Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations appear throughout the IGCSE Mathematics syllabus. Mastering the techniques to solve them is essential for exams and for higher-level study. This guide explains the standard methods, the discriminant, and common pitfalls, with clear worked examples.

二次方程贯穿IGCSE数学大纲。掌握其解法对考试和后续学习都至关重要。本指南将解释标准方法、判别式及常见误区，并配有清晰的例题。

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

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a is not equal to 0. The highest power of the variable is 2, which gives the equation its name.

一元二次方程是指可以写成 ax² + bx + c = 0 形式的方程，其中 a 不等于0。变量的最高次数是2，因此得名。

In this form, a, b, and c are constants, and x is the unknown. For example, x² – 5x + 6 = 0 is a quadratic equation with a = 1, b = -5, c = 6.

在这种形式中，a、b、c 是常数，x 是未知数。例如，x² – 5x + 6 = 0 就是一个 a = 1, b = -5, c = 6 的一元二次方程。

2. The Standard Form and Key Terms | 标准形式与关键术语

The standard form ax² + bx + c = 0 allows us to identify coefficients clearly. The term ax² is the quadratic term, bx is the linear term, and c is the constant term.

标准形式 ax² + bx + c = 0 使我们能清晰地识别各项系数。ax² 称为二次项，bx 称为一次项，c 称为常数项。

When a quadratic equation is not in standard form, we must rearrange it before solving. For instance, 2x² = 8x – 6 must be rewritten as 2x² – 8x + 6 = 0.

当二次方程不是标准形式时，需要先整理成标准形式。例如，2x² = 8x – 6 必须改写为 2x² – 8x + 6 = 0。

3. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method. You look for two numbers that multiply to give a × c and add to give b (when a = 1). Then you rewrite and solve using the zero factor property.

因式分解通常是最快的方法。需要找到两个数，使它们的乘积等于 a × c，和等于 b（当 a = 1 时）。然后利用零因子性质求解。

Example: Solve x² – 5x + 6 = 0. The numbers -2 and -3 multiply to 6 and add to -5. So (x – 2)(x – 3) = 0. Therefore x = 2 or x = 3.

例：解 x² – 5x + 6 = 0。数 -2 和 -3 的乘积为6，和为-5。所以 (x – 2)(x – 3) = 0，因此 x = 2 或 x = 3。

When a is not 1, you may need to use the method of grouping. For example, 2x² + 7x + 3 = 0. Multiply 2 and 3 to get 6. Find two numbers that multiply to 6 and add to 7: that is 6 and 1. Rewrite as 2x² + 6x + x + 3 = 0, then factor by grouping.

当 a 不是1时，可能需要使用分组分解法。例如，2x² + 7x + 3 = 0。2与3相乘得到6。找到两个数相乘为6且和为7，即6和1。改写为 2x² + 6x + x + 3 = 0，再分组因式分解。

2x(x + 3) + 1(x + 3) = 0 → (2x + 1)(x + 3) = 0

Thus x = -1/2 or x = -3.

因此 x = -1/2 或 x = -3。

4. Solving by Completing the Square | 配方法

Completing the square transforms an equation into the form (x + p)² = q. This method always works and gives exactly the same solutions as the formula.

配方法将方程变形为 (x + p)² = q 的形式。这种方法总是有效，所得解与公式法完全一致。

Example: Solve x² + 6x + 2 = 0. First, move the constant: x² + 6x = -2. Then add the square of half the coefficient of x: half of 6 is 3, and 3² = 9. Add 9 to both sides.

例：解 x² + 6x + 2 = 0。先将常数项移项：x² + 6x = -2。然后加上一次项系数一半的平方：6的一半是3，3² = 9。两边同时加9。

x² + 6x + 9 = 7 → (x + 3)² = 7

Take the square root of both sides: x + 3 = ±√7. Hence x = -3 ± √7.

两边开平方：x + 3 = ±√7，因此 x = -3 ± √7。

For equations where the coefficient of x² is not 1, first divide every term by that coefficient.

当x²的系数不为1时，先将每一项除以该系数。

5. Solving by the Quadratic Formula | 公式法

The quadratic formula gives solutions for any quadratic equation ax² + bx + c = 0. It is derived from completing the square and is especially useful when factorisation is difficult.

二次公式适用于任意一元二次方程 ax² + bx + c = 0。它由配方法推导而来，在因式分解困难时尤其有用。

x = (−b ± √(b² − 4ac)) / (2a)

Here the symbol ± means there are two possible values: one with a plus sign and one with a minus sign.

这里的 ± 表示两个可能的值：一个取加号，一个取减号。

Example: Solve 2x² – 3x – 5 = 0. Here a = 2, b = -3, c = -5. Substitute into the formula.

