Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations appear frequently in Edexcel IGCSE Mathematics. This article covers the essential methods for solving them, including factorisation, the quadratic formula, and completing the square. We will also discuss how to interpret the solutions and avoid common exam pitfalls.

在Edexcel IGCSE数学中，二次方程出现频繁。本文涵盖解二次方程的核心方法，包括因式分解、二次公式和配方法。我们还将讨论如何理解解的含义以及避免常见考试陷阱。

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 the variable x is 2, which gives the equation its name ‘quadratic’. In IGCSE Mathematics, you must be able to solve such equations using a variety of techniques.

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。变量的最高次数为 2，因此称为“二次”。在IGCSE数学中，你必须能够使用多种技巧解此类方程。

For example, 2x² + 4x + 1 = 0 is a quadratic equation with a = 2, b = 4 and c = 1. This particular example will be used throughout this article to demonstrate each method.

例如，2x² + 4x + 1 = 0 是一个二次方程，其中 a = 2，b = 4，c = 1。本文将使用该示例来演示每种方法。

2. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method for solving a quadratic equation, provided the equation can be factorised into two linear expressions. For a quadratic with simple integer coefficients, we look for two numbers that multiply to give ac and add to give b. However, not all quadratics can be factorised easily, especially when a ≠ 1.

因式分解通常是解二次方程最快的方法，前提是方程可以分解为两个一次表达式。对于具有简单整数系数的二次方程，我们寻找两个数，它们相乘等于 ac，相加等于 b。然而，并非所有二次方程都能轻松因式分解，尤其是当 a ≠ 1 时。

For the equation 2x² + 4x + 1 = 0, we need two numbers that multiply to 2 × 1 = 2 and add to 4. The pair 2 and 1 gives sum 3, and 4 and 0.5 gives sum 4.5, so no integer pair works. Thus, we cannot factorise this quadratic using simple integers. In such cases, we use the quadratic formula or completing the square.

对于方程 2x² + 4x + 1 = 0，我们需要找到两个数，它们相乘等于 2 × 1 = 2，相加等于 4。2 和 1 的和为 3，4 和 0.5 的和为 4.5，因此没有整数对满足条件。所以，这个二次方程无法用简单整数因式分解。在这种情况下，我们使用二次公式或配方法。

Factorisation works when the equation can be written as (px + q)(rx + s) = 0.

因式分解适用于方程可以写成 (px + q)(rx + s) = 0 的情形。

3. The Quadratic Formula | 二次公式法

The quadratic formula is a universal method that solves any quadratic equation. The formula is derived by completing the square on the general form ax² + bx + c = 0. It is particularly useful when factorisation is difficult or impossible.

二次公式是一种万能方法，可以解任何二次方程。该公式通过对一般形式 ax² + bx + c = 0 进行配方推导得出。当因式分解困难或不可能时，它尤其有用。

x = (−b ± √(b² − 4ac)) / (2a)

For 2x² + 4x + 1 = 0, substitute a = 2, b = 4, c = 1:

对于 2x² + 4x + 1 = 0，代入 a = 2，b = 4，c = 1：

x = (−4 ± √(4² − 4 × 2 × 1)) / (2 × 2) = (−4 ± √(16 − 8)) / 4 = (−4 ± √8) / 4

Since √8 = 2√2, the solutions are x = (−4 + 2√2) / 4 = (−2 + √2) / 2 and x = (−4 − 2√2) / 4 = (−2 − √2) / 2. Numerically, these are approximately −0.293 and −1.707.

因为 √8 = 2√2，解为 x = (−4 + 2√2) / 4 = (−2 + √2) / 2 和 x = (−4 − 2√2) / 4 = (−2 − √2) / 2。近似值为 −0.293 和 −1.707。

You can use the quadratic formula on all quadratic equations, but it is essential to memorise it correctly and be careful with negative signs.

你可以在所有二次方程上使用二次公式，但必须准确记忆公式并注意负号。

4. Completing the Square | 配方法

Completing the square is another algebraic technique that rewrites a quadratic expression in the form a(x + p)² + q. This form is especially useful for identifying the vertex of a parabola and for solving equations, even when factorisation fails.

