Mastering Quadratic Equations for IGCSE Mathematics | IGCSE 数学：掌握二次方程

📚 Mastering Quadratic Equations for IGCSE Mathematics | IGCSE 数学：掌握二次方程

Quadratic equations are one of the most important topics in the IGCSE Mathematics syllabus. They appear in algebra questions, graph-sketching problems, and real-world applications such as projectile motion, area calculations, and optimisation. A strong understanding of quadratic equations will help you solve many different types of exam questions confidently.

二次方程是 IGCSE 数学大纲中最重要的主题之一。它们出现在代数题、画图题以及现实情境应用题中，例如抛体运动、面积计算和最优化问题。扎实掌握二次方程将帮助你自信地解答各类考试题目。

1. Recognising Quadratic Equations | 认识二次方程

A quadratic equation is any equation that can be written in 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 is why it is called quadratic.

二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 是常数，且 a ≠ 0。变量 x 的最高次数是 2，所以称为二次方程。

For example, x² − 5x + 6 = 0 is quadratic, but x³ + 2x = 0 is not quadratic because the highest power is 3.

例如，x² − 5x + 6 = 0 是二次方程，但 x³ + 2x = 0 不是二次方程，因为最高次数是 3。

ax² + bx + c = 0 is called the standard form.
ax² + bx + c = 0 称为标准形式。
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。

2. Standard Form and Coefficients | 标准形式与系数

Before applying any method to solve a quadratic equation, you should rewrite the equation into standard form. This means collecting all terms on one side so that the other side equals zero.

在应用任何解二次方程的方法之前，应先把方程改写为标准形式。这意味着把所有项移到一边，使另一边等于零。

The numbers a, b and c are called the coefficients. In the equation 2x² − 3x + 1 = 0, we have a = 2, b = −3 and c = 1.

数字 a、b、c 称为系数。在方程 2x² − 3x + 1 = 0 中，a = 2，b = −3，c = 1。

Rewriting is essential because factorisation and the quadratic formula both require the equation to be in the form ax² + bx + c = 0.

改写非常关键，因为因式分解法和二次公式法都要求方程写成 ax² + bx + c = 0 的形式。

3. Solving by Factorising | 因式分解法求解

When a quadratic expression can be factorised easily, factorising is often the fastest method. The idea is to write ax² + bx + c as a product of two brackets, then use the zero product property.

当二次式可以容易地因式分解时，因式分解法通常是最快的方法。其思路是把 ax² + bx + c 写成两个括号的乘积，然后利用零乘积性质。

For example, to solve x² − 5x + 6 = 0, factorise to get (x − 2)(x − 3) = 0. Then set each bracket equal to zero: x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3.

例如，要解 x² − 5x + 6 = 0，先因式分解得到 (x − 2)(x − 3) = 0。然后令每个括号分别等于零：x − 2 = 0 或 x − 3 = 0，所以 x = 2 或 x = 3。

Always check that the equation equals zero before factorising.
因式分解前一定要确认方程一边等于零。
Expand the brackets mentally to verify your factorisation.
展开括号进行心算检查因式分解是否正确。

4. The Quadratic Formula | 二次公式求根

The quadratic formula can solve any quadratic equation, even when factorisation is difficult or impossible. For a quadratic equation in standard form, the solutions are given by:

二次公式可以解任何二次方程，即使因式分解困难或不可能。对于标准形式的二次方程，其解由以下公式给出：

x = (−b ± √(b² − 4ac)) ÷ 2a

Where a, b and c are the coefficients from ax² + bx + c = 0.

其中 a、b、c 是来自 ax² + bx + c = 0 的系数。

For example, in 2x² + 3x − 2 = 0, a = 2, b = 3 and c = −2. Substitution gives x = (−3 ± √(3² − 4 × 2 × (−2))) ÷ (2 × 2) = (−3 ± √25) ÷ 4 = (−3 ± 5) ÷ 4. The solutions are x = ½ and x = −2.

例如，在 2x² + 3x − 2 = 0 中，a = 2，b = 3，c = −2。代入公式得到 x = (−3 ± √(3² − 4 × 2 × (−2))) ÷ (2 × 2) = (−3 ± √25) ÷ 4 = (−3 ± 5) ÷ 4。解为 x = ½ 和 x = −2。

5. Completing the Square | 配方法

Completing the square is a method that rewrites a quadratic expression in the form a(x + p)² + q. This form is useful for finding the turning point of a graph and for solving equations that cannot be factorised easily.

配方法是将二次式改写为 a(x + p)² + q 形式的方法。这种形式有助于求图像顶点，也用于求解不易因式分解的方程。

To complete the square for x² + 6x, take half of the coefficient of x, which is 3, and square it to get 9. Therefore x² + 6x = (x + 3)² − 9.

