IGCSE Mathematics: Master Quadratic Equations | IGCSE数学：掌握一元二次方程

📚 IGCSE Mathematics: Master Quadratic Equations | IGCSE数学：掌握一元二次方程

A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in algebra, graphs, geometry, and even calculus at higher levels. Mastering quadratic equations will boost your confidence and your grade.

一元二次方程是 IGCSE 数学中最重要的考点之一。它出现在代数、图像、几何乃至更高级的微积分内容中。掌握好一元二次方程，将直接提升你的信心和成绩。

1. What Is a Quadratic Equation? | 什么是一元二次方程？

A quadratic equation is an 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 unknown is 2.

一元二次方程是可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 是常数，且 a ≠ 0。未知数的最高次数是 2。

x = (-b ± √(b² – 4ac)) / (2a)

For example, 3x² – 5x + 2 = 0 is a quadratic equation. Here a = 3, b = -5, c = 2. Note that a cannot be zero; otherwise the equation becomes linear.

例如，3x² – 5x + 2 = 0 就是一元二次方程。其中 a = 3，b = -5，c = 2。注意 a 不能为 0，否则方程就退化成一次方程了。

The solutions of a quadratic equation are also called roots or zeros. A quadratic equation may have two distinct roots, one repeated root, or no real roots.

一元二次方程的解也叫作根或零点。一个二次方程可能有两个不同的实根、一个重根，或者没有实根。

2. Solving by Factorisation | 用因式分解法求解

Factorisation is the fastest method when the quadratic expression can be written as a product of two linear factors. To solve x² + 5x + 6 = 0, we look for two numbers that multiply to 6 and add to 5. These numbers are 2 and 3.

当二次表达式可以写成两个一次因式的乘积时，因式分解是最快捷的方法。要解 x² + 5x + 6 = 0，我们需要找到两个数，它们相乘等于 6，相加等于 5。这两个数就是 2 和 3。

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

Using the zero product property, if the product of two factors is zero, then at least one factor must be zero. So x + 2 = 0 or x + 3 = 0, giving x = -2 or x = -3.

根据零积性质，如果两个因式的乘积为零，那么至少有一个因式为零。所以 x + 2 = 0 或 x + 3 = 0，解得 x = -2 或 x = -3。

When the coefficient of x² is not 1, factorisation requires more care. For example, solve 2x² – 7x + 3 = 0. We need two factors of the form (px + q)(rx + s) such that pr = 2 and qs = 3, and the cross term sums to -7.

当 x² 的系数不是 1 时，因式分解需要更细心。例如解 2x² – 7x + 3 = 0。我们要找到形如 (px + q)(rx + s) 的两个因式，使得 pr = 2，qs = 3，并且交叉项之和为 -7。

(2x – 1)(x – 3) = 0 ⇒ x = ½ or x = 3

Always expand your factors to check your answer. This prevents careless mistakes and takes only a few seconds.

务必展开因式来检查答案。这一步能避免粗心错误，而且只需要几秒钟。

3. Solving by Completing the Square | 用配方法求解

Completing the square rewrites x² + bx + c as (x + p)² + q. This technique works for every quadratic equation, even when factorisation is difficult.

配方法将 x² + bx + c 改写为 (x + p)² + q 的形式。这种方法适用于所有二次方程，即使是难以因式分解的题目也能处理。

If we have x² + 6x + 2 = 0, we take half of 6, which is 3, and write (x + 3)². Since (x + 3)² = x² + 6x + 9, we must subtract 9 to keep the expression unchanged.

对于 x² + 6x + 2 = 0，我们取 6 的一半，即 3，写成 (x + 3)²。因为 (x + 3)² = x² + 6x + 9，所以需要减去 9 才能保持表达式不变。

x² + 6x + 2 = (x + 3)² – 9 + 2 = (x + 3)² – 7

Set the completed-square expression equal to zero: (x + 3)² = 7. Then x + 3 = ±√7, so x = -3 ± √7. This gives two exact answers, which is especially useful in non-calculator papers.

