114: Solving Quadratic Equations | 114：解二次方程

📚 114: Solving Quadratic Equations | 114：解二次方程

A quadratic equation is a second‑order polynomial equation in a single variable. In the IGCSE Edexcel syllabus, mastering the art of solving such equations is essential because it appears in algebra, graphs, word problems and even geometry. This article provides a complete guide to the methods, the discriminant, graph sketching and common pitfalls, with worked examples aligned to exam style.

二次方程是含有一个变量的二次多项式方程。在 IGCSE Edexcel 考纲中，掌握解二次方程的方法至关重要，因为它出现在代数、函数图像、应用题甚至几何题中。本文将全面讲解解法、判别式、图像草绘和常见错误，并配有贴近考试风格的例题。

1. Standard Form and Key Terms | 1. 标准形式与关键术语

Every quadratic equation can be rearranged into the standard form:

ax² + bx + c = 0, where a ≠ 0

Here, a, b and c are real numbers. a is the coefficient of x², b is the coefficient of x, and c is the constant term. If a = 0, the equation becomes linear, not quadratic. The solutions of the equation are also called the roots or zeros of the quadratic.

每个二次方程都可以化为标准形式：

ax² + bx + c = 0，其中 a ≠ 0

其中 a、b、c 为实数。a 是 x² 的系数，b 是 x 的系数，c 是常数项。若 a = 0，方程退化为一次方程，不再称为二次。方程的解也称为二次方程的根或零点。

2. Solving by Factorisation | 2. 因式分解法

Factorisation is the quickest method when the quadratic has simple rational roots. The idea is to write ax² + bx + c as a product of two linear binomials, then set each factor equal to zero. This works because if the product of two expressions is zero, at least one of them must be zero.

当二次方程具有简单的有理数根时，因式分解法是最快捷的方法。其核心是将 ax² + bx + c 写成两个一次二项式的乘积，然后令每个因式为零。这是因为若两个表达式的乘积为零，则至少有一个表达式的值为零。

Example: Solve x² − 5x + 6 = 0.

Find two numbers that multiply to 6 and add to −5. These are −2 and −3.

找到乘积为 6、和为 −5 的两个数，即 −2 和 −3。
Factorise: (x − 2)(x − 3) = 0.

因式分解：(x − 2)(x − 3) = 0。
Set each factor equal to zero: x − 2 = 0 → x = 2; x − 3 = 0 → x = 3.

令每个因式为零：x − 2 = 0 → x = 2；x − 3 = 0 → x = 3。

For equations with a leading coefficient other than 1, the method becomes slightly longer. For example, 2x² + 5x − 3 = 0 can be factorised as (2x − 1)(x + 3) = 0, giving roots x = ½ and x = −3.

当首项系数不为 1 时，因式分解过程会稍长。例如，2x² + 5x − 3 = 0 可分解为 (2x − 1)(x + 3) = 0，从而得到根 x = ½ 和 x = −3。

3. Solving by Completing the Square | 3. 配方法

Completing the square transforms a quadratic into a perfect square plus a constant. This method is especially useful when the equation cannot be factorised easily, and it also helps in deriving the quadratic formula and in solving max/min problems.

配方法将二次式转化为一个完全平方加上一个常数的形式。当方程不易因式分解时，这种方法非常有用，同时它也用于推导二次公式以及求解最大/最小值问题。

To complete the square for x² + bx:

x² + bx = (x + b/2)² − (b/2)²

For a quadratic ax² + bx + c, first factor out a if a ≠ 1, then apply the formula to the remaining x² + (b/a)x.

对 x² + bx 配方：

x² + bx = (x + b/2)² − (b/2)²

对于 ax² + bx + c，若 a ≠ 1，先提出 a，再对其余部分 x² + (b/a)x 配方。

Example: Solve x² + 6x + 1 = 0 by completing the square.

Move the constant: x² + 6x = −1.

移常数项：x² + 6x = −1。
Add (6/2)² = 9 to both sides: x² + 6x + 9 = 8.

两边同加 (6/2)² = 9：x² + 6x + 9 = 8。
Write as a square: (x + 3)² = 8.

写成完全平方：(x + 3)² = 8。
Take square roots: x + 3 = ±√8 → x = −3 ± 2√2.

开平方：x + 3 = ±√8 → x = −3 ± 2√2。

4. Solving by the Quadratic Formula | 4. 公式法

The quadratic formula is a universal method that works for any quadratic equation. It is derived by completing the square on ax² + bx + c = 0. The formula is:

x = (−b ± √(b² − 4ac)) / 2a

You should memorise this formula and be careful to substitute the values of a, b and c correctly. The expression under the square root, b² − 4ac, is called the discriminant, which we will examine in the next section.

