Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations appear frequently in the IGCSE Edexcel mathematics syllabus. Mastering them is essential for success in algebra, graphs, and problem-solving questions.

二次方程在 IGCSE Edexcel 数学考纲中频繁出现。掌握二次方程是学好代数、函数图像和解决实际问题的基础。

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

A quadratic equation is a polynomial equation of degree 2. The highest power of the variable is 2, and it can be written in the general form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0.

二次方程是次数为 2 的多项式方程。变量的最高次数是 2，其一般形式写为 ax² + bx + c = 0，其中 a、b、c 为实数且 a ≠ 0。

For example, 2x² – 5x + 3 = 0 is a quadratic equation. Here a = 2, b = -5, c = 3.

例如，2x² – 5x + 3 = 0 是一个二次方程，其中 a = 2，b = -5，c = 3。

2. Standard Form and Roots | 标准形式与根

Before solving, a quadratic equation must be rearranged into the standard form ax² + bx + c = 0. This means all terms are on one side, and the other side equals zero.

在求解之前，必须将二次方程整理成标准形式 ax² + bx + c = 0，即所有项都在等号左侧，右侧为零。

The solutions of a quadratic equation are called roots or zeros. A quadratic equation can have two distinct real roots, one repeated real root, or no real roots, depending on the value of the discriminant.

二次方程的解称为根或零点。根据判别式的值，二次方程可以有两个不同的实根、一个重根，或没有实根。

3. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the quadratic expression can be written as a product of two binomials. For example, x² + 5x + 6 = 0 factors as (x + 2)(x + 3) = 0.

当二次表达式能写成两个一次二项式的乘积时，因式分解是最快的方法。例如，x² + 5x + 6 = 0 可以分解为 (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.

利用零积性质，如果两个因式的乘积为零，则至少一个因式为零。所以 x + 2 = 0 或 x + 3 = 0，解得 x = -2 或 x = -3。

For equations with a ≠ 1, such as 2x² – 5x – 3 = 0, find two numbers that multiply to a × c = 2 × (-3) = -6 and add to b = -5. These numbers are -6 and 1, so the expression becomes 2x² – 6x + x – 3 = 0, then factor in groups.

对于 a ≠ 1 的方程，如 2x² – 5x – 3 = 0，找到两个数，它们的乘积等于 a × c = 2 × (-3) = -6，且和为 b = -5。这两个数是 -6 和 1，因此表达式化为 2x² – 6x + x – 3 = 0，再分组因式分解。

4. Solving by Completing the Square | 配方法

Completing the square rewrites ax² + bx + c = 0 in the form a(x + p)² + q = 0. This method is particularly useful when factorisation is not straightforward.

配方法将 ax² + bx + c = 0 改写为 a(x + p)² + q = 0 的形式。当因式分解不直接时，此方法尤为有用。

For example, solve x² + 6x + 2 = 0. Take half of the coefficient of x: (6 ÷ 2) = 3, then square it: 3² = 9. Rewrite as (x + 3)² – 9 + 2 = 0, which simplifies to (x + 3)² – 7 = 0.

例如，解 x² + 6x + 2 = 0。取 x 系数的一半：(6 ÷ 2) = 3，然后平方：3² = 9。改写为 (x + 3)² – 9 + 2 = 0，即 (x + 3)² – 7 = 0。

Then (x + 3)² = 7, so x + 3 = ±√7, giving x = -3 ± √7.

于是 (x + 3)² = 7，所以 x + 3 = ±√7，解得 x = -3 ± √7。

5. The Quadratic Formula | 二次公式

The quadratic formula solves any quadratic equation directly. For ax² + bx + c = 0, the roots are given by the formula below.

二次公式可以直接求解任何二次方程。对于 ax² + bx + c = 0，根由下面的公式给出。

x = (-b ± √(b² – 4ac)) / 2a

To use it, substitute the values of a, b, and c into the formula. For example, solve 3x² – 5x – 2 = 0. Here a = 3, b = -5, c = -2. Then b² – 4ac = (-5)² – 4 × 3 × (-2) = 25 + 24 = 49.

使用时，将 a、b、c 的值代入公式。例如，解 3x² – 5x – 2 = 0。这里 a = 3，b = -5，c = -2。则 b² – 4ac = (-5)² – 4 × 3 × (-2) = 25 + 24 = 49。

So x = (5 ± √49) ÷ (2 × 3) = (5 ± 7) ÷ 6. Therefore x = 12 ÷ 6 = 2, or x = -2 ÷ 6 = -1⁄3.

因此 x = (5 ± √49) ÷ (2 × 3) = (5 ± 7) ÷ 6。所以 x = 12 ÷ 6 = 2，或 x = -2 ÷ 6 = -1⁄3。

6. The Discriminant | 判别式

The expression b² – 4ac is called the discriminant. It tells us the number and type of roots without solving the equation.

表达式 b² – 4ac 称为判别式。它无需解方程即可判断根的个数和类型。

If b² – 4ac > 0, there are two distinct real roots. If b² – 4ac = 0, there is exactly one repeated root. If b² – 4ac < 0, there are no real roots, only complex roots.

