Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations appear everywhere in mathematics, from simple area problems to projectile motion in physics. In the Edexcel IGCSE syllabus, solving quadratics is a fundamental skill that underpins algebra, graphing, and problem-solving. This article breaks down every core method you need, using the number 225 as our recurring example: because 225 = 15², it makes a perfect springboard to understand square roots, factorisation, and the quadratic formula.

二次方程在数学中无处不在，从简单的面积问题到物理中的抛体运动。在 Edexcel IGCSE 大纲中，解二次方程是一项基础技能，支撑着代数、图像和问题求解。本文将 225 作为贯穿例题逐步拆解每一个核心方法：因为 225 = 15²，它是理解平方根、因式分解和二次公式的绝佳起点。

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

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b and c are constants, and a ≠ 0. The highest power of x is 2, which makes the graph a parabola. If a = 0, the equation becomes linear, not quadratic. You must recognise this general form before applying any solving method.

二次方程是指可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数，且 a ≠ 0。x 的最高次数是 2，因此图像是抛物线。如果 a = 0，方程就变成线性的，不再是二次方程。在应用任何解法之前，必须先认出这种一般形式。

2. Solving x² = 225 | 解 x² = 225

The simplest quadratic equation has no linear term and no constant term after rearrangement: x² = k. To solve it, take the square root of both sides. For x² = 225, this gives x = ±√225 = ±15. The ± sign is crucial, because both 15² = 225 and (−15)² = 225. If k is negative, there are no real solutions; for example, x² = −225 has no real roots because no real number squared is negative.

最简单的二次方程经过移项后没有一次项和常数项：x² = k。解它只需两边同时开平方。对于 x² = 225，可以得到 x = ±√225 = ±15。这里的 ± 号是关键的，因为 15² = 225 且 (−15)² = 225。如果 k 为负数，则没有实数解；例如 x² = −225 没有实数根，因为没有任何实数的平方是负数。

3. Square Roots and the ± Sign | 平方根与±号

When you take the square root of a positive number, there are always two results: one positive and one negative. The symbol √225 normally means the positive root (15), but solving an equation requires both roots. Hence we write ≤strong>±√225. This is the single most common place to lose marks: forgetting the negative root. For example, the equation x² = 225 is satisfied by x = 15 and x = −15. Check: (−15)² = 225.

当对一个正数开平方时，总有两个结果：一正一负。符号 √225 通常表示正根（15），但解方程时需要两个根。因此要写 ±√225。这是最常见的失分点：忘记负根。例如，方程 x² = 225 的解是 x = 15 和 x = −15。检验：(−15)² = 225。

4. Factorising Quadratics | 因式分解二次式

Factorising is writing a quadratic as a product of two brackets. For example, x² − 20x + 75 = 0 can be factorised into (x − 5)(x − 15) = 0, because the numbers 5 and 15 multiply to 75 and add to −20. For x² − 225 = 0, use the difference of two squares: x² − 225 = (x − 15)(x + 15) = 0. This pattern is x² − a² = (x − a)(x + a).

因式分解是把二次式写成两个括号的乘积。例如，x² − 20x + 75 = 0 可以分解为 (x − 5)(x − 15) = 0，因为 5 和 15 相乘得 75，相加得 −20。对于 x² − 225 = 0，使用平方差公式：x² − 225 = (x − 15)(x + 15) = 0。这个规律是 x² − a² = (x − a)(x + a)。

5. Solving by Factorisation | 用因式分解求解

If a quadratic factorises, the solution follows from the zero product property: if AB = 0, then A = 0 or B = 0. So from (x − 15)(x + 15) = 0, set x − 15 = 0 or x + 15 = 0, giving x = 15 or x = −15. For a quadratic like x² + 5x + 6 = 0, factorise to (x + 2)(x + 3) = 0, so x = −2 or x = −3. Always expand your brackets to check.

如果二次式可以因式分解，求解就依据零积性质：若 AB = 0，则 A = 0 或 B = 0。所以从 (x − 15)(x + 15) = 0 中，令 x − 15 = 0 或 x + 15 = 0，得到 x = 15 或 x = −15。对于像 x² + 5x + 6 = 0 这样的方程，分解为 (x + 2)(x + 3) = 0，所以 x = −2 或 x = −3。完成后展开括号检查结果。

6. Completing the Square | 配方法

Completing the square rewrites ax² + bx + c as a(x + p)² + q. For x² + 10x − 5 = 0, take half of 10 (which is 5), square it (25), and write (x + 5)² − 25 − 5 = 0, so (x + 5)² = 30. Then x + 5 = ±√30, so x = −5 ± √30. This method always works, even when factorisation is difficult or impossible.

配方法将 ax² + bx + c 改写为 a(x + p)² + q。对于 x² + 10x − 5 = 0，取 10 的一半（即 5），平方得 25，然后写成 (x + 5)² − 25 − 5 = 0，即 (x + 5)² = 30。于是 x + 5 = ±√30，所以 x = −5 ± √30。这种方法总是可行，即使因式分解困难甚至不可能。

7. The Quadratic Formula | 二次公式

The quadratic formula solves any equation of the form ax² + bx + c = 0:

x = (−b ± √(b² − 4ac)) / (2a)

For example, solve 2x² − 8x + 3 = 0. Here a = 2, b = −8, c = 3. Substitute:

x = (8 ± √((−8)² − 4×2×3)) / (2×2) = (8 ± √(64 − 24)) / 4 = (8 ± √40) / 4

Simplify √40 = 2√10, so x = (8 ± 2√10)/4 = 2 ± √10/2. Always write the formula down before substituting, and be careful with negative b.

