Solving Quadratic Equations | 一元二次方程的解法

📚 Solving Quadratic Equations | 一元二次方程的解法

Quadratic equations are one of the most important topics in IGCSE Mathematics. Understanding how to solve them not only helps in algebra but also in geometry, physics, and economics. This article explains the main methods, formulas, and common pitfalls, with step-by-step examples.

一元二次方程是IGCSE数学中最重要的话题之一。理解如何求解它们不仅有助于代数，也有助于几何、物理和经济学。本文将解释主要方法、公式和常见陷阱，并配有逐步示例。

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

A quadratic equation is a polynomial equation of degree 2. In one variable, it can be written in the general form ax² + bx + c = 0, where a, b and c are constants, and a is not equal to zero. The highest power of the variable is 2, which gives the equation its name “quadratic”.

一元二次方程是次数为2的多项式方程。在一个变量下，它可以写成一般形式 ax² + bx + c = 0，其中 a、b、c 是常数，且 a 不等于零。变量的最高次数是2，因此得名“二次”。

For example, 2x² – 5x + 3 = 0 is a quadratic equation. However, 3x – 4 = 0 is linear, and x³ + x² = 0 is not quadratic because the highest power is 3.

例如，2x² – 5x + 3 = 0 是一个一元二次方程。而 3x – 4 = 0 是线性的，x³ + x² = 0 不是二次的，因为最高次数是3。

ax² + bx + c = 0, a ≠ 0

2. Standard Form and Key Terms | 标准形式与关键术语

To solve a quadratic equation, it is often helpful to rewrite it in standard form. The standard form is ax² + bx + c = 0. Here, a is the coefficient of x², b is the coefficient of x, and c is the constant term.

为了解一元二次方程，往往需要将其改写为标准形式。标准形式是 ax² + bx + c = 0。这里，a 是 x² 的系数，b 是 x 的系数，c 是常数项。

The values of x that satisfy the equation are called roots or solutions. A quadratic equation can have two distinct real roots, one repeated real root, or no real roots at all.

满足方程的 x 的值称为根或解。一元二次方程可以有两个不同的实根、一个重根，或者没有实根。

For example, in x² – 5x + 6 = 0, the coefficients are a = 1, b = -5, c = 6. The roots are x = 2 and x = 3.

例如，在 x² – 5x + 6 = 0 中，系数是 a = 1，b = -5，c = 6。根是 x = 2 和 x = 3。

3. Solving by Factorization | 用因式分解法求解

Factorization is often the fastest method when the quadratic expression can be written as a product of two linear factors. The idea is to rewrite ax² + bx + c as (px + q)(rx + s), then set each factor equal to zero.

因式分解通常是当二次表达式可以写成两个一次因式乘积时最快的方法。思路是将 ax² + bx + c 改写为 (px + q)(rx + s)，然后令每个因式等于零。

For example, solve x² – 7x + 12 = 0. We look for two numbers that multiply to 12 and add to -7. They are -3 and -4. So the equation becomes (x – 3)(x – 4) = 0.

例如，解 x² – 7x + 12 = 0。我们寻找两个数，它们相乘得12，相加得 -7。它们是 -3 和 -4。因此方程变为 (x – 3)(x – 4) = 0。

Then set each bracket to zero: x – 3 = 0 or x – 4 = 0. Hence x = 3 or x = 4.

然后令每个括号为零：x – 3 = 0 或 x – 4 = 0。因此 x = 3 或 x = 4。

(x – 3)(x – 4) = 0 ⇒ x = 3 或 x = 4

When the coefficient a is not 1, factorization can require grouping. For example, 2x² + 5x – 3 = 0. Multiply a and c: 2 × (-3) = -6. Find two numbers with product -6 and sum 5: they are 6 and -1. Rewrite the middle term: 2x² + 6x – x – 3 = 0. Group: 2x(x + 3) – 1(x + 3) = 0, giving (2x – 1)(x + 3) = 0. So x = ½ or x = -3.

当系数 a 不为1时，因式分解可能需要分组。例如，2x² + 5x – 3 = 0。将 a 和 c 相乘：2 × (-3) = -6。寻找乘积为 -6 且和为 5 的两个数：它们是 6 和 -1。重写中间项：2x² + 6x – x – 3 = 0。分组：2x(x + 3) – 1(x + 3) = 0，得到 (2x – 1)(x + 3) = 0。所以 x = ½ 或 x = -3。

4. Solving by Completing the Square | 用配方法求解

Completing the square converts a quadratic equation into the form (x + p)² = q. This method is useful when the equation cannot be easily factored, and it also helps to derive the quadratic formula.

配方法将一元二次方程转换为 (x + p)² = q 的形式。当方程不容易因式分解时，这种方法很有用，它也有助于推导求根公式。

Start with x² + bx + c = 0. Move the constant term to the other side: x² + bx = -c. Then add (b/2)² to both sides, because x² + bx + (b/2)² = (x + b/2)².