如果 b² – 4ac > 0，方程有两个不同的实根；如果 b² – 4ac = 0，方程有一个重根；如果 b² – 4ac < 0，方程没有实根，只有复数根。

For example, for 2x² + 4x + 5 = 0, the discriminant is 4² – 4 × 2 × 5 = 16 – 40 = -24 < 0, so there are no real solutions.

例如，对于 2x² + 4x + 5 = 0，判别式为 4² – 4 × 2 × 5 = 16 – 40 = -24 < 0，因此没有实数解。

7. Solving Word Problems | 应用题解法

Many practical problems lead to quadratic equations. Begin by defining the unknown variable, translate the conditions into an equation, then solve it, and finally check whether each root makes sense in the original context.

许多实际问题会导出二次方程。先定义未知数，将条件转化为方程，然后求解，最后检验每个根在原始情境中是否合理。

Example: The area of a rectangle is 48 m² and its length is 4 m more than its width. Let the width be w. Then length = w + 4, and w(w + 4) = 48. This expands to w² + 4w – 48 = 0.

例如：一个矩形面积为 48 m²，其长比宽多 4 m。设宽为 w，则长为 w + 4，且 w(w + 4) = 48。展开得 w² + 4w – 48 = 0。

Factorise: (w + 12)(w – 4) = 0, so w = -12 or w = 4. Since width cannot be negative, the width is 4 m and the length is 8 m.

因式分解：(w + 12)(w – 4) = 0，所以 w = -12 或 w = 4。由于宽度不能为负，因此宽为 4 m，长为 8 m。

8. Graphs of Quadratics | 二次函数图像

The graph of a quadratic function y = ax² + bx + c is a parabola. The sign of a determines the direction: if a > 0, the parabola opens upwards; if a < 0, it opens downwards.

二次函数 y = ax² + bx + c 的图像是一条抛物线。a 的符号决定开口方向：若 a > 0，抛物线开口向上；若 a < 0，抛物线开口向下。

The roots of the equation ax² + bx + c = 0 are the x-intercepts of the parabola. The x-coordinate of the vertex is given by x = -b / (2a), and the y-coordinate is found by substituting this x into the function.

方程 ax² + bx + c = 0 的根是抛物线与 x 轴的交点。顶点的 x 坐标为 x = -b / (2a)，将其代入函数即可得到 y 坐标。

For y = x² – 4x + 3, the roots are x = 1 and x = 3. The vertex x-coordinate is -(-4) / (2×1) = 2, and y = 2² – 4×2 + 3 = -1, so the vertex is (2, -1).

对于 y = x² – 4x + 3，根为 x = 1 和 x = 3。顶点 x 坐标为 -(-4) / (2×1) = 2，y = 2² – 4×2 + 3 = -1，所以顶点为 (2, -1)。

9. Common Mistakes | 常见错误

One common mistake is forgetting to set the equation to zero before factorising. For example, x² = 5x is often wrongly solved by dividing both sides by x, losing the root x = 0.

一个常见错误是忘记在因式分解前将方程化为零。例如，x² = 5x 常被错误地两边同除以 x，从而丢失根 x = 0。

Correct method: x² – 5x = 0, then x(x – 5) = 0, giving x = 0 or x = 5. Never divide by a variable that could be zero.

正确方法：x² – 5x = 0，然后 x(x – 5) = 0，得到 x = 0 或 x = 5。切勿除以可能为零的变量。

Another mistake is misusing the quadratic formula sign. Pay close attention to negative signs inside the formula, especially when b or c is negative.

另一个错误是误用二次公式中的符号。要特别注意公式中的负号，尤其是当 b 或 c 为负时。

10. Practice Questions | 练习

Try these questions to test your understanding. Solve each quadratic equation using the most suitable method.

尝试以下问题来测试你的理解。用最合适的方法解每个二次方程。

English: x² – 7x + 10 = 0

中文：x² – 7x + 10 = 0
English: 2x² + 5x – 3 = 0

中文：2x² + 5x – 3 = 0
English: x² + 4x – 1 = 0

中文：x² + 4x – 1 = 0
English: 9x² – 12x + 4 = 0

中文：9x² – 12x + 4 = 0

Answers: 1) x = 2 or x = 5 2) x = 0.5 or x = -3 3) x = -2 ± √5 4) x = 2⁄3 (repeated root)

答案：1) x = 2 或 x = 5 2) x = 0.5 或 x = -3 3) x = -2 ± √5 4) x = 2⁄3（重根）

11. Summary | 总结

To solve a quadratic equation, always start by writing it in standard form ax² + bx + c = 0. Choose a method: factorisation if simple, completing the square or the quadratic formula otherwise. Use the discriminant to predict the nature of the roots, and always check your solutions in the original equation.

解二次方程时，始终先将其写成标准形式 ax² + bx + c = 0。选择方法：若容易分解则用因式分解法，否则用配方法或二次公式。利用判别式预判根的情况，并始终将解代回原方程检验。

With regular practice, quadratic equations become one of the most reliable scoring areas in the IGCSE mathematics exam.

通过定期练习，二次方程会成为 IGCSE 数学考试中最稳定的得分点之一。

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Solving Quadratic Equations | 解二次方程