Understanding Quadratic Equations | 理解二次方程

📚 Understanding Quadratic Equations | 理解二次方程

Quadratic equations are one of the most important topics in the IGCSE Mathematics syllabus. They appear in algebra, coordinate geometry, and even in problem-solving questions about area and motion. Mastering them is essential for exam success.

二次方程是 IGCSE 数学大纲中最重要的主题之一。它们出现在代数、坐标几何，甚至涉及面积和运动的应用题中。掌握二次方程对考试成功至关重要。

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 constants, and a ≠ 0. The highest power of the variable x is 2.

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

For example, 2x² – 3x + 1 = 0 is a quadratic equation. Here a = 2, b = -3, and c = 1.

例如，2x² – 3x + 1 = 0 是一个二次方程。这里 a = 2，b = -3，c = 1。

A quadratic equation can have two distinct real roots, one repeated root, or no real roots. The number of roots depends on the value of the discriminant.

二次方程可以有两个不同的实数根、一个重根，或者没有实数根。根的个数取决于判别式的值。

2. Solving by Factorisation | 因式分解法解方程

If a quadratic expression can be written as a product of two linear factors, we can solve the equation by setting each factor equal to zero.

如果二次表达式可以写成两个一次因式的乘积，我们可以令每个因式等于零来解方程。

x² + 5x + 6 = 0 ⇒ (x + 2)(x + 3) = 0 ⇒ x = -2 or x = -3

Always check that the factors multiply to give the original expression. The constant term 6 factors into 2 and 3, and their sum is 5, which is the coefficient of x.

始终检查因式相乘是否能得到原表达式。常数项 6 分解为 2 和 3，它们的和为 5，正好是 x 的系数。

Rearrange the equation so that one side is zero.
Factorise the quadratic expression.
Set each factor equal to zero and solve for x.

将方程整理为一侧为零。
对二次表达式进行因式分解。
令每个因式等于零，并解出 x。

3. Solving by Completing the Square | 配方法解方程

Completing the square rewrites a quadratic equation in the form (x + p)² = q, which makes it easy to solve by taking square roots.

配方法将二次方程改写为 (x + p)² = q 的形式，从而可以通过开平方轻松求解。

x² – 6x + 4 = 0 ⇒ (x – 3)² – 9 + 4 = 0 ⇒ (x – 3)² = 5 ⇒ x = 3 ± √5

For an equation of the form x² + bx + c = 0, add and subtract (b/2)² inside the expression.

对于形如 x² + bx + c = 0 的方程，在表达式中加上并减去 (b/2)²。

If the coefficient of x² is not 1, divide the whole equation by that coefficient before completing the square.

如果 x² 的系数不是 1，先对等式两边同时除以该系数，再进行配方。

4. The Quadratic Formula | 二次求根公式

The quadratic formula works for any quadratic equation, including those that cannot be factorised easily.

二次求根公式适用于任何二次方程，包括那些不容易因式分解的方程。

x = (-b ± √(b² – 4ac)) / (2a)

Here b² – 4ac is called the discriminant, often denoted by Δ.

这里的 b² – 4ac 称为判别式，通常用 Δ 表示。

If Δ > 0, there are two distinct real roots.
If Δ = 0, there is one repeated real root.
If Δ < 0, there are no real roots.

如果 Δ > 0，方程有两个不同的实数根。
如果 Δ = 0，方程有一个重实数根。
如果 Δ < 0，方程没有实数根。

5. The Discriminant and the Nature of Roots | 判别式与根的性质

In the IGCSE exam, you may be asked to determine the nature of roots without solving the equation fully. Simply evaluate the discriminant.

在 IGCSE 考试中，你可能会被要求不解方程而判断根的性质。只需计算判别式即可。

For 2x² – 4x + 1 = 0: Δ = (-4)² – 4×2×1 = 16 – 8 = 8 > 0

Since Δ > 0, the equation has two distinct real roots.

因为 Δ > 0，该方程有两个不同的实数根。

If the discriminant is a perfect square, the roots are rational; otherwise, they are irrational.

如果判别式是一个完全平方数，则根是有理数；否则是无理数。

6. Solving Quadratic Equations by Taking Square Roots | 用开平方解二次方程

When a quadratic equation has no linear term (b = 0), it can be solved directly by isolating x².

当二次方程没有一次项（b = 0）时，可以通过孤立 x² 直接求解。

3x² – 12 = 0 ⇒ x² = 4 ⇒ x = ±2

Remember that the square root of a positive number gives two possible values: one positive and one negative.

记住正数的平方根给出两个可能的值：一个正数和一个负数。

If the right-hand side is negative, the equation has no real solutions.

如果等号右边为负数，则方程没有实数解。

7. Graphical Interpretation | 图形解释

The graph of a quadratic function y = ax² + bx + c is a parabola. The roots of the equation ax² + bx + c = 0 correspond to the x-intercepts of the parabola.