对 x² + 6x 进行配方，取 x 的系数一半，即 3，再平方得到 9。因此 x² + 6x = (x + 3)² − 9。

For example, x² + 6x + 2 = 0 becomes (x + 3)² − 7 = 0, so (x + 3)² = 7 and x = −3 ± √7.

例如，x² + 6x + 2 = 0 可化为 (x + 3)² − 7 = 0，所以 (x + 3)² = 7，x = −3 ± √7。

6. Discriminant and Nature of Roots | 判别式与根的性质

The expression b² − 4ac inside the quadratic formula is called the discriminant. It tells you about the nature of the roots without solving the equation completely.

二次公式中的 b² − 4ac 称为判别式。它可以在不完整解方程的情况下判断根的性质。

If b² − 4ac > 0, the equation has two distinct real roots.
如果 b² − 4ac > 0，方程有两个不同的实数根。
If b² − 4ac = 0, the equation has one repeated real root, meaning the graph touches the x-axis.
如果 b² − 4ac = 0，方程有一个重根，即图像与 x 轴相切。
If b² − 4ac < 0, the equation has no real roots, meaning the graph does not cross the x-axis.
如果 b² − 4ac < 0，方程没有实数根，即图像不与 x 轴相交。

Exam questions often ask you to find the number of roots or to determine a condition on a coefficient for a certain number of roots.

考试题经常要求你判断根的个数，或根据根的个数确定某个系数满足的条件。

7. Graphs of Quadratic Functions | 二次函数图像

The graph of y = ax² + bx + c is a parabola. If a > 0, the parabola opens upwards and has a minimum point. If a < 0, the parabola opens downwards and has a maximum point.

y = ax² + bx + c 的图像是抛物线。如果 a > 0，抛物线开口向上，有最低点。如果 a < 0，抛物线开口向下，有最高点。

The roots of the equation ax² + bx + c = 0 are the x-intercepts of the graph. The y-intercept is the constant c, because when x = 0, y = c.

方程 ax² + bx + c = 0 的根是图像与 x 轴的交点。y 轴截距是常数 c，因为当 x = 0 时 y = c。

Sketching a quadratic graph usually requires identifying the turning point, the y-intercept, and the x-intercepts if they exist.

画二次函数图像通常需要确定顶点、y 轴截距，以及 x 轴截距（如果存在）。

8. Turning Point and Axis of Symmetry | 顶点与对称轴

The turning point is the highest or lowest point on a quadratic graph. The vertical line passing through the turning point is the axis of symmetry.

顶点是二次函数图像上的最高点或最低点。经过顶点的竖直线是对称轴。

If the quadratic is written as y = a(x + p)² + q, the turning point is (−p, q). For example, y = (x − 3)² + 2 has a minimum point at (3, 2).

如果二次函数写成 y = a(x + p)² + q 的形式，顶点就是 (−p, q)。例如，y = (x − 3)² + 2 的最小值点为 (3, 2)。

The axis of symmetry can also be found using x = −b ÷ (2a) when the equation is in standard form. This formula gives the x-coordinate of the turning point.

当方程是标准形式时，对称轴也可以用 x = −b ÷ (2a) 求出。这个公式给出顶点的 x 坐标。

9. Applications and Word Problems | 应用与文字题

Quadratic equations often appear in real-life problems. For example, the area of a rectangle can lead to a quadratic equation if the sides involve an unknown length.

二次方程经常出现在现实问题中。例如，如果矩形的边包含未知长度，其面积就可能引出一个二次方程。

Another common example is projectile motion. The height of an object thrown upwards can be modelled by a quadratic expression in time t, and solving the equation tells you when the object reaches a certain height.

另一个常见例子是抛体运动。向上抛出的物体的高度可以用关于时间 t 的二次式来建模，解方程可以告诉你物体何时达到某一高度。

Always define your variable clearly, form the equation carefully, solve it, and then check whether both answers are valid in the context of the problem.

一定要清楚地定义变量，认真建立方程，解出结果，然后检查两个答案在题目情境中是否都合理。

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

Many students lose marks because they forget to rearrange the equation into standard form before using the quadratic formula.

许多学生丢分是因为在使用二次公式前忘记把方程整理成标准形式。

Another common error is making sign mistakes when substituting negative values into the formula. Always use brackets around negative numbers to avoid errors.

另一个常见错误是在公式中代入负值时出现符号错误。一定要在负数外加括号以避免出错。

Always write down a, b and c clearly before substituting.
代入前先清楚地写出 a、b、c。
Check your solutions by substituting them back into the original equation.
把解代回原方程进行检验。
If an answer is impossible in context, such as a negative length, reject it and explain why.
如果答案在情境中不合理，例如长度为负数，应舍去并说明理由。

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Mastering Quadratic Equations for IGCSE Mathematics | IGCSE 数学：掌握二次方程