Mastering Quadratic Equations | 掌握二次方程

📚 Mastering Quadratic Equations | 掌握二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, graphs, and problem-solving questions, and mastering them is essential for exam success. This article provides a complete revision guide with clear explanations, worked examples, and common pitfalls to avoid.

二次方程是IGCSE数学中最重要的话题之一。它们出现在代数、图像和实际问题中，掌握二次方程对考试成功至关重要。本文提供完整的复习指南，包含清晰的解释、典型例题和常见易错点。

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

A quadratic equation is a polynomial equation of degree 2. Its general form is written as:

二次方程是次数为2的多项式方程。它的一般形式写成：

ax² + bx + c = 0, where a ≠ 0

Here, a, b, and c are real numbers, and x is the variable. The coefficient a cannot be zero; otherwise the equation becomes linear.

这里，a、b、c 是实数，x 是变量。系数 a 不能为零，否则方程就变成了线性方程。

Examples of quadratic equations:

二次方程的例子：

x² – 5x + 6 = 0
2x² + 3x – 1 = 0
x² – 9 = 0

The highest power of the variable is 2, and the graph of a quadratic function is a parabola.

变量的最高次数是2，二次函数的图像是一条抛物线。

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 brackets. The principle is: if the product of two factors is zero, then at least one factor must be zero.

因式分解法是当二次表达式可以写成两个线性括号相乘时最快的方法。其原理是：如果两个因子的乘积为零，那么至少有一个因子为零。

For example, solve x² – 5x + 6 = 0. We look for two numbers that multiply to 6 and add to –5. These are –2 and –3.

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

(x – 2)(x – 3) = 0

So x – 2 = 0 or x – 3 = 0, giving x = 2 or x = 3.

因此 x – 2 = 0 或 x – 3 = 0，得到 x = 2 或 x = 3。

Steps for factorisation:

因式分解的步骤：

Write the equation in the form ax² + bx + c = 0.
Factor the left-hand side completely.
Set each factor equal to zero and solve.
检查解是否满足原方程。

When the coefficient of x² is not 1, the factorisation may require the method of grouping or trial and error.

当 x² 的系数不是1时，因式分解可能需要分组法或试错法。

3. Solving by Completing the Square | 配方法

Completing the square is a powerful algebraic technique that transforms a quadratic expression into a perfect square plus an adjustment. It works for any quadratic equation, even when factorisation is difficult.

配方法是一种强大的代数技巧，它将二次表达式转化为一个完全平方加上一个调整项。它适用于任何二次方程，即使在因式分解困难时也能使用。

The general formula for completing the square on x² + bx is:

对 x² + bx 配方的一般公式是：

x² + bx = (x + b/2)² – (b/2)²

For example, solve x² + 6x – 7 = 0 by completing the square.

例如，用配方法解 x² + 6x – 7 = 0。

First, move the constant to the other side:

首先，将常数移到另一边：

x² + 6x = 7

Add (6/2)² = 9 to both sides:

两边加上 (6/2)² = 9：

x² + 6x + 9 = 16

Then (x + 3)² = 16, so x + 3 = ±4. This gives x = 1 or x = –7.

于是 (x + 3)² = 16，所以 x + 3 = ±4。得到 x = 1 或 x = –7。

Completing the square is also useful for finding the turning point of a parabola.

配方法在求抛物线顶点时也很有用。

4. The Quadratic Formula | 二次公式

The quadratic formula is the most general method for solving any quadratic equation. It is derived from completing the square and is especially useful when the expression cannot be factorised easily.

二次公式是解任何二次方程最通用的方法。它由配方法推导而来，在表达式无法轻易因式分解时特别有用。

For ax² + bx + c = 0, the solutions are given by:

对于 ax² + bx + c = 0，解由以下公式给出：

x = [–b ± √(b² – 4ac)] / (2a)

Example: solve 2x² + 3x – 2 = 0.

