Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, graphs, co-ordinate geometry, and many exam-style problems. This article explains the key methods for solving quadratic equations and the reasoning behind each step.

二次方程是 IGCSE 数学中最重要的主题之一。它出现在代数、图像、坐标几何以及许多考试题中。本文将讲解解二次方程的核心方法，并解释每一步背后的原理。

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

A quadratic equation is an equation where the highest power of the variable is 2. Its general form can be written as:

二次方程是变量最高次数为 2 的方程。它的一般形式可以写成：

ax² + bx + c = 0, 其中 a ≠ 0

Here, a, b and c are constants. The coefficient a cannot be zero, because if a = 0, the equation becomes linear, not quadratic.

这里 a、b、c 都是常数。系数 a 不能为零，因为如果 a = 0，方程就变成一次方程，而不是二次方程。

For example, x² + 3x – 4 = 0 and 2x² – 5x + 1 = 0 are both quadratic equations. In an IGCSE exam, you may be asked to solve a quadratic equation by factorising, by using the quadratic formula, or by completing the square.

例如，x² + 3x – 4 = 0 和 2x² – 5x + 1 = 0 都是二次方程。在 IGCSE 考试中，你可能会被要求通过因式分解、使用求根公式或配方法来解二次方程。

2. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the quadratic expression can be written as a product of two linear factors. The key idea is the null factor law: if the product of two expressions is zero, then at least one of them must be zero.

当二次表达式可以写成两个一次因式的乘积时，因式分解是最快的方法。核心思想是零因子法则：如果两个表达式的乘积为零，那么至少其中一个表达式必须为零。

若 (px + q)(rx + s) = 0，则 px + q = 0 或 rx + s = 0

Let us solve x² + 5x + 6 = 0. We need two numbers that multiply to 6 and add to 5. The numbers 2 and 3 work, because 2 × 3 = 6 and 2 + 3 = 5. Therefore:

我们来解 x² + 5x + 6 = 0。我们需要两个数，它们的乘积为 6，和为 5。数字 2 和 3 满足条件，因为 2 × 3 = 6 且 2 + 3 = 5。因此：

x² + 5x + 6 = (x + 2)(x + 3) = 0

Using the null factor law, x + 2 = 0 or x + 3 = 0, so x = –2 or x = –3. Always check your solutions by substituting them back into the original equation.

根据零因子法则，x + 2 = 0 或 x + 3 = 0，所以 x = –2 或 x = –3。务必把解代回原方程进行验算。

When a is not 1, factorising requires a little more care. For example, 2x² + 7x + 3 = 0 can be factorised as (2x + 1)(x + 3) = 0, giving x = –½ or x = –3.

当 a 不等于 1 时，因式分解需要更细心。例如，2x² + 7x + 3 = 0 可以分解为 (2x + 1)(x + 3) = 0，得到 x = –½ 或 x = –3。

3. Solving by the Quadratic Formula | 用公式法解二次方程

When factorisation is difficult or impossible, you can use the quadratic formula. This formula gives the exact solutions of any quadratic equation ax² + bx + c = 0.

当因式分解困难或无法进行时，你可以使用求根公式。这个公式给出任何二次方程 ax² + bx + c = 0 的精确解。

x = (–b ± √(b² – 4ac)) / (2a)

Let us apply it to 2x² + 3x – 2 = 0. Here a = 2, b = 3 and c = –2. Substituting these values gives:

我们将它应用于 2x² + 3x – 2 = 0。这里 a = 2，b = 3，c = –2。代入这些值得到：

x = (–3 ± √(3² – 4 × 2 × –2)) / (2 × 2) = (–3 ± √(9 + 16)) / 4 = (–3 ± √25) / 4

Since √25 = 5, we have x = (–3 + 5) / 4 = 2 / 4 = ½, or x = (–3 – 5) / 4 = –8 / 4 = –2. So the solutions are x = ½ and x = –2.

因为 √25 = 5，所以 x = (–3 + 5) / 4 = 2 / 4 = ½，或 x = (–3 – 5) / 4 = –8 / 4 = –2。因此解为 x = ½ 和 x = –2。

In an exam, write down the formula first, then substitute the values carefully. Be especially careful with negative signs and with the order of operations.

在考试中，先写出公式，然后仔细代入数值。要特别注意负号和运算顺序。

4. Solving by Completing the Square | 配方法

Completing the square is another algebraic technique that rewrites a quadratic in the form a(x – h)² + k. This method is very useful for finding maximum or minimum values and for deriving the quadratic formula.