二次公式可以解任何 ax² + bx + c = 0 形式的方程：

x = (−b ± √(b² − 4ac)) / (2a)

例如，解 2x² − 8x + 3 = 0。这里 a = 2，b = −8，c = 3。代入：

x = (8 ± √((−8)² − 4×2×3)) / (2×2) = (8 ± √(64 − 24)) / 4 = (8 ± √40) / 4

化简 √40 = 2√10，所以 x = (8 ± 2√10)/4 = 2 ± √10/2。代入前一定要写出公式，并小心负数 b。

8. The Discriminant | 判别式

The expression b² − 4ac is called the discriminant. It tells you the nature of the roots without solving the equation:

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.

For x² = 225, which can be written as x² + 0x − 225 = 0, the discriminant is 0² − 4×1×(−225) = 900 > 0, so two real roots. For x² + 2x + 5 = 0, discriminant = 4 − 20 = −16, so no real roots.

表达式 b² − 4ac 称为判别式。它无需解方程就能判断根的性质：

如果 b² − 4ac > 0，方程有两个不同的实数根。
如果 b² − 4ac = 0，方程有一个重复的实数根。
如果 b² − 4ac < 0，方程没有实数根。

对于 x² = 225，可写成 x² + 0x − 225 = 0，判别式为 0² − 4×1×(−225) = 900 > 0，所以有两个实数根。对于 x² + 2x + 5 = 0，判别式为 4 − 20 = −16，所以没有实数根。

9. Quadratic Graphs | 二次函数图像

The graph of y = ax² + bx + c is a parabola. The roots of the equation are the x-coordinates where the graph crosses the x-axis. For y = x² − 225, the graph crosses at x = −15 and x = 15. The line of symmetry is halfway between the roots, x = 0, and the vertex is at (0, −225). If a > 0, the parabola opens upwards; if a < 0, it opens downwards. If the discriminant is negative, the graph never touches the x-axis.

y = ax² + bx + c 的图像是抛物线。方程的根就是图像与 x 轴交点的 x 坐标。对于 y = x² − 225，图像在 x = −15 和 x = 15 处穿过 x 轴。对称轴位于两根中间，即 x = 0，顶点为 (0, −225)。如果 a > 0，抛物线开口向上；如果 a < 0，开口向下。如果判别式为负，图像完全不接触 x 轴。

10. Word Problems and Applications | 应用题

Quadratic equations often appear in area, geometry, and motion problems. For example: “A square has area 225 cm². Find its side length.” Let side = x, then x² = 225, so x = 15 cm (reject −15 because length is positive). In realistic problems, sometimes one root is discarded because it does not make sense in the context. Always read the question and state your final answer in the correct unit.

二次方程常出现在面积、几何和运动问题中。例如：“一个正方形的面积为 225 cm²，求它的边长。”设边长为 x，则 x² = 225，所以 x = 15 cm（舍去 −15，因为长度为正）。在现实问题中，有时需要丢弃一个根，因为它不符合实际意义。务必仔细读题，并用正确单位写出最终答案。

11. Common Mistakes | 常见错误

Students often make these errors when solving quadratics:

Forgetting the ± sign when taking square roots.
Writing 10x² as 10x° (confusing powers).
Factorising incorrectly by not checking signs.
Substituting wrongly into the quadratic formula, especially b when b is negative.
Thinking x² = 225 has only one solution, x = 15.

To avoid these, always check by substituting your solutions back into the original equation. For x = −15, (−15)² = 225, so it is valid.

学生在解二次方程时常犯以下错误：

开平方时忘记 ± 号。
将 10x² 误写成 10x°（混淆指数）。
因式分解时没有检查符号而分解错误。
代入二次公式时出错，尤其是 b 为负数时。
认为 x² = 225 只有一个解 x = 15。

为避免这些错误，务必把解代回原方程检验。对于 x = −15，(−15)² = 225，所以它是有效的。

12. Practice Questions | 练习

1. Solve: x² = 196
2. Factorise and solve: x² − 8x + 15 = 0
3. Solve by completing the square: x² + 6x + 2 = 0
4. Use the quadratic formula: 3x² − 5x − 2 = 0
5. Find the discriminant of x² − 225 = 0 and state the nature of the roots.

Answers: 1. x = ±14; 2. (x − 3)(x − 5) = 0, x = 3 or 5; 3. x = −3 ± √7; 4. x = 2 or −1/3; 5. discriminant = 900 > 0, two distinct real roots.

1. 解：x² = 196
2. 因式分解并求解：x² − 8x + 15 = 0
3. 配方求解：x² + 6x + 2 = 0
4. 使用二次公式：3x² − 5x − 2 = 0
5. 求 x² − 225 = 0 的判别式，并说明根的性质。

答案：1. x = ±14；2. (x − 3)(x − 5) = 0，x = 3 或 5；3. x = −3 ± √7；4. x = 2 或 −1/3；5. 判别式 = 900 > 0，两个不同的实数根。

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Solving Quadratic Equations | 解二次方程