Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations appear in almost every IGCSE Mathematics paper. Whether you are asked to solve them algebraically, interpret their graphs, or use them in problem-solving, a clear understanding of the standard methods is essential.

二次方程几乎出现在每一份 IGCSE 数学试卷中。无论是要求你用代数方法求解、解读它们的图像，还是在应用题中使用它们，清晰掌握标准方法都至关重要。

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 real numbers, and a ≠ 0. The highest power of the unknown x is 2.

二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b、c 是实数，且 a ≠ 0。未知数 x 的最高次数是 2。

If a = 0, the equation becomes linear, not quadratic.

如果 a = 0，方程就变成一次方程，而不是二次方程。
The values of x that make the equation true are called roots or solutions.

使方程成立的 x 值称为根或解。
A quadratic equation can have at most two real roots.

一个二次方程至多有两个实数根。

2. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method when the quadratic expression can be written as a product of two linear factors.

当二次表达式可以写成两个一次因式相乘时，因式分解通常是最快的方法。

Consider x² + 5x + 6 = 0. We look for two numbers that multiply to give 6 and add to give 5. These numbers are 2 and 3.

考虑 x² + 5x + 6 = 0。我们寻找两个数，它们相乘得到 6，相加得到 5。这两个数是 2 和 3。

(x + 2)(x + 3) = 0

Using the zero-product property, either x + 2 = 0 or x + 3 = 0, so x = -2 or x = -3.

根据零乘性质，要么 x + 2 = 0，要么 x + 3 = 0，因此 x = -2 或 x = -3。

When the coefficient of x² is not 1, for example 2x² + 7x + 3 = 0, you may need to use the ‘ac’ method or simply test factor pairs carefully.

当 x² 的系数不是 1 时，例如 2x² + 7x + 3 = 0，你可能需要使用“ac 法”或仔细检验因子对。

3. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form p(x + q)² + r. This is especially useful for finding turning points and solving equations.

配方法将二次式改写为 p(x + q)² + r 的形式。这在求顶点坐标和解方程时尤其有用。

For x² + 6x + 1 = 0, take half of 6, which is 3, and square it to get 9. Rewrite as (x + 3)² – 9 + 1 = 0.

对于 x² + 6x + 1 = 0，取 6 的一半为 3，平方得 9。改写为 (x + 3)² – 9 + 1 = 0。

(x + 3)² – 8 = 0

Then solve: (x + 3)² = 8, so x + 3 = ±√8, hence x = -3 ± 2√2.

然后求解：(x + 3)² = 8，所以 x + 3 = ±√8，因此 x = -3 ± 2√2。

For expressions with a coefficient on x², factor that coefficient out first before completing the square.

对于 x² 有系数的表达式，先提取该系数，再配方。

4. The Quadratic Formula | 二次公式

The quadratic formula works for every quadratic equation, even when factorisation is difficult or impossible.

二次公式适用于所有二次方程，即使无法因式分解或用因式分解很困难。

x = (-b ± √(b² – 4ac)) / 2a

Given ax² + bx + c = 0, substitute the values of a, b and c carefully.

给定 ax² + bx + c = 0，认真代入 a、b、c 的值。

Example: Solve 2x² – 3x – 5 = 0. Here a = 2, b = -3, c = -5.

示例：解 2x² – 3x – 5 = 0。这里 a = 2，b = -3，c = -5。

x = (3 ± √(9 + 40)) / 4 = (3 ± 7) / 4

So x = (3 + 7)/4 = 2.5, or x = (3 – 7)/4 = -1.

所以 x = (3 + 7)/4 = 2.5，或 x = (3 – 7)/4 = -1。

5. The Discriminant | 判别式

The discriminant is the part of the quadratic formula under the square root: Δ = b² – 4ac. It tells us how many real roots exist without solving the equation.

判别式是二次公式中根号下的部分：Δ = b² – 4ac。它无需解方程就能告诉我们方程有多少个实数根。

Value of Δ	Number of real roots	Graph interpretation
Δ > 0	Two distinct real roots	The curve crosses the x-axis twice
Δ = 0	One repeated real root	The curve touches the x-axis once
Δ < 0	No real roots	The curve does not intersect the x-axis

For example, 3x² – 6x + 2 = 0 has Δ = 36 – 24 = 12 > 0, so it has two real roots.