配方法是另一种代数技巧，它把二次式改写成 a(x – h)² + k 的形式。这种方法非常适用于求最大值或最小值，也用于推导求根公式。

Start with x² + 6x + 2 = 0. Halve the coefficient of x, which is 6, to get 3. Then write x² + 6x as (x + 3)² – 9, because (x + 3)² = x² + 6x + 9. So the equation becomes:

从 x² + 6x + 2 = 0 开始。把 x 的系数 6 除以 2，得到 3。然后将 x² + 6x 写成 (x + 3)² – 9，因为 (x + 3)² = x² + 6x + 9。于是方程变为：

(x + 3)² – 9 + 2 = 0 ⇒ (x + 3)² = 7

Taking the square root of both sides gives x + 3 = ±√7, so x = –3 ± √7.

两边开平方得到 x + 3 = ±√7，所以 x = –3 ± √7。

If the coefficient of x² is not 1, divide through by a before completing the square. For example, 2x² + 8x + 3 = 0 becomes x² + 4x + 3/2 = 0, then (x + 2)² – 4 + 3/2 = 0, so (x + 2)² = 5/2.

如果 x² 的系数不是 1，先除以 a，再配方。例如，2x² + 8x + 3 = 0 变为 x² + 4x + 3/2 = 0，然后 (x + 2)² – 4 + 3/2 = 0，所以 (x + 2)² = 5/2。

5. The Discriminant | 判别式

The expression b² – 4ac in the quadratic formula is called the discriminant. Its sign tells us how many real roots the quadratic equation has.

求根公式中的 b² – 4ac 称为判别式。它的符号告诉我们可以得到多少个实数根。

判别式 Δ = b² – 4ac	实数根的情况
Δ > 0	两个不同实数根
Δ = 0	一个重复实数根（两相等实根）
Δ < 0	无实数根

For the equation x² – 4x + 4 = 0, the discriminant is (–4)² – 4 × 1 × 4 = 16 – 16 = 0. This means the equation has one repeated root, x = 2.

对于方程 x² – 4x + 4 = 0，判别式为 (–4)² – 4 × 1 × 4 = 16 – 16 = 0。这意味着方程有一个重根，x = 2。

The discriminant also tells you the number of x-intercepts of the parabola y = ax² + bx + c. A positive discriminant gives two intercepts, a zero discriminant gives one intercept, and a negative discriminant gives no real intercepts.

判别式也告诉你抛物线 y = ax² + bx + c 与 x 轴的交点个数。判别式为正时有两个交点，为零时有一个交点，为负时没有实数交点。

6. Quadratic Graphs and Roots | 二次函数图像与根

The solutions of a quadratic equation ax² + bx + c = 0 correspond to the x-values where the graph of y = ax² + bx + c crosses the x-axis. These points are often called roots, zeros, or x-intercepts.

二次方程 ax² + bx + c = 0 的解对应函数 y = ax² + bx + c 的图像与 x 轴交点的 x 坐标。这些点通常称为根、零点或 x 截距。

A quadratic graph is a parabola. If a > 0, it opens upwards; if a < 0, it opens downwards. The axis of symmetry is found by x = –b / (2a), and the vertex lies on this axis.

二次函数图像是抛物线。当 a > 0 时开口向上；当 a < 0 时开口向下。对称轴由 x = –b / (2a) 给出，顶点位于对称轴上。

For example, y = x² – 2x – 3 has roots x = –1 and x = 3. The axis of symmetry is x = 1, halfway between –1 and 3. Substituting x = 1 gives y = 1 – 2 – 3 = –4, so the vertex is (1, –4).

例如，y = x² – 2x – 3 的根为 x = –1 和 x = 3。对称轴为 x = 1，正好位于 –1 和 3 的中间。代入 x = 1 得 y = 1 – 2 – 3 = –4，所以顶点为 (1, –4)。

When writing a quadratic in completed-square form, y = a(x – h)² + k, the vertex is directly read as (h, k). This is helpful when sketching graphs or finding minimum and maximum values.

当二次函数写成配方式 y = a(x – h)² + k 时，可直接读出顶点为 (h, k)。这在画图和求最大、最小值时非常方便。

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

IGCSE problems often require setting up a quadratic equation from a real-life situation. After solving, check which answer makes sense in the context.

IGCSE 考题经常需要从实际情境中建立二次方程。解完方程后，要判断哪个答案符合实际背景。

Example: The area of a rectangle is 40 cm², and its length is 3 cm longer than its width. Let the width be x cm. Then the length is x + 3 cm, so the area equation is:

例：一个矩形的面积为 40 cm²，长比宽多 3 cm。设宽为 x cm，则长为 x + 3 cm，面积方程为：

x(x + 3) = 40 ⇒ x² + 3x – 40 = 0

Factorise: (x + 8)(x – 5) = 0, so x = –8 or x = 5. Since a width cannot be negative, the width is 5 cm and the length is 8 cm.

