Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations appear frequently in the Edexcel IGCSE Mathematics syllabus. Mastering them is essential for success in both Paper 1 and Paper 2, as questions may range from simple factorisation to complex word problems.

二次方程在 Edexcel IGCSE 数学考纲中频繁出现。掌握它是 Paper 1 和 Paper 2 取得高分的关键，题目可能从简单的因式分解到复杂的应用题不等。

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 ≠ 0. The highest power of the variable is 2. Here, a, b and c are real numbers, and x is the unknown.

二次方程是形如 ax² + bx + c = 0 的方程，其中 a ≠ 0。变量的最高次数是 2。这里 a、b、c 是实数，x 是未知数。

For example, 2x² – 3x + 1 = 0 is a quadratic equation, while x³ – 2x = 0 is not, because the highest power is 3.

例如，2x² – 3x + 1 = 0 是二次方程，而 x³ – 2x = 0 不是，因为最高次数是 3。

2. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method when the quadratic expression can be written as a product of two linear factors. The principle is: if p × q = 0, then p = 0 or q = 0.

当二次表达式可以写成两个一次因式的乘积时，因式分解通常是最快的方法。其原理是：若 p × q = 0，则 p = 0 或 q = 0。

Example: Solve x² – 5x + 6 = 0. Factorise to (x – 2)(x – 3) = 0. Therefore x = 2 or x = 3.

例：解 x² – 5x + 6 = 0。因式分解为 (x – 2)(x – 3) = 0。因此 x = 2 或 x = 3。

For expressions with a leading coefficient not equal to 1, such as 2x² + 5x – 3 = 0, you may need to use the “ac method” or trial and error.

对于首项系数不为 1 的表达式，如 2x² + 5x – 3 = 0，你可能需要使用“ac 法”或尝试法。

Always rearrange the equation so one side is 0 before factorising.
Check your answers by substituting back into the original equation.

分解前务必将方程整理为一边等于 0。
将解代回原方程验证答案。

3. Completing the Square | 配方法

Completing the square rewrites a quadratic expression in the form a(x + p)² + q. This is especially useful for finding turning points and solving equations that do not factorise neatly.

配方法将二次表达式改写为 a(x + p)² + q 的形式。这在寻找顶点以及解不易因式分解的方程时特别有用。

Example: Solve x² + 6x + 2 = 0 by completing the square.

例：用配方法解 x² + 6x + 2 = 0。

Start with x² + 6x = –2. Add (6/2)² = 9 to both sides: (x + 3)² = 7. Then x + 3 = ±√7, so x = –3 ± √7.

先得到 x² + 6x = –2。两边加上 (6/2)² = 9：(x + 3)² = 7。于是 x + 3 = ±√7，所以 x = –3 ± √7。

x² + bx = (x + b/2)² – (b/2)²

This formula is the key to completing the square.

这个公式是配方法的关键。

4. The Quadratic Formula | 二次求根公式

The quadratic formula works for all quadratic equations, including those that cannot be factorised. It is derived from completing the square.

二次求根公式适用于所有二次方程，包括那些无法因式分解的方程。它由配方法推导而来。

x = (–b ± √(b² – 4ac)) / (2a)

For the equation ax² + bx + c = 0, substitute the values of a, b and c carefully.

对于方程 ax² + bx + c = 0，小心代入 a、b、c 的值。

Example: Solve 2x² – 4x – 3 = 0. Here a = 2, b = –4, c = –3.

例：解 2x² – 4x – 3 = 0。这里 a = 2，b = –4，c = –3。

x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = (2 ± √10) / 2

Remember to simplify surds where possible.

记得尽可能化简根式。

5. The Discriminant | 判别式

The discriminant, Δ = b² – 4ac, tells us the nature of the roots without solving the equation.

判别式 Δ = b² – 4ac 无需解方程即可判断根的性质。

If Δ > 0: two distinct real roots
If Δ = 0: one repeated real root
If Δ < 0: no real roots (two complex roots)

若 Δ > 0：两个不相等的实数根
若 Δ = 0：一个重根
若 Δ < 0：没有实数根（两个复数根）

Example: For x² – 4x + 4 = 0, Δ = 16 – 16 = 0, so there is one repeated root x = 2.

例：对于 x² – 4x + 4 = 0，Δ = 16 – 16 = 0，所以有一个重根 x = 2。

In IGCSE questions, you may be asked to find the range of values of k for which an equation has real roots. Set the discriminant ≥ 0 and solve the resulting inequality.

在 IGCSE 题目中，你可能需要求使方程有实数根时 k 的取值范围。令判别式 ≥ 0 并解所得不等式。

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

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

函数 y = ax² + bx + c 的图像是抛物线。若 a > 0，开口向上且有最小值点；若 a < 0，开口向下且有最大值点。

The roots of the equation ax² + bx + c = 0 are the x-intercepts of the graph. The turning point can be found by completing the square or using x = –b/(2a).