从 x² + bx + c = 0 开始。将常数项移到另一边：x² + bx = -c。然后在两边加上 (b/2)²，因为 x² + bx + (b/2)² = (x + b/2)²。

Example: solve x² + 6x – 7 = 0. First, x² + 6x = 7. Add (6/2)² = 9 to both sides: x² + 6x + 9 = 16. This gives (x + 3)² = 16. Taking square roots gives x + 3 = ±4, so x = 1 or x = -7.

示例：解 x² + 6x – 7 = 0。首先，x² + 6x = 7。两边加上 (6/2)² = 9：x² + 6x + 9 = 16。这得到 (x + 3)² = 16。开平方得 x + 3 = ±4，所以 x = 1 或 x = -7。

(x + 3)² = 16 ⇒ x + 3 = ±4 ⇒ x = 1 或 x = -7

For the general form ax² + bx + c = 0, divide all terms by a before completing the square.

对于一般形式 ax² + bx + c = 0，在配方前先将所有项除以 a。

5. The Quadratic Formula | 求根公式

The quadratic formula is a universal method that works for any quadratic equation. It is derived from completing the square and is written as follows:

求根公式是一种通用的方法，适用于任何一元二次方程。它由配方法推导而来，写法如下：

x = (-b ± √(b² – 4ac)) / (2a)

Here, b² – 4ac is called the discriminant. The symbol ± means that the formula gives two solutions: one with the plus sign and one with the minus sign.

这里，b² – 4ac 称为判别式。符号 ± 表示公式给出两个解：一个取加号，一个取减号。

Example: solve 2x² – 4x – 6 = 0. Here a = 2, b = -4, c = -6. Substitute into the formula:

示例：解 2x² – 4x – 6 = 0。这里 a = 2，b = -4，c = -6。代入公式：

x = (4 ± √((-4)² – 4×2×(-6))) / (2×2) = (4 ± √(16 + 48)) / 4 = (4 ± √64) / 4 = (4 ± 8) / 4

So x = (4 + 8)/4 = 3, and x = (4 – 8)/4 = -1. The solutions are 3 and -1.

所以 x = (4 + 8)/4 = 3，x = (4 – 8)/4 = -1。解为 3 和 -1。

6. The Discriminant | 判别式

The discriminant Δ = b² – 4ac tells us the number and type of roots without solving the equation. This is a key idea in IGCSE exams.

判别式 Δ = b² – 4ac 可以在不解方程的情况下告诉我们根的数量和类型。这是IGCSE考试中的一个关键概念。

If Δ > 0, the equation has two distinct real roots. If Δ = 0, it has exactly one repeated real root. If Δ < 0, it has no real roots, but two complex roots.

如果 Δ > 0，方程有两个不同的实根。如果 Δ = 0，方程有一个重根。如果 Δ < 0，方程没有实根，但有两个复数根。

Δ = b² – 4ac

For example, in x² – 2x + 5 = 0, a = 1, b = -2, c = 5, so Δ = (-2)² – 4×1×5 = 4 – 20 = -16. Since Δ < 0, there are no real roots.

例如，在 x² – 2x + 5 = 0 中，a = 1，b = -2，c = 5，所以 Δ = (-2)² – 4×1×5 = 4 – 20 = -16。因为 Δ < 0，所以没有实根。

Also, if the discriminant is a perfect square and a, b, c are rational numbers, the roots are rational and can be found by factoring.

另外，如果判别式是一个完全平方数，且 a、b、c 是有理数，那么根是有理数，可以通过因式分解找到。

7. Solving by Using Graphs | 用图像法求解

The roots of a quadratic equation ax² + bx + c = 0 are the x-coordinates of the points where the graph of y = ax² + bx + c crosses the x-axis. This graphical interpretation is often tested in IGCSE.

一元二次方程 ax² + bx + c = 0 的根是曲线 y = ax² + bx + c 与 x 轴交点的 x 坐标。这种图像解释在IGCSE中经常被考查。

If the graph cuts the x-axis at two points, the equation has two distinct real roots. If it touches the x-axis at one point, the root is repeated. If it does not cross the x-axis, there are no real roots.

如果图像与 x 轴相交于两个点，方程有两个不同的实根。如果图像与 x 轴相切于一个点，则根是重根。如果不穿越 x 轴，则没有实根。

For example, y = x² – 4x + 3 can be plotted. The curve crosses the x-axis at (1, 0) and (3, 0), so the solutions of x² – 4x + 3 = 0 are x = 1 and x = 3.