例：解 2x² + 3x – 2 = 0。

Here a = 2, b = 3, c = –2. Substitute into the formula:

这里 a = 2，b = 3，c = –2。代入公式：

x = [–3 ± √(9 + 16)] / 4 = [–3 ± √25] / 4

So x = (–3 + 5)/4 = 1/2 or x = (–3 – 5)/4 = –2.

因此 x = (–3 + 5)/4 = 1/2 或 x = (–3 – 5)/4 = –2。

You must memorise the quadratic formula carefully, including the ± sign and the discriminant inside the square root.

你必须牢记二次公式，包括 ± 号以及根号内的判别式。

5. The Discriminant | 判别式

The discriminant is the expression under the square root in the quadratic formula: b² – 4ac. It tells us the nature of the roots without solving the equation completely.

判别式是二次公式中根号内的表达式：b² – 4ac。它告诉我们在不完全解方程的情况下根的性质。

Discriminant	Nature of Roots
b² – 4ac > 0	Two distinct real roots
b² – 4ac = 0	One repeated real root (two equal roots)
b² – 4ac < 0	No real roots (two complex roots)

判别式	根的性质
b² – 4ac > 0	两个不同的实数根
b² – 4ac = 0	一个重根（两个相等的实数根）
b² – 4ac < 0	没有实数根（两个复数根）

For example, the equation x² – 4x + 4 = 0 has discriminant 16 – 16 = 0, so it has exactly one solution x = 2.

例如，方程 x² – 4x + 4 = 0 的判别式为 16 – 16 = 0，所以它只有一个解 x = 2。

The discriminant also helps determine how many intersections a parabola has with the x-axis.

判别式还可以帮助确定抛物线与x轴有多少个交点。

6. The Graph of a Quadratic Function | 二次函数图像

The graph of y = ax² + bx + c is a parabola. The sign of a determines its orientation:

y = ax² + bx + c 的图像是抛物线。a 的符号决定开口方向：

If a > 0, the parabola opens upwards (U-shaped).
如果 a > 0，抛物线开口向上（U形）。
If a < 0, the parabola opens downwards (n-shaped).
如果 a < 0，抛物线开口向下（n形）。

The roots of the equation are the x-intercepts of the graph. The y-intercept is simply the value of c.

方程的根是图像与x轴的交点。y轴截距就是 c 的值。

To sketch a parabola, you need three key features:

绘制抛物线草图需要三个关键特征：

The x-intercepts (roots), if any.
x轴截距（根），如果有的话。
The y-intercept.
y轴截距。
The turning point (vertex).
顶点（转向点）。

7. Finding the Vertex | 求顶点

The vertex is the maximum or minimum point of the parabola. For the function y = ax² + bx + c, the x-coordinate of the vertex is:

顶点是抛物线的最大值或最小值点。对于函数 y = ax² + bx + c，顶点的 x 坐标为：

x = –b / (2a)

Substitute this value back into the equation to find the y-coordinate.

将这个值代回方程即可求出 y 坐标。

Alternatively, if the quadratic is written in the completed square form y = p(x + q)² + r, the vertex is at (–q, r).

另一种方式，如果二次函数写成配方形式 y = p(x + q)² + r，顶点就是 (–q, r)。

For example, y = x² – 6x + 5. Here a = 1, b = –6, so x = 6/2 = 3. Then y = 9 – 18 + 5 = –4. The vertex is (3, –4).

例如，y = x² – 6x + 5。这里 a = 1，b = –6，所以 x = 6/2 = 3。然后 y = 9 – 18 + 5 = –4。顶点为 (3, –4)。

8. Solving Quadratic Inequalities | 解二次不等式

Quadratic inequalities involve expressions like ax² + bx + c > 0 or ax² + bx + c ≤ 0. Solving them requires an understanding of the parabola’s shape.

