Solving Quadratic Equations: Factorising, Formula and Completing the Square | 解二次方程：因式分解、公式法与配方法

📚 Solving Quadratic Equations: Factorising, Formula and Completing the Square | 解二次方程：因式分解、公式法与配方法

Quadratic equations appear in almost every IGCSE Mathematics paper. Whether you are solving them by factorising, using the quadratic formula, or completing the square, a clear step-by-step approach will help you avoid mistakes and gain full marks.

二次方程几乎出现在每一份IGCSE数学试卷中。无论你是通过因式分解、公式法还是配方法来解题，清晰的分步思路都能帮助你避免错误、拿到满分。

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

A quadratic equation is a polynomial equation of degree 2. Its highest power of the unknown is x², and it can be written in the form ax² + bx + c = 0, where a, b and c are real numbers and a ≠ 0.

二次方程是未知数最高次数为2的多项式方程，可写成 ax² + bx + c = 0，其中a、b、c为实数，且a ≠ 0。

• If a = 0, the equation becomes linear, not quadratic. | 若a = 0，方程变为一次方程，而不是二次方程。
• The values of x that satisfy the equation are called roots or solutions. | 满足方程的x值称为根或解。

ax² + bx + c = 0 (a ≠ 0)

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2. Standard Form and Rearranging | 标准形式与整理方程

Before solving, always rearrange the equation so that one side is zero. For example, x² = 5x – 6 becomes x² – 5x + 6 = 0.

求解前，一定要将方程整理成一边为零的形式。例如，x² = 5x – 6 应写成 x² – 5x + 6 = 0。

Expanding brackets and collecting like terms are essential skills. Do not try to factorise before the equation is in standard form.

展开括号、合并同类项是基本技能。在方程化为标准形式之前不要急于因式分解。

Example: (x – 2)(x + 3) = 4. Expand first, then subtract 4: x² + x – 6 = 4, so x² + x – 10 = 0.

例如：(x – 2)(x + 3) = 4。先展开，然后两边减4：x² + x – 6 = 4，即 x² + x – 10 = 0。

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3. Solving by Factorising (when a = 1) | 因式分解法（a=1时）

When a = 1, look for two numbers that multiply to give c and add to give b. Then write (x + p)(x + q) = 0 and set each factor to zero.

当a = 1时，寻找两个数，使其乘积为c、和为b。然后写成 (x + p)(x + q) = 0，并令每个因式等于零。

For x² – 5x + 6 = 0, the numbers are -2 and -3 because (-2) × (-3) = 6 and (-2) + (-3) = -5. So (x – 2)(x – 3) = 0, giving x = 2 or x = 3.

对于 x² – 5x + 6 = 0，两个数是 -2 和 -3，因为 (-2) × (-3) = 6，(-2) + (-3) = -5。所以 (x – 2)(x – 3) = 0，解得 x = 2 或 x = 3。

• Always check that p + q = b and p × q = c. | 务必检查p + q = b 且 p × q = c。
• If c is negative, one factor is positive and the other is negative. | 若c为负，则一个因式为正，另一个为负。

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4. Solving by Factorising (when a ≠ 1) | 因式分解法（a≠1时）

When a ≠ 1, factorising is more challenging. One method is to multiply a and c, find two numbers whose product is ac and sum is b, then split the middle term and factor by grouping.

当a ≠ 1时，因式分解更具挑战性。一种方法是先计算a与c的乘积，找到两个数使它们的积为ac、和为b，然后拆分中间项并分组因式分解。

Example: 2x² + 7x + 3 = 0. Here ac = 2 × 3 = 6. The numbers are 1 and 6 because 1 × 6 = 6 and 1 + 6 = 7. Split 7x as x + 6x:

例如：2x² + 7x + 3 = 0。这里 ac = 2 × 3 = 6。两个数是1和6，因为1 × 6 = 6，1 + 6 = 7。将7x拆为x + 6x：

2x² + x + 6x + 3 = 0 → x(2x + 1) + 3(2x + 1) = 0 → (x + 3)(2x + 1) = 0

Thus x = -3 or x = -½.

因此 x = -3 或 x = -½。

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5. Solving by the Quadratic Formula | 公式法

The quadratic formula works for any quadratic equation, even when factorising is difficult. For ax² + bx + c = 0,

公式法适用于任何二次方程，即使难以因式分解。对于 ax² + bx + c = 0，

x = ( -b ± √(b² – 4ac) ) / (2a)

Substitute the values of a, b and c. Simplify the square root first, then calculate the two solutions.

代入a、b、c的值。先化简根号，再计算两个解。

Example: 2x² – 4x – 3 = 0. Here a = 2, b = -4, c = -3.

