Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in algebra, geometry, and even in real-world problem solving. This article will guide you through every essential method and concept you need to master quadratic equations, with clear English explanations followed by Chinese translations.

二次方程是 IGCSE 数学中最重要的主题之一。它不仅出现在代数中，还出现在几何和实际应用题里。本文将用中英对照的方式，带你掌握解二次方程的所有关键方法与概念。

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

A quadratic equation is a polynomial equation of degree 2. This means the highest power of the variable is 2. The general form is:

二次方程是次数为 2 的多项式方程，意味着变量的最高次幂是 2。它的一般形式是：

ax² + bx + c = 0

Here, a, b, and c are constants, and a ≠ 0. If a = 0, the equation becomes linear, not quadratic.

其中 a、b、c 是常数，且 a ≠ 0。如果 a = 0，方程就变成一次方程，而不是二次方程。

x is the unknown variable | x 是未知数
a is the coefficient of x² | a 是 x² 的系数
b is the coefficient of x | b 是 x 的系数
c is the constant term | c 是常数项

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

Before solving, you should rearrange any quadratic equation into standard form:

在求解之前，你需要将任何二次方程整理成标准形式：

ax² + bx + c = 0

For example, the equation 2x² + 5x = 3 must be rewritten as 2x² + 5x − 3 = 0.

例如，方程 2x² + 5x = 3 必须改写为 2x² + 5x − 3 = 0。

Roots / Solutions – values of x that make the equation true | 根/解 – 使方程成立的 x 值
Quadratic expression – the left-hand side when the equation is in standard form | 二次表达式 – 标准形式下的左侧部分
Monic quadratic – when a = 1 | 首一二次式 – 当 a = 1 时

Always check that the equation is set to zero before applying any solving method.

在应用任何解法之前，务必确认方程右边等于零。

3. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the quadratic has simple integer roots. The idea is to write the quadratic as a product of two linear factors.

当二次方程具有简单的整数根时，因式分解是最快的方法。其思路是将二次式写成两个一次因式的乘积。

For a monic quadratic x² + bx + c, find two numbers whose product is c and whose sum is b.

对于首一二次式 x² + bx + c，找到两个数，使它们的乘积为 c，和为 b。

x² + 7x + 12 = (x + 3)(x + 4)

Because 3 × 4 = 12 and 3 + 4 = 7. Then set each factor to zero:

因为 3 × 4 = 12 且 3 + 4 = 7。然后令每个因式为零：

x + 3 = 0 → x = −3, x + 4 = 0 → x = −4

For non-monic quadratics like 2x² + 5x − 3 = 0, use the product-sum method with the leading coefficient, or factor by grouping.

对于非首一二次式如 2x² + 5x − 3 = 0，可结合首项系数使用乘积-求和法，或使用分组分解法。

2x² + 5x − 3 = (2x − 1)(x + 3)

Then x = ½ or x = −3.

因此 x = ½ 或 x = −3。

4. The Quadratic Formula | 二次公式法

When factorisation is difficult or impossible, the quadratic formula always works. It is derived from completing the square and is valid for any quadratic equation.

当因式分解困难或无法进行时，二次公式始终有效。它由配方法推导而来，适用于任何二次方程。

x = (−b ± √(b² − 4ac)) / (2a)

For example, solve 2x² + 3x − 2 = 0. Here a = 2, b = 3, c = −2.

例如，解 2x² + 3x − 2 = 0。这里 a = 2，b = 3，c = −2。

x = (−3 ± √(3² − 4 × 2 × (−2))) / (2 × 2) = (−3 ± √25) / 4

So x = (−3 + 5)/4 = ½ or x = (−3 − 5)/4 = −2.

所以 x = (−3 + 5)/4 = ½ 或 x = (−3 − 5)/4 = −2。

Always write a, b, c first | 先写出 a、b、c 的值
Carefully substitute into the formula | 仔细代入公式
Simplify the square root first | 先化简平方根

5. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² + q. This method is useful for finding turning points and solving equations.

配方法将二次式改写为 (x + p)² + q 的形式。这种方法对求极值点和解方程很有用。

Start with x² + bx. Add and subtract (b/2)².

从 x² + bx 开始，加上并减去 (b/2)²。

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

To solve the equation x² + 6x + 2 = 0, set the completed square equal to zero:

要解方程 x² + 6x + 2 = 0，令配方式等于零：

(x + 3)² − 7 = 0 → (x + 3)² = 7 → x + 3 = ±√7

x = −3 ± √7

For a non-monic quadratic, factor out the leading coefficient first.

对于非首一二次式，先提取首项系数。

6. The Discriminant | 判别式

The discriminant is the part of the quadratic formula under the square root: b² − 4ac. It tells us the nature of the roots without solving the equation.

判别式是二次公式中根号下的部分：b² − 4ac。它可以在不解方程的情况下告诉我们根的性质。

Value of b² − 4ac	Nature of roots
Positive ( > 0 )	Two distinct real roots
Zero ( = 0 )	One repeated real root
Negative ( < 0 )	No real roots

If the discriminant is a perfect square, the roots are rational; if not, they are irrational.

如果判别式是一个完全平方数，则根为有理数；否则根为无理数。

Example: For x² − 4x + 4 = 0, the discriminant is (−4)² − 4 × 1 × 4 = 0, so there is exactly one repeated root.

