Solving Quadratic Equations for IGCSE | IGCSE 数学：解二次方程

📚 Solving Quadratic Equations for IGCSE | IGCSE 数学：解二次方程

Quadratic equations are one of the most important algebra topics in IGCSE Mathematics. You will meet them in factorisation, in graphs and in real-life problems such as area, projectile motion and profit modelling.

二次方程是 IGCSE 数学代数学中最重要的课题之一。你会在因式分解、图像以及面积、抛体运动和利润建模等实际情境中遇到它们。

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

e highest power of the unknown is 2. For example, x² + 3x − 4 = 0 is a quadratic equation.

二次方程是未知数最高次数为 2 的方程。例如，x² + 3x − 4 = 0 就是一个二次方程。

equation in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

二次方程必须能写成标准形式 ax² + bx + c = 0，其中 a、b、c 为常数且 a ≠ 0。

ax² + bx + c = 0, a ≠ 0

0, the equation becomes linear rather than quadratic, so the condition a ≠ 0 is essential.

如果 a = 0，方程就变成一次方程而不是二次方程，因此条件 a ≠ 0 至关重要。

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2. Standard Form and Coefficients | 标准式与系数

Always rearrange the equation to make one side equal to zero before choosing a solution method.

在选择解法之前，一定要先把方程整理为一边等于 0 的形式。

Example: 3x² − 5 = 2x + 7 becomes 3x² − 2x − 12 = 0. Here a = 3, b = −2 and c = −12.

示例：3x² − 5 = 2x + 7 移项后得到 3x² − 2x − 12 = 0。此时 a = 3，b = −2，c = −12。

Equation	a	b	c
3x² − 2x − 12 = 0	3	−2	−12

Getting the signs of a, b and c right is very important for the quadratic formula.

正确识别 a、b、c 的正负号对二次公式的计算非常重要。

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3. The Zero Product Principle | 零乘积原理

Factorisation relies on the zero product principle.

因式分解法依赖于零乘积原理。

If pq = 0, then p = 0 or q = 0, or both.

如果 pq = 0，则 p = 0 或 q = 0，或两者都成立。

actor a quadratic to (x + p)(x + q) = 0, each bracket can be set equal to zero separately.

这意味着一旦你把二次方程分解为 (x + p)(x + q) = 0，就可以分别令每个括号等于零。

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4. Solving by Factorisation | 因式分解法

To solve by factorisation, first write the equation in standard form, then find two numbers whose product equals ac and whose sum equals b.

因式分解法的步骤为：先写成标准形式，然后找出两个数，使它们的乘积等于 ac，和等于 b。

Example: Solve x² + 5x + 6 = 0.

示例：解方程 x² + 5x + 6 = 0。

The two numbers that multiply to 6 and add to 5 are 2 and 3.

乘积为 6、和为 5 的两个数是 2 和 3。

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

Therefore 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。

Always expand your factors mentally to check that they give the original left-hand side.

始终在脑海中展开因式，检查它们是否能还原为等式左边原来的表达式。

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5. Special Case: Difference of Two Squares | 特殊情形：平方差

A common quadratic pattern is the difference of two squares.

一个常见的二次方程模式是平方差。

x² − a² = (x − a)(x + a)

Example: Solve x² − 16 = 0.

示例：解方程 x² − 16 = 0。

Factorise to (x − 4)(x + 4) = 0, so x = 4 or x = −4, often written as x = ±4.

将其分解为 (x − 4)(x + 4) = 0，所以 x = 4 或 x = −4，通常写作 x = ±4。

Do not write x = 4 only when solving x² = 16; the negative root is also valid.

在解 x² = 16 时不要只写 x = 4；负根同样成立。

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6. Completing the Square | 配方法

Completing the square is useful when the quadratic does not factorise easily, and it also helps with graphing.

