Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 二次方程：因式分解、配方法与求根公式

📚 Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 二次方程：因式分解、配方法与求根公式

Quadratic equations appear throughout the IGCSE Mathematics syllabus, from simple factorising exercises to curve sketching, optimisation problems and real-life applications such as area, projectile motion and finance. This revision guide explains the standard form, the main algebraic solution methods, the discriminant, graphical interpretation and common exam-style applications.

二次方程贯穿 IGCSE 数学课程，从简单的因式分解练习到曲线绘图、优化问题以及面积、抛体运动和金融等实际应用。本复习指南讲解标准形式、主要代数解法、判别式、图像解释以及常见考试题型。

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

A quadratic equation is a polynomial equation in which the highest power of the unknown, usually x, is 2. In its standard form it can be written as ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

二次方程是未知数（通常为 x）的最高次数为 2 的多项式方程。标准形式可写作 ax² + bx + c = 0，其中 a、b、c 为常数且 a ≠ 0。

If a = 0, the equation becomes linear rather than quadratic. The word ‘quadratic’ comes from ‘quadratus’, the Latin word for square, because the variable is squared.

如果 a = 0，方程就变成一次方程而非二次方程。“二次”一词来源于拉丁语“quadratus”，意为平方，因为变量被平方。

For example, x² – 5x + 6 = 0, 2x² + 3x – 8 = 0 and -x² + 7x = 0 are all quadratic equations. Some equations such as (x – 1)(x + 4) = 0 are also quadratic once expanded.

例如，x² – 5x + 6 = 0、2x² + 3x – 8 = 0 和 -x² + 7x = 0 都是二次方程。有些方程如 (x – 1)(x + 4) = 0 展开后也是二次方程。

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2. Standard Form and Terminology | 标准形式与术语

In the standard form ax² + bx + c = 0, the number a is the coefficient of x², b is the coefficient of x, and c is the constant term. The value of a must not be zero, but b and c can be zero.

在标准形式 ax² + bx + c = 0 中，a 是 x² 的系数，b 是 x 的系数，c 是常数项。a 不能为零，但 b 和 c 可以为零。

For example, in 3x² – 5x + 2 = 0, we have a = 3, b = -5 and c = 2. In x² – 9 = 0, we have a = 1, b = 0 and c = -9. Being able to identify a, b and c quickly is essential for using the quadratic formula and the discriminant.

例如，在 3x² – 5x + 2 = 0 中，a = 3，b = -5，c = 2。在 x² – 9 = 0 中，a = 1，b = 0，c = -9。能够快速识别 a、b、c 对使用求根公式和判别式非常重要。

The leading coefficient a controls the width and direction of the parabola y = ax² + bx + c, while c gives the y-intercept of the graph.

首项系数 a 决定抛物线 y = ax² + bx + c 的开口方向和宽窄，而 c 给出图像与 y 轴的截距。

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3. The Zero Product Principle | 零乘积原理

The zero product principle states that if the product of two factors is zero, then at least one of the factors must be zero. In symbols, if p × q = 0, then p = 0 or q = 0.

零乘积原理指出，如果两个因式的乘积为零，那么至少有一个因式必须为零。用符号表示为：如果 p × q = 0，则 p = 0 或 q = 0。

This principle is the foundation of solving quadratic equations by factorising. Once a quadratic is written as a product of two linear factors equal to zero, each factor can be set equal to zero to find the possible values of x.

这个原理是因式分解法解二次方程的基础。一旦将二次方程写成两个一次因式的乘积等于零的形式，就可以令每个因式等于零，从而求出 x 的可能值。

It is important to remember that the product must equal zero. If the product equals, say, 5, the zero product principle cannot be applied directly.

必须记住，乘积必须等于零。如果乘积等于例如 5，则不能直接应用零乘积原理。

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4. Solving by Factorising | 因式分解法

Factorising is usually the fastest method when the quadratic has integer roots and the coefficients are reasonably simple. The general steps are: rearrange the equation into the form ax² + bx + c = 0, factorise the left-hand side, then set each factor equal to zero.

当二次方程有整数根且系数比较简单时，因式分解通常是最快的方法。一般步骤是：将方程整理为 ax² + bx + c = 0，将左边因式分解，然后令每个因式等于零。

Worked example: solve x² + 5x + 6 = 0. The left-hand side factorises to (x + 2)(x + 3), so (x + 2)(x + 3) = 0. Setting each factor to zero gives x + 2 = 0 or x + 3 = 0, so x = -2 or x = -3.

例题：解 x² + 5x + 6 = 0。左边因式分解为 (x + 2)(x + 3)，所以 (x + 2)(x + 3) = 0。令每个因式为零，得到 x + 2 = 0 或 x + 3 = 0，因此 x = -2 或 x = -3。

Another example: solve 2x² – 3x – 5 = 0. Factorising gives (2x – 5)(x + 1) = 0. Therefore 2x – 5 = 0 gives x = 5/2, and x + 1 = 0 gives x = -1.