因式分解：(x + 8)(x – 5) = 0，所以 x = –8 或 x = 5。宽度不能是负数，因此宽为 5 cm，长为 8 cm。

Another common type involves consecutive integers. For example, find two consecutive positive integers whose product is 42. Let the integers be n and n + 1. Then n(n + 1) = 42, so n² + n – 42 = 0, which gives n = 6 or n = –7. The positive integer is 6, so the integers are 6 and 7.

另一类常见的题型涉及连续整数。例如，求两个连续正整数，其乘积为 42。设两个整数为 n 和 n + 1，则 n(n + 1) = 42，即 n² + n – 42 = 0，解得 n = 6 或 n = –7。正整数为 6，因此这两个整数是 6 和 7。

8. Common Mistakes | 常见错误

Not rearranging the equation to zero before factorising. Always write the equation in the form ax² + bx + c = 0 first.

因式分解前没有把方程整理为零。一定要先将方程写成 ax² + bx + c = 0 的形式。
Losing one of the two solutions. Every quadratic equation has two solutions, but they may be repeated. Do not divide both sides by x unless you are sure x is not zero, because this can remove the root x = 0.

丢掉了其中一个解。每个二次方程都有两个解，但也可能相等。不要随便两边同除以 x，除非你确定 x 不为零，否则会丢掉 x = 0 这个根。
Sign errors when substituting into the quadratic formula. Use brackets when substituting negative values.

代入求根公式时出现符号错误。代入负值时要使用括号。
Mistaking the discriminant. The discriminant is b² – 4ac, not b² + 4ac or b² – 4a c without proper signs.

弄错判别式。判别式是 b² – 4ac，而不是 b² + 4ac，也不能漏掉符号。
Forgetting that if the discriminant is negative, the equation has no real solutions. In IGCSE, unless complex numbers are introduced, write ‘no real roots’.

忘记当判别式为负时方程没有实数解。在 IGCSE 中，如果没有引入复数，应写”无实数根”。

9. Practice Questions | 练习

Try these questions on your own, then check the answers and review your methods.

请独立尝试这些题目，然后对照答案并回顾你的方法。

Question 1: Solve x² – 7x + 12 = 0.

题目 1：解方程 x² – 7x + 12 = 0。
Question 2: Solve 3x² + 5x – 2 = 0 using the quadratic formula.

题目 2：使用求根公式解方程 3x² + 5x – 2 = 0。
Question 3: Show that x² + 4x + 7 = 0 has no real roots.

题目 3：证明 x² + 4x + 7 = 0 没有实数根。
Question 4: A square has its side increased by 2 cm. If the new area is 49 cm², find the original side length.

题目 4：一个正方形边长增加 2 cm 后，新面积为 49 cm²，求原边长。

Answers: 1. x = 3 or x = 4. 2. x = 1/3 or x = –2. 3. Discriminant = 16 – 28 = –12 < 0. 4. Let original side be x; (x + 2)² = 49, so x = 5 cm.

参考答案：1. x = 3 或 x = 4。2. x = 1/3 或 x = –2。3. 判别式 = 16 – 28 = –12 < 0。4. 设原边长为 x；(x + 2)² = 49，所以 x = 5 cm。

10. Conclusion | 总结

Solving quadratic equations is a fundamental skill for IGCSE Mathematics. You should be confident in factorising, using the quadratic formula, completing the square, and interpreting the discriminant. With regular practice, these methods will become automatic.

解二次方程是 IGCSE 数学的基本技能。你应该熟练掌握因式分解、求根公式、配方法以及判别式的应用。通过规律练习，这些方法会变得非常熟练。

Always read the question carefully and choose the most suitable method. If a quadratic can be factorised easily, use factorisation; otherwise, use the quadratic formula. Completing the square is especially useful when working with graphs or solving harder problems.

做题时一定要仔细审题，选择最合适的方法。如果可以轻松因式分解，就使用因式分解法；否则使用求根公式。在研究图像或解决较难问题时，配方法尤其有用。

Remember to check your solutions and understand why each method works, not just how. This deeper understanding will help you earn full marks in exams and build a strong foundation for future mathematics.

记住要检查你的解，并理解每种方法背后的原理，而不只是会操作。这种深入的理解能帮助你在考试中获得满分，并为今后的数学学习打下坚实基础。

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Solving Quadratic Equations | 解二次方程