令配方后的表达式等于零：(x + 3)² = 7。于是 x + 3 = ±√7，所以 x = -3 ± √7。这样得到两个精确解，在不允许使用计算器的考试中尤其好用。

The general result is: x² + bx = (x + b/2)² – (b/2)². Make sure you square the half-coefficient correctly.

一般公式是：x² + bx = (x + b/2)² – (b/2)²。注意要把一半系数正确地平方。

4. The Quadratic Formula | 求根公式

The quadratic formula can solve any quadratic equation directly. It is derived from completing the square and is especially useful when factors are not obvious. The formula is stated below.

求根公式可以直接求解任何二次方程。它由配方法推导而来，当因式不容易看出来时特别有用。公式如下：

x = (-b ± √(b² – 4ac)) / (2a), where a ≠ 0

To use the formula, first rewrite the equation in the standard form ax² + bx + c = 0. Then substitute the values of a, b, and c carefully.

使用公式前，先把方程写成标准形式 ax² + bx + c = 0。然后仔细代入 a、b、c 的值。

For example, solve 2x² + 3x – 5 = 0. Here a = 2, b = 3, c = -5. Substitute into the formula:

例如，解 2x² + 3x – 5 = 0。这里 a = 2，b = 3，c = -5。代入公式得：

x = (-3 ± √(3² – 4 × 2 × (-5))) / (2 × 2) = (-3 ± √(9 + 40)) / 4 = (-3 ± 7) / 4

So x = (-3 + 7)/4 = 1, or x = (-3 – 7)/4 = -2.5. Always check that your final roots satisfy the original equation.

因此 x = (-3 + 7)/4 = 1，或者 x = (-3 – 7)/4 = -2.5。最后务必检验所求的根是否满足原方程。

5. The Discriminant | 判别式

The expression b² – 4ac inside the square root of the quadratic formula is called the discriminant. It tells us the nature of the roots without actually solving the equation.

求根公式根号内的表达式 b² – 4ac 称为判别式。它无需解方程就能告诉我们根的性质。

Discriminant 判别式	Nature of Roots 根的性质
b² – 4ac > 0	Two distinct real roots 两个不同的实根
b² – 4ac = 0	One repeated real root 一个重根（两个相等的实根）
b² – 4ac < 0	No real roots 没有实根

For example, in x² – 4x + 4 = 0, the discriminant is (-4)² – 4 × 1 × 4 = 16 – 16 = 0. Therefore the equation has exactly one repeated root: x = 2.

例如，在 x² – 4x + 4 = 0 中，判别式为 (-4)² – 4 × 1 × 4 = 16 – 16 = 0。因此方程只有一个重根：x = 2。

The discriminant also tells us whether the graph of y = ax² + bx + c crosses the x-axis, touches the x-axis, or never meets the x-axis.

判别式还告诉我们 y = ax² + bx + c 的图像与 x 轴相交、相切，还是完全没有交点。

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

The graph of y = ax² + bx + c is a parabola. If a > 0, it opens upwards like a cup. If a < 0, it opens downwards like a cap.

y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，抛物线开口向上；如果 a < 0，抛物线开口向下。

The roots of the equation ax² + bx + c = 0 are the x-coordinates where the graph crosses or touches the x-axis.

方程 ax² + bx + c = 0 的根，就是图像与 x 轴相交或相切处的 x 坐标。
The axis of symmetry is the vertical line x = -b/(2a). It passes through the vertex of the parabola.

对称轴是竖直线 x = -b/(2a)，它经过抛物线的顶点。
The vertex (turning point) lies on the axis of symmetry. Its x-coordinate is -b/(2a); substitute this back into the equation to find the y-coordinate.

顶点（转向点）位于对称轴上，其 x 坐标为 -b/(2a)；把该 x 值代回方程即可求出 y 坐标。

For example, y = x² – 2x – 3 has roots x = 3 and x = -1, because x² – 2x – 3 = (x – 3)(x + 1). The axis of symmetry is x = (3 + (-1))/2 = 1.