配方法是另一种代数技巧，它将二次表达式改写为 a(x + p)² + q 的形式。这种形式特别适用于确定抛物线的顶点和求解方程，即使因式分解失败也能使用。

To complete the square for 2x² + 4x + 1 = 0, first factor out the coefficient of x²:

要对 2x² + 4x + 1 = 0 配方，首先提出 x² 的系数：

2(x² + 2x) + 1 = 0

Now complete the square inside the brackets. For x² + 2x, half the coefficient of x is 1, so write (x + 1)² − 1:

现在对括号内部配方。对于 x² + 2x，x 系数的一半是 1，所以写成 (x + 1)² − 1：

2[(x + 1)² − 1] + 1 = 0 → 2(x + 1)² − 2 + 1 = 0 → 2(x + 1)² − 1 = 0

Then solve for x:

然后解出 x：

2(x + 1)² = 1 → (x + 1)² = ½ → x + 1 = ±√(½) → x = −1 ± √(½)

Notice that √(½) = √2 / 2, so x = −1 + √2 / 2 = (−2 + √2) / 2, which matches the quadratic formula result.

注意 √(½) = √2 / 2，所以 x = −1 + √2 / 2 = (−2 + √2) / 2，这与二次公式的结果一致。

5. Solving by Taking Square Roots | 开平方法

If a quadratic equation can be written in the form (x + p)² = q, you can solve it directly by taking square roots. This is essentially the final step of completing the square, but it can also be used directly for equations without a linear term.

如果二次方程可以写成 (x + p)² = q 的形式，你可以直接通过开平方求解。这本质上是配方法的最后一步，但也可直接用于没有一次项的方程。

For example, consider (2x − 3)² = 16. Taking the square root of both sides gives:

例如，考虑 (2x − 3)² = 16。对两边开平方得：

2x − 3 = ±4 → 2x = 3 ± 4 → x = (3 ± 4) / 2

So x = 7/2 = 3.5 or x = −1/2 = −0.5.

因此 x = 7/2 = 3.5 或 x = −1/2 = −0.5。

When using this method, remember that the square root of a positive number has two values: one positive and one negative.

使用此方法时，请记住正数的平方根有两个值：一正一负。

6. Understanding the Discriminant | 理解判别式

The discriminant is the part of the quadratic formula under the square root sign: Δ = b² − 4ac. It tells us the nature of the solutions without solving the equation fully.

判别式是二次公式中根号下的部分：Δ = b² − 4ac。它能在不完全解方程的情况下告诉我们解的性质。

For 2x² + 4x + 1 = 0, Δ = 4² − 4 × 2 × 1 = 16 − 8 = 8. Since Δ > 0 and not a perfect square, the equation has two distinct irrational roots.

对于 2x² + 4x + 1 = 0，Δ = 4² − 4 × 2 × 1 = 16 − 8 = 8。因为 Δ > 0 且不是完全平方数，方程有两个不同的无理根。

The three cases to remember are:

需要记住的三种情况是：

If Δ > 0, there are two real and distinct roots.

如果 Δ > 0，有两个不同实数根。
If Δ = 0, there is one real repeated root (a double root).

如果 Δ = 0，有一个实数重根（二重根）。
If Δ < 0, there are no real roots (two complex roots, not required at IGCSE).

如果 Δ < 0，没有实数根（两个复数根，IGCSE不要求）。

7. Solving Quadratic Inequalities | 解二次不等式

Although the main topic is solving equations, IGCSE often asks to solve quadratic inequalities using a sign diagram or graph. For example, solve x² − x − 6 > 0.

虽然主题是解方程，但IGCSE经常要求使用符号图或图像解二次不等式。例如，解 x² − x − 6 > 0。

First factorise: (x − 3)(x + 2) = 0, so critical points are x = −2 and x = 3. The parabola opens upwards, so the inequality is satisfied when x < −2 or x > 3.

首先因式分解：(x − 3)(x + 2) = 0，所以关键点为 x = −2 和 x = 3。抛物线开口向上，因此不等式在 x < −2 或 x > 3 时成立。

x < −2 or x > 3 (or in set notation: {x : x < −2} ∪ {x : x > 3})

x < −2 或 x > 3（或用集合记号：{x : x < −2} ∪ {x : x > 3}）

Remember to check the direction of the inequality sign when multiplying by a negative number, though this is not needed when solving quadratics directly.

请注意，当乘以负数时不等式方向要改变，但在直接解二次不等式时不需要这样做。

8. Graphs of Quadratic Equations | 二次方程的图像

The graph of a quadratic equation y = ax² + bx + c is a parabola. The solutions of ax² + bx + c = 0 are the x-coordinates where the parabola crosses the x-axis. If the discriminant is negative, the parabola does not touch the x-axis.