例如：2x² – 4x – 3 = 0，其中 a = 2，b = -4，c = -3。

x = ( 4 ± √(16 + 24) ) / 4 = ( 4 ± √40 ) / 4 = ( 4 ± 2√10 ) / 4 = ( 2 ± √10 ) / 2

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6. Solving by Completing the Square | 配方法

Completing the square writes a quadratic as a(x + p)² + q. For x² + bx, add and subtract (b/2)².

配方法将二次式写成 a(x + p)² + q 的形式。对于 x² + bx，需要加上并减去 (b/2)²。

Example: x² + 6x + 2 = 0. Since (6/2)² = 9, rewrite as (x + 3)² – 9 + 2 = 0, so (x + 3)² = 7. Then x + 3 = ±√7, giving x = -3 ± √7.

例如：x² + 6x + 2 = 0。因为 (6/2)² = 9，改写为 (x + 3)² – 9 + 2 = 0，即 (x + 3)² = 7。于是 x + 3 = ±√7，得 x = -3 ± √7。

• Remember the ± symbol when taking square roots. | 开平方时不要忘记±号。
• This method is especially useful for finding the turning point of a parabola. | 该方法特别适用于求抛物线顶点。

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7. The Discriminant | 判别式

The expression b² – 4ac is called the discriminant, often denoted by Δ. It tells you how many real roots the quadratic has.

表达式 b² – 4ac 称为判别式，常用 Δ 表示。它告诉我们二次方程有几个实数根。

Discriminant Δ | 判别式 Δ	Roots | 根的情况
Δ > 0	Two distinct real roots | 两个不同实数根
Δ = 0	Two equal real roots (a repeated root) | 两个相等实数根（重根）
Δ < 0	No real roots | 无实数根

For example, the equation x² + 2x + 3 = 0 has Δ = 4 – 12 = -8 < 0, so it has no real roots.

例如，方程 x² + 2x + 3 = 0 的 Δ = 4 – 12 = -8 < 0，因此没有实数根。

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8. Quadratic Graphs and Roots | 二次函数图像与根

The solutions of ax² + bx + c = 0 are the x-intercepts of the graph y = ax² + bx + c. The graph is a parabola that opens upward if a > 0 and downward if a < 0.

方程 ax² + bx + c = 0 的解就是图像 y = ax² + bx + c 的x轴交点。图像是抛物线：a > 0 时开口向上，a < 0 时开口向下。

• If Δ > 0, the graph crosses the x-axis at two points. | 若Δ > 0，图像与x轴有两个交点。
• If Δ = 0, the graph touches the x-axis at one point (the vertex). | 若Δ = 0，图像与x轴相切于一点（顶点）。
• If Δ < 0, the graph does not intersect the x-axis. | 若Δ < 0，图像与x轴没有交点。

x-coordinate of vertex = -b / (2a)

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9. Word Problems Using Quadratics | 二次方程应用题

Many IGCSE questions require you to form a quadratic equation from a real-life situation. Common contexts include area, motion, and number problems.

许多IGCSE题目要求你从实际情境中建立二次方程。常见背景包括面积、运动和数字问题。

Example: The length of a rectangle is 3 cm more than its width, and its area is 70 cm². Let width = x, then length = x + 3. So x(x + 3) = 70, giving x² + 3x – 70 = 0. Factorising: (x + 10)(x – 7) = 0, so x = -10 (reject) or x = 7. Width = 7 cm, length = 10 cm.

例：一个长方形的长比宽多3 cm，面积为70 cm²。设宽为x，则长为x + 3。因此 x(x + 3) = 70，得 x² + 3x – 70 = 0。因式分解：(x + 10)(x – 7) = 0，所以 x = -10（舍去）或 x = 7。宽=7 cm，长=10 cm。

Always reject negative answers if they do not make sense in the context.

如果负数答案在情境中没有意义，一定要舍去。

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10. Common Mistakes and Exam Tips | 常见错误与考试技巧

Below is a list of frequent errors and how to avoid them.

以下是一些常见错误及避免方法。

• Forgetting to rearrange the equation to zero before factorising. | 因式分解前忘记将方程整理成一边为零。
• Sign errors when substituting into the quadratic formula: pay attention to negative values of b and c. | 代入公式法时出现符号错误：注意b和c为负值的情况。
• Forgetting ± when taking square roots. | 开平方时忘记±。
• Not simplifying surds such as √40 to 2√10. | 没有将√40等根式化简为2√10。
• Not checking solutions by substituting back into the original equation. | 没有将解代回原方程进行检验。

Exam tip: If a question says “give your answer correct to 3 significant figures”, use the quadratic formula or a calculator rather than factorising.

考试技巧：如果题目要求“答案保留3位有效数字”，应使用公式法或计算器，而不是因式分解。

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