例如：对于 x² − 4x + 4 = 0，判别式为 (−4)² − 4 × 1 × 4 = 0，所以只有一个重根。

7. Roots and Coefficients | 根与系数的关系

For a quadratic equation ax² + bx + c = 0 with roots α and β, the following relations hold:

对于具有根 α 和 β 的二次方程 ax² + bx + c = 0，以下关系成立：

α + β = −b/a

αβ = c/a

These formulas allow you to find the sum and product of roots directly from the coefficients.

这些公式允许你直接从系数求出根的和与积。

For example, for 2x² − 4x + 1 = 0, the sum of roots = −(−4)/2 = 2, and the product = 1/2.

例如，对于 2x² − 4x + 1 = 0，根的和 = −(−4)/2 = 2，根的积 = 1/2。

This is very useful when checking answers or constructing a quadratic from given roots.

这在检查答案或根据已知根构造二次方程时非常有用。

8. Word Problems | 应用题

Many IGCSE problems involve forming a quadratic equation from a context. You must translate the words into mathematical statements.

许多 IGCSE 题目涉及从实际情境中建立二次方程。你必须将文字转化为数学表达式。

Example: The area of a rectangle is 24 cm². Its length is 5 cm more than its width. Find the width.

例如：一个矩形的面积是 24 cm²，长比宽多 5 cm。求宽。

Let the width be x. Then the length is x + 5. Area = x(x + 5) = 24.

设宽为 x，则长为 x + 5。面积 = x(x + 5) = 24。

x² + 5x − 24 = 0

Factorise: (x + 8)(x − 3) = 0, so x = 3 (reject x = −8 because width cannot be negative).

因式分解：(x + 8)(x − 3) = 0，所以 x = 3（拒绝 x = −8，因为宽不能为负）。

Define the variable clearly | 清楚地定义变量
Form the equation from the given conditions | 根据条件建立方程
Solve and interpret the roots in the context | 解方程并结合实际解释根

9. Graphing Quadratics | 二次函数图像

A quadratic function y = ax² + bx + c produces a parabola. The graph of a quadratic equation ax² + bx + c = 0 corresponds to the x-intercepts of the parabola.

二次函数 y = ax² + bx + c 的图像是抛物线。二次方程 ax² + bx + c = 0 的解对应于抛物线与 x 轴的交点。

If a > 0, the parabola opens upwards | 若 a > 0，抛物线开口向上
If a < 0, it opens downwards | 若 a < 0，抛物线开口向下
The axis of symmetry is x = −b/(2a) | 对称轴为 x = −b/(2a)
The vertex (turning point) lies on this axis | 顶点（极值点）位于对称轴上

The number of x-intercepts matches the discriminant: two, one, or zero.

x 轴交点的个数与判别式对应：两个、一个或零个。

Sketching the graph helps visualise the roots and the minimum or maximum value.

画出图像有助于直观理解根以及最小值或最大值。

10. Common Mistakes | 常见错误

Students often lose marks in quadratic equations due to avoidable errors. Here are the most common ones:

学生在二次方程中常因可避免的错误而失分。以下是最常见的错误：

Forgetting to rearrange the equation to zero | 忘记将方程整理为零
Incorrect sign when moving terms | 移项时符号错误
Mistaking a, b, c in the quadratic formula | 在二次公式中弄错 a、b、c
Missing the ± sign when taking square roots | 开平方时遗漏 ± 号
Not simplifying roots fully | 没有完全化简根式
Rejecting valid roots in word problems without reason | 在应用题中无缘无故舍去有效根

Always check your solutions by substituting back into the original equation.

务必通过代入原方程来检查你的解。

11. Practice Questions | 练习

Try these questions to test your understanding:

尝试以下问题来测试你的理解：

Solve x² − 5x + 6 = 0 by factorisation. | 用因式分解解 x² − 5x + 6 = 0。
Solve 3x² + 4x − 2 = 0 using the quadratic formula. Give your answer correct to 2 decimal places. | 用二次公式解 3x² + 4x − 2 = 0，答案保留两位小数。
Find the discriminant of 2x² − 3x + 5 = 0 and state the nature of its roots. | 求 2x² − 3x + 5 = 0 的判别式，并说明根的性质。
Write x² + 8x + 3 in the form (x + p)² + q. | 将 x² + 8x + 3 写成 (x + p)² + q 的形式。
A right-angled triangle has hypotenuse 13 cm and one side 7 cm longer than the other. Find the lengths of the two shorter sides. | 一个直角三角形的斜边为 13 cm，其中一条直角边比另一条长 7 cm。求两条直角边的长度。

Answers: 1. x = 2 or x = 3. 2. x ≈ 0.39 or x ≈ −1.72. 3. Discriminant = −31, no real roots. 4. (x + 4)² − 13. 5. 5 cm and 12 cm.

答案：1. x = 2 或 x = 3。2. x ≈ 0.39 或 x ≈ −1.72。3. 判别式 = −31，无实数根。4. (x + 4)² − 13。5. 5 cm 和 12 cm。

12. Summary | 总结

To master quadratic equations, you must be able to identify the standard form, choose an appropriate method, and interpret the roots correctly.

要掌握二次方程，你必须能够识别标准形式，选择合适的方法，并正确解释根的含义。

Use factorisation when roots are rational | 当根为有理数时用因式分解
Use the quadratic formula when factorisation is difficult | 当因式分解困难时用二次公式
Use completing the square for turning points or when needed | 需要极值点时用配方法
Always check the discriminant to know the nature of roots | 始终检查判别式以了解根的性质

With regular practice, quadratic equations will become one of the easiest topics in IGCSE Mathematics.

通过定期练习，二次方程将成为 IGCSE 数学中最简单的主题之一。

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Solving Quadratic Equations | 解二次方程