadratic Inequalities | 二次不等式

Quadratic inequalities require you to find the range of x values that make the quadratic expression positive or negative. A graph or a number line is helpful.

二次不等式要求找出使二次表达式为正或负的 x 的取值范围。借助图像或数轴会更清晰。

Example: Solve x² − x − 6 > 0.

示例：解 x² − x − 6 > 0。

First factorise: (x − 3)(x + 2) > 0. The roots are x = 3 and x = −2.

先因式分解： (x − 3)(x + 2) > 0。根为 x = 3 和 x = −2。

Sketch the parabola y = x² − x − 6. It opens upwards. The inequality is > 0 where the curve is above the x-axis, which is to the left of −2 or to the right of 3.

画出抛物线 y = x² − x − 6。它开口向上。不等式 > 0 对应曲线在 x 轴上方的部分，即 −2 的左侧或 3 的右侧。

x < −2 or x > 3

If the inequality were < 0, the solution would be −2 < x < 3.

如果不等式为 < 0，则解为 −2 < x < 3。

Remember to reverse the inequality signs if you multiply or divide by a negative number while moving terms, but with quadratics it is usually safer to keep the x² coefficient positive.

注意，在移项过程中如果乘以或除以负数，需要改变不等号方向；但对于二次不等式，最好保持 x² 的系数为正。

8. Quadratic Graphs and Turning Points | 二次函数图像与顶点

The graph of y = ax² + bx + c is a parabola. If a > 0, it has a U shape with a minimum point. If a < 0, it has an upside-down U shape with a maximum point.

函数 y = ax² + bx + c 的图像是抛物线。若 a > 0，呈 U 形，有最小值点；若 a < 0，呈倒 U 形，有最大值点。

The x-coordinate of the turning point is given by x = −b / (2a).

顶点的 x 坐标为 x = −b / (2a)。

Example: For y = 2x² − 8x + 3, the turning point x-coordinate is:

示例：对于 y = 2x² − 8x + 3，顶点的 x 坐标为：

x = −(−8) / (2×2) = 8 / 4 = 2

Substitute x = 2 into the equation to find y:

将 x = 2 代入方程求 y：

y = 2(2)² − 8(2) + 3 = 8 − 16 + 3 = −5

So the minimum point is (2, −5).

因此最小值点为 (2, −5)。

This coordinate is also used to write the quadratic in completed square form: y = 2(x − 2)² − 5.

该坐标也可用于将二次函数写成配方法形式：y = 2(x − 2)² − 5。

9. Solving Quadratics by Using a Graph | 利用图像解二次方程

When a quadratic equation cannot be solved easily, you may be asked to estimate the roots by drawing the graph of y = ax² + bx + c and reading the x-intercepts.

当二次方程不易求解时，题目可能会要求你画出 y = ax² + bx + c 的图像，并通过读取与 x 轴的交点来估算根。

To draw a quadratic graph, choose a set of x-values, calculate the corresponding y-values, plot the points, and draw a smooth curve.

画二次函数图像时，选取一组 x 值，计算对应的 y 值，描点并连成平滑曲线。

If the graph does not cross the x-axis, the equation has no real solutions. If it just touches the x-axis, there is one repeated solution.

若图像不与 x 轴相交，则方程无实数解；若图像与 x 轴相切，则有一个重根。

10. Applications: Word Problems | 应用：文字题

Quadratic equations are often used in real-life problems. Common contexts include area, projectile motion, and consecutive number problems.

二次方程常常用于实际生活问题中。常见情境包括面积、抛体运动、连续整数问题等。

Example: A rectangle has length 4 cm more than its width. Its area is 60 cm². Find the width.

示例：一个矩形的长比宽多 4 cm，面积为 60 cm²。求宽。

Let the width be w cm. Then the length is w + 4 cm. Area = w(w + 4) = 60.

设宽为 w cm，则长为 w + 4 cm。面积 = w(w + 4) = 60。

w² + 4w − 60 = 0

Factorise: (w + 10)(w − 6) = 0, so w = −10 or w = 6. Since width cannot be negative, the width is 6 cm.

因式分解得 (w + 10)(w − 6) = 0，所以 w = −10 或 w = 6。由于宽度不能为负，故宽为 6 cm。

Always reject any solution that does not make sense in the context of the problem.

始终要舍弃在问题情境中没有意义的解。

11. Choosing the Best Method | 选择最佳方法

Which method should you use? Here is a quick guide:

应该用哪种方法？这里有一个快速指南：

Situation	Recommended method
Simple factorisation possible	Factorising
Coefficient of x² is 1 and b is even	Completing the square
Cannot factorise, or leading coefficient is not 1 and not easily simplified	Quadratic formula
Need to sketch the graph or solve an inequality	Completed square form

情况：可直接因式分解 → 用因式分解法；x² 系数为 1 且 b 为偶数 → 用配方法；无法因式分解或系数较复杂 → 用二次公式；需要画图或解不等式 → 用配方形式。

In exam questions, always look for the easiest path. Factorising is fast, but the quadratic formula always works. Completing the square is especially useful for finding turning points.

在考试中，尽量选择最简单的路径。因式分解速度快，但二次公式永远有效。配方法特别适用于求顶点坐标。

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

Students often make the following mistakes:

学生常犯以下错误：

Forgetting to rearrange the equation into the form ax² + bx + c = 0 before solving.
Oversight of the ± sign when taking square roots.
Sign errors when substituting negative b into the quadratic formula.
Dropping a solution when factorising: remember that if ab = 0, then a = 0 or b = 0.
Not checking answers by substitution, especially when solving word problems.

求解前忘记将方程整理为 ax² + bx + c = 0 的形式。
开平方时遗漏 ± 号。
将负 b 代入二次公式时符号出错。
因式分解丢解：记住若 ab = 0，则 a = 0 或 b = 0。
没有通过代入检查答案，尤其在做应用题时。

Exam tips:

考试技巧：

Always show your working; even if your final answer is wrong, you may earn method marks.
If the quadratic formula is required, write it down first, then substitute.
Use the discriminant to check whether you expect two, one, or zero real roots.
In multiple-choice or calculator papers, use your calculator to verify your answer quickly.

务必写出步骤；即使最终答案错误，也可能获得方法分。
如需用二次公式，先写下公式，再代入数值。
用判别式判断应有几个实数根，以检验结果。
在选择题或允许使用计算器的试卷中，用计算器快速验证答案。

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Solving Quadratic Equations | 求解二次方程