二次不等式涉及如 ax² + bx + c > 0 或 ax² + bx + c ≤ 0 的表达式。解它们需要理解抛物线的形状。

Steps:

步骤：

Solve the corresponding quadratic equation ax² + bx + c = 0 to find critical points.
Sketch the parabola or test intervals between the roots.
Determine which regions satisfy the inequality.
解对应的二次方程 ax² + bx + c = 0 找到临界点。
画出抛物线草图或测试根之间的区间。
判断哪些区域满足不等式。

Example: solve x² – 3x + 2 > 0. The roots are x = 1 and x = 2. Since a > 0, the parabola opens upwards, so the graph is above the x-axis when x < 1 or x > 2.

例：解 x² – 3x + 2 > 0。根是 x = 1 和 x = 2。由于 a > 0，抛物线开口向上，因此当 x < 1 或 x > 2 时图像在x轴上方。

The solution is x < 1 or x > 2.

解为 x < 1 或 x > 2。

9. Applications in Real Life | 实际应用

Quadratic equations are used to model many real-world situations, such as projectile motion, area optimisation, and profit maximisation.

二次方程被用于建模许多现实世界的情境，例如抛体运动、面积优化和利润最大化。

For example, the height h(t) of an object thrown upwards can be modelled as h(t) = –5t² + 20t + 2. To find when it hits the ground, set h(t) = 0 and solve for t.

例如，一个物体向上抛出的高度 h(t) 可以建模为 h(t) = –5t² + 20t + 2。要找出它何时落地，令 h(t) = 0 并解出 t。

In geometry, if a rectangle has perimeter 20 cm and area 24 cm², the dimensions can be found by forming the quadratic equation x² – 10x + 24 = 0, which gives x = 4 or x = 6.

在几何中，如果一个矩形的周长为20厘米、面积为24平方厘米，可以通过构造二次方程 x² – 10x + 24 = 0 来求尺寸，解得 x = 4 或 x = 6。

10. Common Mistakes to Avoid | 常见易错点

Students often lose marks on quadratic equations due to small errors. Watch out for the following:

学生在二次方程上经常因为小错误而失分。注意以下几点：

Forgetting to rearrange the equation to the form ax² + bx + c = 0 before solving.
在求解前忘记将方程化为 ax² + bx + c = 0 的形式。
Sign errors when applying the quadratic formula, especially with negative values of b or c.
应用二次公式时的符号错误，特别是 b 或 c 为负值时。
Incorrectly stating that √(a² + b²) = a + b. This is false!
错误地认为 √(a² + b²) = a + b。这是错误的！
Losing the ± sign when taking square roots.
取平方根时丢失 ± 号。
Mixing up the discriminant with the roots.
把判别式和根混淆。

11. Practice Questions | 练习题

Try the following questions to test your understanding. Solve each equation or graph carefully.

尝试以下问题来测试你的理解。仔细解每个方程或画图。

Solve x² + 7x + 10 = 0 using factorisation.
用因式分解法解 x² + 7x + 10 = 0。
Solve 2x² – 5x – 3 = 0 using the quadratic formula.
用二次公式解 2x² – 5x – 3 = 0。
Find the vertex of y = x² – 8x + 12.
求 y = x² – 8x + 12 的顶点。
Determine the nature of roots for 3x² + 4x + 2 = 0.
判断 3x² + 4x + 2 = 0 根的性质。
Sketch the graph of y = –x² + 4x, labelling intercepts and turning point.
画出 y = –x² + 4x 的草图，标注截距和顶点。

Check your answers with a calculator, then review the steps above.

用计算器检查答案，然后回顾上述步骤。

12. Summary | 总结

Quadratic equations are a central part of IGCSE Mathematics. You should be comfortable with all three solving methods: factorisation, completing the square, and the quadratic formula. Understanding the discriminant and the graph of a parabola will help you solve a wide range of problems.

二次方程是IGCSE数学的核心部分。你应该熟练掌握三种解法：因式分解法、配方法以及二次公式。理解判别式和抛物线图像将帮助你解决各种问题。

Remember to always check whether your answer makes sense, especially when applying equations to real-life contexts. Practise regularly, and you will master this topic in no time.

记住，在将方程应用于现实情境时，始终检查你的答案是否合理。定期练习，你一定能很快掌握这个主题。

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Mastering Quadratic Equations | 掌握二次方程