当二次方程不易因式分解时，配方法非常有用，同时它也有助于描绘图像。

The key identity is:

关键恒等式为：

x² + bx = (x + b/2)² − (b/2)²

Example: Solve x² + 6x − 7 = 0.

示例：解方程 x² + 6x − 7 = 0。

Move the constant: x² + 6x = 7. Add (6/2)² = 9 to both sides: x² + 6x + 9 = 16.

先移常数项：x² + 6x = 7。两边同时加上 (6/2)² = 9：x² + 6x + 9 = 16。

(x + 3)² = 16

Take square roots: x + 3 = ±4, so x = 1 or x = −7.

开平方得：x + 3 = ±4，所以 x = 1 或 x = −7。

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7. The Quadratic Formula | 二次公式

The quadratic formula works for any quadratic equation, even when factorisation is difficult or impossible using integers.

二次公式适用于任何二次方程，即使很难或无法用整数进行因式分解时也有效。

x = (−b ± √b² − 4ac) ÷ 2a

Example: Solve 2x² − 3x − 5 = 0.

示例：解方程 2x² − 3x − 5 = 0。

Here a = 2, b = −3 and c = −5. The discriminant is (−3)² − 4 × 2 × (−5) = 9 + 40 = 49.

这里 a = 2，b = −3，c = −5。判别式为 (−3)² − 4 × 2 × (−5) = 9 + 40 = 49。

x = (3 ± √49) ÷ 4 = (3 ± 7) ÷ 4

So x = 10/4 = 2.5 or x = −4/4 = −1.

所以 x = 10/4 = 2.5 或 x = −4/4 = −1。

Always simplify the final answers and write them exactly unless the question asks for decimals.

始终化简最终答案；除非题目要求写成小数，否则应写出精确值。

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8. Discriminant and Number of Roots | 判别式与根的个数

The expression b² − 4ac inside the quadratic formula is called the discriminant.

二次公式中的 b² − 4ac 称为判别式。

D = b² − 4ac

Discriminant	Number of real roots
D > 0	Two distinct real roots
D = 0	One repeated real root
D < 0	No real roots

If a question asks how many roots a quadratic has, you do not need to solve it fully; just calculate D.

如果题目问二次方程有多少个根，你不需要完全解出方程；只需计算 D。

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9. Graphical Interpretation | 图像解释

abola. The roots of the equation are the x-coordinates where the graph crosses the x-axis.

y = ax² + bx + c 的图像是一条抛物线。方程的根就是图像与 x 轴交点的 x 坐标。

axis twice. If D = 0, it touches the x-axis once. If D < 0, it does not cut the x-axis.

如果 D > 0，抛物线与 x 轴有两个交点；如果 D = 0，抛物线与 x 轴相切于一点；如果 D < 0，抛物线与 x 轴没有交点。

When a > 0 the parabola opens upwards, and when a < 0 it opens downwards.

当 a > 0 时抛物线开口向上，当 a < 0 时抛物线开口向下。

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

• Forgetting to set the equation equal to zero before factorising. Always rearrange first.

  因式分解前忘记让方程等于零。务必先移项整理。

• Using the wrong sign for b in the quadratic formula. Write a, b and c clearly first.

  在二次公式中把 b 的符号写错。请先清楚地写出 a、b、c。

• Solving x² = 25 incorrectly as x = 5 only. Remember the negative root: x = ±5.

  解 x² = 25 时错误地只写 x = 5。请记住还有负根：x = ±5。

• sing the root x = 0. Never divide by an unknown expression unless you know it is non-zero.

  两边同时除以 x 而丢失根 x = 0。除非确定表达式不为零，否则不要除以未知表达式。

To succeed in IGCSE quadratic equations, practise identifying a, b and c quickly, choose factorisation first when possible, and always check your roots by substitution.

要在 IGCSE 二次方程中取得好成绩，请练习快速识别 a、b、c，优先使用因式分解法，并始终用代入法检验你的根。

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