二次公式是解任意二次方程的通用方法，它通过对 ax² + bx + c = 0 配方而导出。公式为：

x = (−b ± √(b² − 4ac)) / 2a

应当熟记此公式，并注意正确代入 a、b、c 的数值。根号内的表达式 b² − 4ac 称为判别式，我们将在下一节讨论。

Example: Solve 2x² − 4x − 3 = 0 using the formula.

Here a = 2, b = −4, c = −3. Substitution gives:

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

Hence x = (4 + √40)/4 or x = (4 − √40)/4. These can be simplified to x = 1 ± √10/2.

例：用公式解 2x² − 4x − 3 = 0。

这里 a = 2，b = −4，c = −3。代入得：

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

因此 x = (4 + √40)/4 或 x = (4 − √40)/4，可化简为 x = 1 ± √10/2。

5. The Discriminant and the Nature of Roots | 5. 判别式与根的性质

The discriminant, usually written as Δ, is defined as Δ = b² − 4ac. Its value tells us how many real roots the quadratic equation has and whether they are distinct or repeated.

判别式通常记作 Δ，定义为 Δ = b² − 4ac。它的值告知我们该二次方程有多少个实根，以及它们是相异根还是重根。

Discriminant Δ	Nature of Roots	Graphical Interpretation
Δ > 0	Two distinct real roots	Parabola crosses the x‑axis twice
Δ = 0	One repeated real root	Parabola touches the x‑axis at one point
Δ < 0	No real roots (two complex roots)	Parabola does not meet the x‑axis

For example, the equation x² − 4x + 4 = 0 has Δ = 16 − 16 = 0, so it has one repeated root, x = 2. The equation x² + x + 1 = 0 has Δ = 1 − 4 = −3, so it has no real roots.

例如，方程 x² − 4x + 4 = 0 的 Δ = 16 − 16 = 0，因此有一个重根 x = 2。方程 x² + x + 1 = 0 的 Δ = 1 − 4 = −3，因此没有实根。

6. Sketching Quadratic Graphs | 6. 二次函数图像草绘

A quadratic function y = ax² + bx + c has a parabolic graph. The sign of a determines whether the parabola opens upward (a > 0) or downward (a < 0). The vertex is found by completing the square: y = a(x − h)² + k gives the vertex at (h, k).

二次函数 y = ax² + bx + c 的图像是抛物线。a 的符号决定抛物线开口方向：a > 0 时开口向上，a < 0 时开口向下。顶点可通过配方得到：y = a(x − h)² + k 的顶点为 (h, k)。

To sketch the graph accurately, follow these steps:

Find the roots (where y = 0) by solving the equation. If the discriminant is positive, there are two x‑intercepts; if zero, one intercept; if negative, none.

求根：解方程 y = 0。若判别式为正，有两个 x 轴交点；为零，有一个；为负，则没有。
Find the y‑intercept by putting x = 0, giving the point (0, c).

令 x = 0 求 y 轴截距，得到点 (0, c)。
Find the vertex using x = −b/(2a), then substitute to get y.

用 x = −b/(2a) 求顶点横坐标，再代入求得 y。

Example: Sketch y = x² − 2x − 3.

Roots: (x − 3)(x + 1) = 0, so x = 3 and x = −1. y‑intercept: (0, −3). Vertex: x = −(−2)/(2×1) = 1, y = 1 − 2 − 3 = −4, so vertex is (1, −4). The parabola opens upward because a = 1 > 0.

例：草绘 y = x² − 2x − 3。

根：(x − 3)(x + 1) = 0，故 x = 3 和 x = −1。y 轴截距：(0, −3)。顶点：x = −(−2)/(2×1) = 1，y = 1 − 2 − 3 = −4，所以顶点为 (1, −4)。因 a = 1 > 0，抛物线开口向上。

7. Word Problems Involving Quadratics | 7. 二次方程应用题

Many real‑world problems lead to quadratic equations, such as finding dimensions when the area is given, calculating projectile motion, or determining break‑even points. The key is to translate the problem into an equation using the given conditions.

许多实际问题会导出二次方程，例如已知面积求几何尺寸、计算抛射体的运动、分析损益平衡点等。关键在于根据题目条件将问题转化为二次方程。

Example: A rectangle has length 3 cm more than its width. Its area is 70 cm². Find the dimensions.