二次方程 y = ax² + bx + c 的图像是一条抛物线。ax² + bx + c = 0 的解是抛物线与 x 轴交点的横坐标。如果判别式为负，抛物线不与 x 轴相交。

For y = 2x² + 4x + 1, the vertex can be found by completing the square: y = 2(x + 1)² − 1. So the vertex is (−1, −1). Since a = 2 > 0, the parabola opens upwards. The x-intercepts are approximately −0.293 and −1.707, as found earlier.

对于 y = 2x² + 4x + 1，可以通过配方找到顶点：y = 2(x + 1)² − 1。所以顶点为 (−1, −1)。由于 a = 2 > 0，抛物线开口向上。x 截距约为 −0.293 和 −1.707，如前所述。

Being able to sketch the graph quickly helps you check whether your algebraic solutions are reasonable, especially in multiple-choice questions.

能够快速画出草图有助于你检查代数解是否合理，尤其是在选择题中。

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

Many IGCSE problems describe a real-life situation that leads to a quadratic equation. For example, a rectangle has a length 3 cm longer than its width, and its area is 40 cm². Let the width be x cm, then length = (x + 3) cm, so x(x + 3) = 40.

许多IGCSE问题描述一个实际情境，导致二次方程。例如，一个长方形的长比宽长 3 cm，面积为 40 cm²。设宽为 x cm，则长 = (x + 3) cm，所以 x(x + 3) = 40。

x² + 3x − 40 = 0 → (x + 8)(x − 5) = 0 → x = 5 or x = −8

Since width cannot be negative, discard x = −8. The width is 5 cm and the length is 8 cm.

因为宽度不能为负，舍去 x = −8。所以宽度为 5 cm，长度为 8 cm。

When solving word problems, always consider the context of the problem and reject any solution that does not make sense in that context.

解应用题时，始终考虑问题的实际背景，并舍弃在该背景下不合理的解。

10. Exam Tips and Common Mistakes | 考试技巧与常见错误

Here are some key points to remember for the exam:

以下是一些考试中需要记住的关键点：

Arrange the equation in the form ax² + bx + c = 0 before solving.

先将方程整理为 ax² + bx + c = 0 的形式再求解。
If factorisation is not obvious, use the quadratic formula. You are allowed a calculator, but ensure you input the numbers correctly.

如果因式分解不明显，使用二次公式。考试允许使用计算器，但确保输入数字正确。
Do not forget the ± sign when taking square roots.

开平方时不要忘记 ± 符号。
When a = 1, factorisation is often straightforward; when a ≠ 1, look for common factors first.

当 a = 1 时，因式分解通常简单；当 a ≠ 1 时，先寻找公因式。
Check your answers by substituting them back into the original equation.

通过将解代回原方程来检查答案。

For the equation 2x² + 4x + 1 = 0, substituting x = (−2 + √2) / 2 gives:

对于方程 2x² + 4x + 1 = 0，代入 x = (−2 + √2) / 2 得到：

2[(−2+√2)/2]² + 4[(−2+√2)/2] + 1 = 2[(6−4√2)/4] + (−4+2√2) + 1 = (3−2√2) −4+2√2 +1 = 0

The simplification confirms the solution.

这个化简验证了解的正确性。

11. Practice Questions | 练习题

Solve the following quadratic equations using any method:

使用任意方法解下列二次方程：

1) x² − 7x + 12 = 0

1) x² − 7x + 12 = 0
2) 3x² − 5x − 2 = 0

2) 3x² − 5x − 2 = 0
3) 4x² + 12x + 9 = 0

3) 4x² + 12x + 9 = 0
4) 2x² − 4x + 1 = 0

4) 2x² − 4x + 1 = 0

Answers: 1) x = 3 or x = 4; 2) x = −1/3 or x = 2; 3) x = −3/2 (repeated); 4) x = 1 ± √2/2.

答案：1) x = 3 或 x = 4；2) x = −1/3 或 x = 2；3) x = −3/2（重根）；4) x = 1 ± √2/2。

12. Summary | 总结

You have now reviewed the main methods for solving quadratic equations: factorisation, the quadratic formula, and completing the square. Each method has its place, and the discriminant tells you the nature of the roots. With regular practice, you can confidently tackle any quadratic equation in your IGCSE exam.

你已经复习了解二次方程的主要方法：因式分解、二次公式和配方法。每种方法都有其适用场合，判别式告诉你根的性质。通过定期练习，你可以自信地应对IGCSE考试中的任何二次方程。

Remember the example 2x² + 4x + 1 = 0 – it shows how even an unfactorable quadratic can be solved beautifully with the quadratic formula.

记住例子 2x² + 4x + 1 = 0——它展示了即使一个无法因式分解的二次方程也能用二次公式很好地求解。

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Solving Quadratic Equations | 解二次方程