例如，y = x² – 2x – 3 的根是 x = 3 和 x = -1，因为 x² – 2x – 3 = (x – 3)(x + 1)。对称轴为 x = (3 + (-1))/2 = 1。

Sketching a quadratic graph requires three key features: the roots, the y-intercept, and the vertex. The y-intercept is simply the value of c.

画二次函数草图需要三个关键信息：根、y 轴截距和顶点。y 轴截距就是常数项 c 的值。

7. Solving Quadratic Inequalities | 解一元二次不等式

IGCSE Extended candidates may be asked to solve quadratic inequalities such as x² – 5x + 6 < 0. The first step is to solve the corresponding quadratic equation to find critical values.

IGCSE 拓展课程可能会要求解二次不等式，例如 x² – 5x + 6 < 0。第一步是解对应的二次方程，找到临界值。

x² – 5x + 6 = (x – 2)(x – 3) = 0 gives x = 2 or x = 3. These values divide the number line into three intervals: x < 2, 2 < x < 3, and x > 3.

x² – 5x + 6 = (x – 2)(x – 3) = 0，解得 x = 2 或 x = 3。这两个值把数轴分成三个区间：x < 2，2 < x < 3，x > 3。

Test a point in each interval. For x = 0, the expression is 6 > 0. For x = 2.5, the expression is -0.25 < 0. For x = 4, the expression is 2 > 0. Therefore the solution is 2 < x < 3.

在每个区间取一点检验。x = 0 时，表达式为 6 > 0；x = 2.5 时，表达式为 -0.25 < 0；x = 4 时，表达式为 2 > 0。因此解集为 2 < x < 3。

Remember the graphical interpretation: the parabola is below the x-axis between the two roots when a > 0. For x² – 5x + 6 > 0, the solution would be x < 2 or x > 3.

记住图像的直观理解：当 a > 0 时，抛物线在两个根之间位于 x 轴下方。对于 x² – 5x + 6 > 0，解集则为 x < 2 或 x > 3。

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 sum of the roots is α + β = -b/a, and the product is αβ = c/a. These relationships are very useful for checking answers and for constructing equations.

对于以 α 和 β 为根的二次方程 ax² + bx + c = 0，两根之和为 α + β = -b/a，两根之积为 αβ = c/a。这些关系常用于检验答案和构造方程。

For example, in 2x² – 8x + 6 = 0, the sum of the roots is -(-8)/2 = 4, and the product is 6/2 = 3. The actual roots are 1 and 3, and indeed 1 + 3 = 4, 1 × 3 = 3.

例如，在 2x² – 8x + 6 = 0 中，两根之和为 -(-8)/2 = 4，两根之积为 6/2 = 3。实际根是 1 和 3，确实有 1 + 3 = 4，1 × 3 = 3。

If you are given the sum S and product P of the roots, you can form the equation x² – Sx + P = 0. This is a common exam question.

如果已知两根之和 S 与两根之积 P，就可以构造方程 x² – Sx + P = 0。这是常见考题。

9. Word Problems and Applications | 应用题与实际应用

Quadratic equations appear in real-world problems such as projectile motion, area calculations, and number puzzles. The key is to translate the English sentence into a mathematical equation.

一元二次方程广泛出现在实际问题中，例如抛体运动、面积计算和数字谜题。关键在于把题目语言转化为数学方程。

Example: A rectangle has length 4 cm longer than its width. Its area is 45 cm². Let the width be x cm. Then the length is x + 4 cm, and the area equation is x(x + 4) = 45.

例：一个长方形的长比宽多 4 cm，面积为 45 cm²。设宽为 x cm，则长为 x + 4 cm，面积方程为 x(x + 4) = 45。

x² + 4x – 45 = 0 ⇒ (x + 9)(x – 5) = 0 ⇒ x = -9 or x = 5

Since width cannot be negative, x = 5 cm. The length is 9 cm. Always reject solutions that do not make sense in the context of the problem.