方程 ax² + bx + c = 0 的根就是图像与 x 轴的交点。顶点可通过配方法或 x = –b/(2a) 求得。

Δ	x-axis intersections	与 x 轴交点
Δ > 0	two points	两个交点
Δ = 0	one point (touches)	一个交点（相切）
Δ < 0	no intersection	没有交点

Sketching a quadratic graph requires the roots, the y-intercept, and the turning point.

画二次函数草图需要根、y 截距和顶点。

7. Solving Quadratic Equations by Graph | 图像法解二次方程

To solve ax² + bx + c = 0 graphically, draw the graph of y = ax² + bx + c and read the x-coordinates where the curve crosses the x-axis.

要用图像法解 ax² + bx + c = 0，画出 y = ax² + bx + c 的图像，读取曲线与 x 轴交点的 x 坐标。

If the equation is given in the form ax² + bx + c = k, you can either draw the line y = k and find the intersections, or rearrange to make one side equal 0.

如果方程是 ax² + bx + c = k，你可以画直线 y = k 并找交点，或者移项使一边等于 0。

Graphical solutions are often approximations. The IGCSE exam may ask you to estimate roots to one decimal place.

图像解通常是近似值。IGCSE 考试可能要求你估计到一位小数。

8. Solving Word Problems | 解应用题

Quadratic equations can model areas, projectile motion, and number problems. Read the problem carefully and define a variable for the unknown quantity.

二次方程可以建模面积、抛体运动以及数字问题。仔细阅读题目，为未知量定义一个变量。

Example: The area of a rectangle is 50 cm². Its length is 5 cm more than its width. Find the width.

例：一个矩形的面积是 50 cm²。它的长比宽多 5 cm。求宽。

Let the width be x. Then length = x + 5. Area = x(x + 5) = 50. This gives x² + 5x – 50 = 0. Factorising: (x + 10)(x – 5) = 0. Since width cannot be negative, x = 5 cm.

设宽为 x。则长为 x + 5。面积 = x(x + 5) = 50。得到 x² + 5x – 50 = 0。因式分解：(x + 10)(x – 5) = 0。由于宽不能为负，所以 x = 5 cm。

Always reject solutions that make no sense in the context, such as negative lengths or times.

始终舍弃在情境中无意义的解，例如负的长度或时间。

9. Common Mistakes | 常见错误

Students often make the following errors when solving quadratic equations:

学生在解二次方程时常犯以下错误：

Forgetting to set the equation to 0 before factorising.
Misapplying the quadratic formula by mixing up signs.
Confusing (x + 3)² with x² + 9 — they are not equal.
Losing a solution when using square roots: x² = 9 gives x = ±3, not just x = 3.

因式分解前忘记将方程化为等于 0 的形式。
代入求根公式时弄错正负号。
混淆 (x + 3)² 与 x² + 9 —— 它们不相等。
开平方时丢失一个解：x² = 9 应得 x = ±3，而不只是 x = 3。

Always check your solutions by substitution.

务必通过代入检验你的解。

10. Practice Questions | 练习题目

Try these questions to test your understanding:

尝试以下题目来测试你的理解：

Solve x² – 7x + 12 = 0.
Solve 3x² + 2x – 8 = 0 using the quadratic formula.
Find the value of k such that x² + kx + 9 = 0 has a repeated root.
Sketch the graph of y = –x² + 4x – 3, labelling the roots and turning point.

解 x² – 7x + 12 = 0。
用求根公式解 3x² + 2x – 8 = 0。
求 k 的值，使 x² + kx + 9 = 0 有重根。
画出 y = –x² + 4x – 3 的草图，标出根和顶点。

Answers: 1) x = 3 or 4. 2) x = 4/3 or –2. 3) k = ±6. 4) Roots at x = 1 and 3; vertex at (2, 1).

答案：1) x = 3 或 4。2) x = 4/3 或 –2。3) k = ±6。4) 根在 x = 1 和 3；顶点在 (2, 1)。

11. Summary | 总结

You now have a complete toolkit for solving quadratic equations: factorisation, completing the square, the quadratic formula, and graphical methods. The discriminant helps you predict the nature of the roots without solving fully.

你现在已经掌握了解决二次方程的完整工具：因式分解、配方法、求根公式和图像法。判别式帮助你无需完全求解即可预判根的性质。

For the Edexcel IGCSE exam, practise all methods and recognise which is most efficient for each type of equation. Pay attention to standard form, signs, and simplification of surds.

针对 Edexcel IGCSE 考试，练习所有方法，并识别每种方程类型最有效的方法。注意标准形式、符号以及根式的化简。

With consistent practice, quadratic equations will become one of the most reliable scoring areas in your mathematics paper.

通过持续练习，二次方程将成为你数学试卷中最稳定的得分点之一。

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Solving Quadratic Equations | 解二次方程