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

x = (−(−3) ± √((−3)² − 4×2×(−5))) / (2×2)

x = (3 ± √(9 + 40)) / 4 = (3 ± √49) / 4 = (3 ± 7) / 4

So x = (3 + 7)/4 = 10/4 = 2.5 or x = (3 − 7)/4 = −4/4 = −1.

所以 x = (3 + 7)/4 = 10/4 = 2.5 或 x = (3 − 7)/4 = −4/4 = −1。

6. The Discriminant | 判别式

The expression b² − 4ac inside the quadratic formula is called the discriminant, often denoted by Δ (the Greek letter delta). It tells us how many real roots an equation has.

二次公式中的 b² − 4ac 称为判别式，常用 Δ（希腊字母）表示。它可以告诉我们方程有几个实根。

If Δ > 0, there are two distinct real roots.

如果 Δ > 0，方程有两个不相等的实根。
If Δ = 0, there is exactly one repeated real root.

如果 Δ = 0，方程有一个重根（两个相等的实根）。
If Δ < 0, there are no real roots (the roots are complex).

如果 Δ < 0，方程没有实根（根为复数）。

Example: For x² + 4x + 5 = 0, Δ = 16 − 20 = −4, so there are no real roots.

例：对于 x² + 4x + 5 = 0，Δ = 16 − 20 = −4，所以没有实根。

7. Solving Word Problems | 解应用题

Many real-world problems lead to quadratic equations. The steps are: define the unknown, form a quadratic equation from the given information, solve it, and check which answers fit the context.

许多实际问题会导出二次方程。步骤包括：设未知数，根据已知条件列出二次方程，求解，并检查哪些答案符合实际背景。

Example: The area of a rectangle is 36 cm² and its length is 5 cm longer than its width. Let the width be x. Then the length is x + 5.

例：一个矩形的面积为36 cm²，长比宽长5 cm。设宽为 x，则长为 x + 5。

x(x + 5) = 36 → x² + 5x − 36 = 0

Factorise: (x + 9)(x − 4) = 0. So x = −9 or x = 4. Since width cannot be negative, the width is 4 cm and the length is 9 cm.

因式分解：(x + 9)(x − 4) = 0，所以 x = −9 或 x = 4。因为宽不能为负，所以宽为4 cm，长为9 cm。

8. Common Mistakes and Tips | 常见错误与技巧

One common mistake is forgetting to set the equation to zero before factorising. Always rearrange to ax² + bx + c = 0 first.

常见错误之一是在因式分解前忘记将方程化为零。务必先整理成 ax² + bx + c = 0。

Another mistake is misusing the quadratic formula when b is negative. Be careful with signs: subtracting a negative produces addition.

另一个错误是在 b 为负数时误用二次公式。注意符号：减去负数相当于加上正数。

When using a calculator, estimate your answer first. For example, if the roots are 2 and 3, then (x − 2)(x − 3) = 0 expands to x² − 5x + 6 = 0, which matches the original.

使用计算器时，先估计答案。例如，若根为2和3，则 (x − 2)(x − 3) = 0 展开为 x² − 5x + 6 = 0，与原式相符。

A helpful tip is to check your solutions by substitution into the original equation. This catches sign errors and arithmetic slips.

一个有用的技巧是将解代入原方程验证。这能发现符号错误和计算失误。

9. Practice Questions | 练习题

Try these questions on your own, then check the answers below.

请自行尝试以下题目，然后核对下方答案。

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

用因式分解法解 x² − 7x + 10 = 0。
Solve 3x² + 5x − 2 = 0 using the quadratic formula.

用公式法解 3x² + 5x − 2 = 0。
Find the discriminant of 2x² − 4x + 1 = 0 and state the number of real roots.

求 2x² − 4x + 1 = 0 的判别式，并说明实根个数。

Answers: 1) x = 2 or x = 5 2) x = 1/3 or x = −2 3) Δ = 8, two distinct real roots

答案：1) x = 2 或 x = 5 2) x = 1/3 或 x = −2 3) Δ = 8，有两个不相等的实根

10. Summary | 总结

You now know four approaches to solving quadratic equations: factorisation, completing the square, the quadratic formula, and the discriminant for root analysis. Choose the method that suits the problem best.

现在你已经掌握了解一元二次方程的四种方法：因式分解法、配方法、公式法以及用于分析根的判别式。根据题目特点选择最合适的方法。

For IGCSE exams, practising all methods is vital. Remember to present your working clearly, as method marks are often awarded alongside correct answers.

对于IGCSE考试，练习所有方法至关重要。请记住清晰展示解题过程，因为评分时通常会按步骤给分。

Keep practising, and quadratic equations will become straightforward.

持续练习，二次方程会变得简单易解。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Solving Quadratic Equations | 解一元二次方程