因为宽度不能为负数，所以 x = 5 cm，长为 9 cm。务必舍去在题目情境中不合理的解。

For projectile motion, the height h of an object at time t is often modelled by h = -5t² + vt + h₀. Setting h = 0 gives the time when the object hits the ground.

在抛体运动中，物体在 t 时刻的高度常表示为 h = -5t² + vt + h₀。令 h = 0 即可求出物体落地的时间。

10. Common Pitfalls and Exam Tips | 常见错误与应试技巧

Many students lose marks in quadratic equation questions due to avoidable errors. Here are the most common mistakes and how to avoid them.

很多学生在二次方程题目上丢分，原因是本可避免的错误。以下是最常见的错误以及避免方法。

Forgetting to set the equation to zero before factorising or using the formula. Always rewrite as ax² + bx + c = 0 first.

在因式分解或使用公式前忘记把方程化为零。一定要先把方程写成 ax² + bx + c = 0 的形式。
Losing the negative sign in front of coefficients. Write brackets around negative values when substituting into the quadratic formula.

弄丢负号。代入求根公式时，负值要用括号括起来。
Confusing (x + p)² with x² + p². Remember (x + p)² = x² + 2px + p².

混淆 (x + p)² 与 x² + p²。记住 (x + p)² = x² + 2px + p²。
Not checking roots by substitution. One minute of checking can save two marks.

不代入原方程检验根。花一分钟检验，可以挽回两分。

Read the question carefully: does it ask for exact values, or values to one decimal place? Use the required form in your final answer.

仔细审题：题目要求精确值，还是保留一位小数？最终答案要按要求的形式书写。

11. Practice Questions | 练习题

Try these questions on your own before looking at the solutions. They cover factorisation, completing the square, the quadratic formula, and the discriminant.

请先独立尝试以下题目，再看解答。它们涵盖因式分解、配方法、求根公式和判别式。

Solve x² – 7x + 10 = 0 by factorisation.

用因式分解法解 x² – 7x + 10 = 0。
Solve x² + 4x – 1 = 0 by completing the square, giving your answers in exact form.

用配方法解 x² + 4x – 1 = 0，答案用精确形式表示。
Use the quadratic formula to solve 3x² + 5x – 2 = 0.

用求根公式解 3x² + 5x – 2 = 0。
Find the discriminant of 2x² – 4x + 3 = 0 and state the number of real roots.

求 2x² – 4x + 3 = 0 的判别式，并判断实根的个数。
A rectangle’s length is 3 times its width, and its area is 48 cm². Find the width.

一个长方形的长是宽的 3 倍，面积为 48 cm²。求宽。

Answers: 1. x = 2 or x = 5. 2. x = -2 ± √5. 3. x = ½ or x = -2. 4. Discriminant = (-4)² – 4 × 2 × 3 = -8 < 0, so no real roots. 5. Width = 4 cm.

参考答案：1. x = 2 或 x = 5。2. x = -2 ± √5。3. x = ½ 或 x = -2。4. 判别式 = (-4)² – 4 × 2 × 3 = -8 < 0，所以没有实根。5. 宽 = 4 cm。

12. Summary | 要点总结

Quadratic equations can be solved by factorisation, completing the square, or the quadratic formula. Factorisation is fastest but not always possible; completing the square and the formula always work.

一元二次方程可以用因式分解、配方或求根公式求解。因式分解最快但并非总能实现；配方法和求根公式则始终适用。

Use the discriminant b² – 4ac to determine the nature of the roots before solving. Relate the roots to the graph of y = ax² + bx + c for a deeper understanding.

求解前先用判别式 b² – 4ac 判断根的性质。把根与 y = ax² + bx + c 的图像联系起来，能帮助你更深入地理解。

Practice every method until it becomes automatic. In the IGCSE exam, quadratic equations are guaranteed to appear, so mastering them is one of the best investments of your revision time.

反复练习每一种方法，直到熟练掌握。在 IGCSE 考试中，二次方程几乎必考，因此掌握它是复习时间中回报率最高的投入之一。

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IGCSE Mathematics: Master Quadratic Equations | IGCSE数学：掌握一元二次方程