例如，3x² – 6x + 2 = 0 的 Δ = 36 – 24 = 12 > 0，因此它有两个实数根。

6. Sum and Product of Roots | 根的和与积

For any quadratic equation ax² + bx + c = 0 with roots α and β, the following relationships hold:

对于任意二次方程 ax² + bx + c = 0，若其根为 α 和 β，则以下关系成立：

α + β = -b/a, αβ = c/a

This is useful when you need to find a new equation with given roots, or to verify your solutions.

当需要构造一个具有给定根的二次方程，或验证你的解时，这个关系非常有用。

Example: If the roots are 2 and -5, then the sum is -3 and the product is -10. The equation is x² + 3x – 10 = 0.

示例：如果根为 2 和 -5，则和为 -3，积为 -10。方程为 x² + 3x – 10 = 0。

7. Graphical Interpretation | 图像意义

The graph of y = ax² + bx + c is a parabola. The roots of the equation ax² + bx + c = 0 are exactly the x-intercepts of this parabola.

y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的根正好是这条抛物线与 x 轴的交点。

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

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

The turning point can be found by completing the square: for y = (x – h)² + k, the turning point is (h, k).

顶点可以通过配方法求得：对于 y = (x – h)² + k，顶点为 (h, k)。

The axis of symmetry is the vertical line x = -b/(2a), which passes through the turning point.

对称轴是垂直线 x = -b/(2a)，它经过顶点。

8. Solving Word Problems | 应用题求解

Many IGCSE questions present quadratic equations in a real-world context. The key is to translate the words into an equation correctly.

许多 IGCSE 题目以实际情境呈现二次方程。关键是要正确地将文字转化为方程。

Example: The product of two consecutive positive integers is 156. Let the smaller integer be n. Then n(n + 1) = 156, so n² + n – 156 = 0.

示例：两个连续正整数的积是 156。设较小的整数为 n。则 n(n + 1) = 156，即 n² + n – 156 = 0。

Factorise: (n + 13)(n – 12) = 0. Since n > 0, n = 12. The integers are 12 and 13.

因式分解：(n + 13)(n – 12) = 0。因为 n > 0，所以 n = 12。这两个整数是 12 和 13。

Always check whether a negative root is valid in the given context. Sometimes it must be rejected.

始终检查负根在特定情境中是否有效。有时必须舍去负根。

9. Common Pitfalls | 常见陷阱

Students often make small errors that cost marks. Here are the most frequent ones:

学生常犯小错误而导致失分。以下是最常见的陷阱：

Forgetting to write the equation in standard form ax² + bx + c = 0 before factorising or applying the formula.

在因式分解或使用公式之前，忘记将方程写成标准形式 ax² + bx + c = 0。
Sign errors when substituting negative values into the quadratic formula.

将负值代入二次公式时出现符号错误。
Dividing both sides by x when x may be zero, which loses a root.

两边同时除以 x，而 x 可能为零，导致丢失一个根。
Misinterpreting “no real roots” as “no roots at all” — actually complex roots may exist.

将“没有实数根”误解为“根本没有根”——实际上可能存在复数根。
Forgetting to simplify surds like √8 = 2√2.

忘记化简根式，例如 √8 = 2√2。

10. Practice for Success | 成功需要练习

To master quadratic equations, you need to recognise which method is most efficient for each question. Start with factorisation, then use completing the square or the quadratic formula when needed.

要掌握二次方程，你需要识别每个问题中哪种方法最高效。先从因式分解开始，需要时再使用配方法或二次公式。

Try solving the same equation by two different methods to verify your answer. This builds confidence and reduces calculation errors.

尝试用两种不同方法解同一个方程来验证答案。这能增强信心并减少计算错误。

Also practise drawing rough sketches of parabolas using the roots, the y-intercept, and the turning point. This reinforces the link between algebra and geometry.

同时练习根据根、y 轴截距和顶点大致画出抛物线草图。这有助于强化代数与几何之间的联系。

Remember: every quadratic can be solved by the quadratic formula, but factorisation is often quicker. Develop your instinct for choosing wisely.

记住：所有二次方程都可以用二次公式求解，但因式分解通常更快。培养你做出明智选择的直觉。

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Solving Quadratic Equations | 解一元二次方程