另一个例子：解 2x² – 3x – 5 = 0。因式分解得到 (2x – 5)(x + 1) = 0。因此 2x – 5 = 0 得到 x = 5/2，x + 1 = 0 得到 x = -1。

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5. Completing the Square | 配方法

Completing the square is a method that rewrites a quadratic expression as a squared binomial plus or minus a constant. The key identity for a monic quadratic is x² + bx = (x + b/2)² – (b/2)².

配方法是一种将二次表达式改写为一个平方二项式加上或减去一个常数的方法。对于首项系数为 1 的二次式，关键恒等式为 x² + bx = (x + b/2)² – (b/2)²。

For example, to solve x² + 6x + 2 = 0, rewrite x² + 6x as (x + 3)² – 9. The equation becomes (x + 3)² – 9 + 2 = 0, so (x + 3)² – 7 = 0. Then (x + 3)² = 7, giving x + 3 = ±√7, so x = -3 ± √7.

例如，解 x² + 6x + 2 = 0，将 x² + 6x 改写为 (x + 3)² – 9。方程变为 (x + 3)² – 9 + 2 = 0，所以 (x + 3)² – 7 = 0。然后 (x + 3)² = 7，得到 x + 3 = ±√7，因此 x = -3 ± √7。

If a ≠ 1, first divide the whole equation by a or factor a out of the variable terms. Completing the square is particularly useful for deriving the quadratic formula and for writing quadratics in vertex form.

如果 a ≠ 1，需要先将整个方程除以 a，或将变量项中的 a 提出来。配方法在推导求根公式和将二次式写成顶点式时特别有用。

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6. The Quadratic Formula | 求根公式

The quadratic formula works for every quadratic equation, even when factorising is difficult or impossible using integers. For ax² + bx + c = 0, the solutions are given by:

求根公式适用于所有二次方程，即使不能用整数进行因式分解。对于 ax² + bx + c = 0，解由以下公式给出：

x = (-b ± √(b² – 4ac)) / 2a

Worked example: solve 2x² – 3x – 1 = 0. Here a = 2, b = -3 and c = -1. Substituting into the formula gives x = (3 ± √((-3)² – 4 × 2 × (-1))) / 4 = (3 ± √(9 + 8)) / 4 = (3 ± √17) / 4. The solutions are x = (3 + √17)/4 or x = (3 – √17)/4.

例题：解 2x² – 3x – 1 = 0。这里 a = 2，b = -3，c = -1。代入公式得到 x = (3 ± √((-3)² – 4 × 2 × (-1))) / 4 = (3 ± √(9 + 8)) / 4 = (3 ± √17) / 4。解为 x = (3 + √17)/4 或 x = (3 – √17)/4。

When using the formula, always write down the values of a, b and c first, and use brackets around negative numbers to avoid sign errors.

使用公式时，一定要先写出 a、b、c 的值，并在负数处使用括号，以避免符号错误。

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7. The Discriminant | 判别式

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

判别式是求根公式中平方根下的部分：Δ = b² – 4ac。它可以在不完全解方程的情况下告诉我们根的性质。

• 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, meaning the parabola touches the x-axis.
• 如果 Δ = 0，方程有一个重复实根，即抛物线与 x 轴相切。
• If Δ < 0, there are no real roots, so the parabola does not cross the x-axis.
• 如果 Δ < 0，方程没有实根，因此抛物线不与 x 轴相交。

For example, for x² – 4x + 5 = 0, Δ = (-4)² – 4 × 1 × 5 = 16 – 20 = -4, so the equation has no real solutions. This is a very common IGCSE question type.

例如，对于 x² – 4x + 5 = 0，Δ = (-4)² – 4 × 1 × 5 = 16 – 20 = -4，所以该方程没有实数解。这是 IGCSE 中非常常见的题型。

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8. Graphical Interpretation | 图像解释

The graph of y = ax² + bx + c is a parabola. The solutions of the equation ax² + bx + c = 0 are the x-coordinates where the parabola intersects the x-axis. These points are also called the roots or x-intercepts.

y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的解就是抛物线与 x 轴交点的 x 坐标。这些点也称为根或 x 轴截距。

If the discriminant is positive, the parabola crosses the x-axis twice. If it is zero, the parabola touches the x-axis once. If it is negative, the parabola lies entirely above or entirely below the x-axis, depending on the sign of a.

如果判别式为正，抛物线与 x 轴相交两次。如果判别式为零，抛物线与 x 轴相切一次。如果判别式为负，抛物线完全位于 x 轴上方或下方，具体取决于 a 的符号。

The axis of symmetry is x = -b / 2a, and the vertex has x-coordinate -b / 2a. Completing the square gives the vertex form y = a(x – h)² + k, where (h, k) is the vertex.