例如，可以画出 y = x² – 4x + 3 的图像。曲线在 (1, 0) 和 (3, 0) 处穿过 x 轴，因此 x² – 4x + 3 = 0 的解是 x = 1 和 x = 3。

Graphical methods may also be used to solve equations such as x² = x + 2, by drawing y = x² and y = x + 2 and finding their intersection points.

图像法也可用于解如 x² = x + 2 这样的方程，通过绘制 y = x² 和 y = x + 2 并找到它们的交点。

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

Quadratic equations often appear in real-life problems involving areas, projectile motion, profit, and numbers. The key is to translate the words into an equation.

二次方程经常出现在涉及面积、抛体运动、利润和数字的现实问题中。关键是把文字转化为方程。

Example: The length of a rectangle is 3 cm more than its width. Its area is 40 cm². Let the width be w. Then the length is w + 3. The equation is w(w + 3) = 40, or w² + 3w – 40 = 0.

示例：一个矩形的长比宽多3厘米。它的面积是40平方厘米。设宽为 w。则长为 w + 3。方程为 w(w + 3) = 40，即 w² + 3w – 40 = 0。

Factor: (w + 8)(w – 5) = 0. So w = -8 or w = 5. Since width cannot be negative, the width is 5 cm and the length is 8 cm.

因式分解：(w + 8)(w – 5) = 0。所以 w = -8 或 w = 5。因为宽不能为负，所以宽为5厘米，长为8厘米。

w(w + 3) = 40 ⇒ w² + 3w – 40 = 0 ⇒ w = 5

Always check whether the roots make sense in the original context. Reject negative or zero values if they are not possible.

始终检查根在原始情境中是否有意义。如果负数或零不可能，就舍去它们。

9. Common Mistakes and Tips | 常见错误与技巧

One common mistake is forgetting to rewrite the equation in standard form before applying the quadratic formula. For example, the equation 2x² + 3x = 5 must first become 2x² + 3x – 5 = 0.

一个常见错误是在应用求根公式前忘记将方程改写为标准形式。例如，方程 2x² + 3x = 5 必须先变为 2x² + 3x – 5 = 0。

Another mistake is dividing both sides by a variable that may be zero. For instance, do not cancel x from x² = 5x unless you consider that x = 0 is a solution. The correct result is x = 0 or x = 5.

另一个错误是除以一个可能为零的变量。例如，不要从 x² = 5x 中消去 x，除非你考虑到 x = 0 是解。正确结果是 x = 0 或 x = 5。

When using completing the square, remember to add the same value to both sides. Also, make sure the coefficient of x² is 1 before completing the square.

使用配方法时，记得在两边加上相同的值。同时，在配方前确保 x² 的系数为1。

Use a calculator to check your roots by substituting them back into the original equation. If the result is zero, the root is correct.

使用计算器将根代回原方程进行检验。如果结果为零，则根是正确的。

10. Practice Questions | 练习

Try the following questions to test your understanding. Solve each equation and state the nature of the roots where requested.

尝试以下问题来测试你的理解。解每个方程，并在要求时说明根的性质。

Solve x² – 9x + 20 = 0.
Solve 3x² + 5x – 2 = 0 using the quadratic formula.
Find the value of k such that x² + kx + 9 = 0 has a repeated root.
Write x² + 4x + 1 in the form (x + p)² + q.
Sketch the graph of y = x² – 2x – 3 and state the roots.

解 x² – 9x + 20 = 0。
使用求根公式解 3x² + 5x – 2 = 0。
求 k 的值，使得 x² + kx + 9 = 0 有重根。
将 x² + 4x + 1 写成 (x + p)² + q 的形式。
画出 y = x² – 2x – 3 的图像并说明根。

Answers: 1) x = 4, 5. 2) x = ⅓ or x = -2. 3) k = 6 or k = -6. 4) (x + 2)² – 3. 5) roots at x = -1 and x = 3.

答案：1) x = 4，5。2) x = ⅓ 或 x = -2。3) k = 6 或 k = -6。4) (x + 2)² – 3。5) 根为 x = -1 和 x = 3。

11. Summary | 总结

Quadratic equations can be solved by factorization, completing the square, using the quadratic formula, or by reading the graph. Each method has its advantages, and the discriminant helps you know what type of roots to expect.

一元二次方程可以通过因式分解、配方法、求根公式或图像法来求解。每种方法都有其优点，判别式帮助你预期根的类型。

To be successful in IGCSE exams, practise all methods and pay attention to details such as standard form, signs, and reject impossible answers in word problems.

为了在IGCSE考试中取得成功，练习所有方法并注意细节，如标准形式、符号，以及在应用题中舍去不可能的答案。

With regular practice, solving quadratic equations will become a reliable skill that appears in many parts of mathematics.

通过定期练习，解一元二次方程将成为一个可靠的技能，它出现在数学的许多部分。

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Solving Quadratic Equations | 一元二次方程的解法