二次函数 y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的根对应于抛物线与 x 轴的交点。

If the discriminant is positive, the parabola crosses the x-axis at two points. If it is zero, the parabola touches the x-axis at one point. If it is negative, the parabola does not intersect the x-axis.

如果判别式为正，抛物线与 x 轴有两个交点。如果为零，抛物线在一点与 x 轴相切。如果为负，抛物线不与 x 轴相交。

Discriminant 判别式	Roots 根	Graph 图像
Δ > 0	Two real roots 两个实根	Intersects x-axis twice 与 x 轴相交两点
Δ = 0	One repeated root 一个重根	Touches x-axis 与 x 轴相切
Δ < 0	No real roots 无实根	No intersection 无交点

8. Sum and Product of Roots | 根的和与积

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

对于具有根 α 和 β 的二次方程 ax² + bx + c = 0，有以下关系成立：

α + β = -b/a

αβ = c/a

These formulas are useful when you need to find a new equation given its roots, or to check your solutions quickly.

这些公式在已知根求新方程，或快速检验解答时非常有用。

For example, if one root of x² – 5x + k = 0 is 2, then the other root is 3, because the sum of roots is 5. Therefore k = 2 × 3 = 6.

例如，如果 x² – 5x + k = 0 的一个根是 2，则另一个根是 3，因为根的和为 5。因此 k = 2 × 3 = 6。

9. Solving Word Problems with Quadratic Equations | 用二次方程解应用题

Many IGCSE problems involve forming a quadratic equation from a real-world situation, such as area, projectile motion, or consecutive integers.

许多 IGCSE 问题涉及从现实情境中建立二次方程，例如面积、抛体运动或连续整数。

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 length = x + 3, and area = x(x + 3) = 40.

示例：一个矩形的面积是 40 平方厘米，其长度比宽度长 3 厘米。设宽度为 x 厘米。则长度 = x + 3，面积 = x(x + 3) = 40。

x² + 3x – 40 = 0 ⇒ (x + 8)(x – 5) = 0 ⇒ x = 5 (or x = -8, rejected)

Always check whether each solution makes sense in the context of the problem. Negative lengths or distances are usually rejected.

始终检查每个解在问题情境中是否有意义。负的长度或距离通常被舍去。

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

One common mistake is forgetting to write the equation in the standard form ax² + bx + c = 0 before factorising or using the formula.

一个常见错误是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。

Another mistake is misapplying the quadratic formula signs, especially when b is negative. Use brackets carefully.

另一个错误是误用二次公式中的符号，特别是当 b 为负数时。谨慎使用括号。

Always show your method clearly, especially when using the quadratic formula.
Check if the answer should be rounded to a specific degree of accuracy.
For multiple-choice questions, test the answers by substituting back into the original equation.

始终清晰展示你的解题过程，尤其是使用二次公式时。
检查答案是否需要精确到特定精度。
对于选择题，将答案代回原方程进行检验。

11. Practice Questions | 练习题目

Try these questions to test your understanding. Solutions are provided below.

尝试以下问题来测试你的理解。解答在下方。

1. Solve x² – 7x + 12 = 0 by factorisation.

1. 用因式分解法解 x² – 7x + 12 = 0。

2. Solve 2x² + 3x – 5 = 0 using the quadratic formula.

2. 使用二次公式解 2x² + 3x – 5 = 0。

3. Determine the nature of roots of 4x² – 4x + 1 = 0.

3. 判断 4x² – 4x + 1 = 0 的根的性质。

Solutions: 1. (x – 3)(x – 4) = 0, so x = 3 or 4. 2. x = (-3 ± √(9 + 40)) / 4 = (-3 ± 7) / 4, so x = 1 or x = -2.5. 3. Δ = 16 – 16 = 0, so one repeated root.

解答：1. (x – 3)(x – 4) = 0，所以 x = 3 或 4。2. x = (-3 ± √(9 + 40)) / 4 = (-3 ± 7) / 4，所以 x = 1 或 x = -2.5。3. Δ = 16 – 16 = 0，所以有一个重根。

12. Summary | 总结

Quadratic equations can be solved by factorisation, completing the square, using the quadratic formula, or taking square roots. The discriminant tells us about the nature of the roots without solving the equation.

二次方程可以通过因式分解、配方法、二次求根公式或开平方来求解。判别式让我们无需解方程就能判断根的性质。

Be sure to practise all methods and check your answers for reasonableness. With consistent revision, quadratic equations will become one of the easiest scoring topics in your IGCSE exam.

务必练习所有方法并检查答案的合理性。通过持续复习，二次方程将成为你 IGCSE 考试中容易得分的考点之一。

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Understanding Quadratic Equations | 理解二次方程