对称轴为 x = -b / 2a，顶点的 x 坐标也是 -b / 2a。配方法可得到顶点式 y = a(x – h)² + k，其中 (h, k) 为顶点。

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9. Choosing the Best Method | 选择最佳方法

Different methods are suited to different situations. Factorising is best for simple quadratics with integer roots. The quadratic formula is best for non-factorable quadratics and for checking answers. Completing the square is best for finding the vertex or deriving the quadratic formula.

不同的方法适用于不同情况。因式分解最适合有整数根的简单二次方程。求根公式最适合不能因式分解的二次方程以及检查答案。配方法最适合求顶点或推导求根公式。

Method	Best when	Limitations
Factorising | 因式分解	Integer roots, simple coefficients | 整数根、简单系数	Not always possible | 并非总能分解
Quadratic formula | 求根公式	Any quadratic | 任何二次方程	More arithmetic steps | 计算步骤较多
Completing the square | 配方法	Vertex form, proofs | 顶点式、证明	Longer for messy a values | a 值复杂时较繁琐

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10. Common Mistakes and Exam Tips | 常见错误与考试提示

One common mistake is dividing both sides by x. For example, in x² = 5x, dividing by x gives x = 5, but the solution x = 0 is lost. Instead, write x² – 5x = 0, then factorise to get x(x – 5) = 0, so x = 0 or x = 5.

一个常见错误是两边同时除以 x。例如，在 x² = 5x 中，除以 x 得到 x = 5，但解 x = 0 丢失了。正确做法是写成 x² – 5x = 0，然后因式分解得到 x(x – 5) = 0，所以 x = 0 或 x = 5。

Another common error is mishandling negative signs in the quadratic formula, especially when b is negative. Always use brackets: x = (-(-3) ± √((-3)² – 4(2)(-1))) / 4.

另一个常见错误是在求根公式中处理负号不当，尤其是当 b 为负数时。始终使用括号：x = (-(-3) ± √((-3)² – 4(2)(-1))) / 4。

Also remember to rearrange the equation to standard form before identifying a, b and c. If the equation is given as 3x² = 2x – 1, rewrite it as 3x² – 2x + 1 = 0 first.

此外，在识别 a、b、c 之前，要先将方程整理为标准形式。如果方程给出为 3x² = 2x – 1，应先改写为 3x² – 2x + 1 = 0。

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11. Worked Examples | 解题示例

Example 1: solve x² – 7x + 10 = 0 by factorising. We need two numbers that multiply to 10 and add to -7: these are -2 and -5. So (x – 2)(x – 5) = 0, giving x = 2 or x = 5.

示例 1：用因式分解法解 x² – 7x + 10 = 0。我们需要两个数，乘积为 10，和为 -7：这两个数是 -2 和 -5。因此 (x – 2)(x – 5) = 0，得到 x = 2 或 x = 5。

Example 2: a rectangle has length 3 cm more than its width, and its area is 54 cm². Let the width be w cm, so the length is w + 3 cm. Area gives w(w + 3) = 54. Expanding gives w² + 3w – 54 = 0. Factorising gives (w + 9)(w – 6) = 0, so w = 6 or w = -9. Since width must be positive, w = 6 cm and the length is 9 cm.

示例 2：一个矩形的长比宽多 3 cm，面积为 54 cm²。设宽为 w cm，则长为 w + 3 cm。面积给出 w(w + 3) = 54。展开得到 w² + 3w – 54 = 0。因式分解得到 (w + 9)(w – 6) = 0，所以 w = 6 或 w = -9。由于宽必须为正，因此 w = 6 cm，长为 9 cm。

Example 3: use the discriminant to show that x² – 2x + 7 = 0 has no real roots. Here Δ = (-2)² – 4 × 1 × 7 = 4 – 28 = -24. Since Δ < 0, the equation has no real solutions.

示例 3：用判别式证明 x² – 2x + 7 = 0 没有实根。这里 Δ = (-2)² – 4 × 1 × 7 = 4 – 28 = -24。由于 Δ < 0，该方程没有实数解。

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12. Quick Revision Checklist | 快速复习清单

• Can you write a quadratic equation in the form ax² + bx + c = 0?
• 你能否将二次方程写成 ax² + bx + c = 0 的形式？
• Can you factorise simple and harder quadratics accurately?
• 你能否准确地对简单和较难的二次式进行因式分解？
• Can you apply the zero product principle correctly?
• 你能否正确应用零乘积原理？
• Can you complete the square and write the vertex form?
• 你能否完成配方并写出顶点式？
• Can you use the quadratic formula with correct substitution?
• 你能否正确代入并运用求根公式？
• Can you calculate and interpret the discriminant?
• 你能否计算并解释判别式？
• Can you connect roots to the graph of a parabola?
• 你能否将根与抛物线图像联系起来？

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