Let width = x cm, then length = (x + 3) cm. The area is x(x + 3) = 70, so x² + 3x − 70 = 0. Factorising gives (x + 10)(x − 7) = 0. Hence x = 7 (since x cannot be negative). The rectangle is 7 cm by 10 cm.

例：一个长方形的长比宽多 3 厘米，面积为 70 平方厘米。求它的长与宽。

设宽为 x 厘米，则长为 (x + 3) 厘米。面积为 x(x + 3) = 70，即 x² + 3x − 70 = 0。因式分解得 (x + 10)(x − 7) = 0，所以 x = 7（舍去负根）。长方形为 7 厘米 × 10 厘米。

8. Quadratic Equations with No Real Roots | 8. 无实根的二次方程

When the discriminant is negative, the equation has no real solutions. For example, x² + 2x + 5 = 0. The graph of y = x² + 2x + 5 lies entirely above the x‑axis, meaning it never reaches y = 0.

当判别式为负时，方程没有实数解。例如 x² + 2x + 5 = 0，其函数图像 y = x² + 2x + 5 完全位于 x 轴上方，即永远不取 y = 0。

In the IGCSE syllabus, you are not required to solve complex equations. However, you may be asked to state that there are no real roots, or to find the range of values of k for which a given equation has real roots. For instance, for kx² + 4x + 1 = 0 to have real roots, the discriminant must be ≥ 0: 16 − 4k ≥ 0 → k ≤ 4.

在 IGCSE 考纲中，不要求解复数方程。但你可能会被要求说明方程无实数根，或求使给定方程有实数根时参数 k 的取值范围。例如，要使 kx² + 4x + 1 = 0 有实数根，判别式须满足：16 − 4k ≥ 0 → k ≤ 4。

9. Common Mistakes and How to Avoid Them | 9. 常见错误及避免方法

Students often lose marks on quadratic equations due to avoidable slips. Let us examine the most frequent errors and the correct approaches.

学生在解二次方程时经常因可避免的失误而丢分。下面我们分析最常见的错误及正确做法。

Not rearranging to standard form before factorising. Always write ax² + bx + c = 0 first.

因式分解前未化为标准形式。务必先将方程写成 ax² + bx + c = 0。
Forgetting that a negative number squared is positive. When using the formula, −b becomes positive if b is negative.

忘记负数的平方为正。使用公式时，若 b 为负，则 −b 为正。
Dropping one root when solving by taking square roots. Remember to write ±√.

开平方时丢掉一个根。记住要写 ±√。
Simplifying fractions incorrectly. For example, (4 ± √40)/4 cannot be simplified to 1 ± √40; you must simplify √40 to 2√10 and then divide each term.

化简分数出错。例如 (4 ± √40)/4 不能化为 1 ± √40；应先将 √40 化简为 2√10，再逐项除以 4。
Confusing the vertex with the roots. The vertex is the turning point, not an x‑intercept (unless the graph touches the axis).

混淆顶点与根。顶点是转向点，不是 x 轴交点（除非图像与 x 轴相切）。

10. Exam Tips and Summary | 10. 考试技巧与总结

In the Edexcel IGCSE exam, quadratic equations appear in both paper 1 and paper 2. Always show all working steps; a correct answer without supporting method may not receive full marks. Read the question carefully: sometimes it asks for the answer correct to 2 decimal places, which means you should use the quadratic formula or completing the square, not factorisation if the roots are irrational.

在 Edexcel IGCSE 考试中，二次方程在 Paper 1 和 Paper 2 中都会出现。务必写出全部解题步骤；仅有正确答案而没有方法可能得不到满分。仔细读题：若题目要求答案精确到小数点后两位，应使用公式法或配方法，特别是当根为无理数时。

To summarise, the three main analytical methods are:

Method	When to Use
Factorisation	When the roots are simple integers or rational numbers (easy to spot).
Completing the square	When you need to find the vertex or the equation has a negative discriminant (to show no real roots).
Quadratic formula	Always works; ideal for irrational roots where factorisation fails.

总之，三种主要的分析方法为：

方法	适用时机
因式分解法	根为简单整数或有理数时（易观察）。
配方法	需要求顶点，或需要说明判别式为负时方程无实根。
公式法	通用方法；当根为无理数且因式分解失效时尤为合适。

Remember: practice is the key to mastering quadratic equations. Work through past exam questions, check your discriminant before choosing a method, and always verify your roots by substituting them back into the original equation.

记住：练习是掌握二次方程的关键。多做历年真题，在选取解法前先算判别式，并将求得的根代回原方程进行检验。

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114: Solving Quadratic Equations | 114：解二次方程