📚 Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | IGCSE 数学：二次方程解法全解
In IGCSE Mathematics, quadratic equations appear across algebra, graphs, mensuration and word problems. A quadratic equation is any equation of degree 2, and mastering three solution methods - factorising, completing the square and the quadratic formula - gives you a reliable toolkit for both Core and Extended papers. This article explains each method step by step, with examples, common mistakes and exam tips.
在 IGCSE 数学中，二次方程广泛出现在代数、图像、几何测量和应用题中。二次方程是次数为 2 的方程，掌握因式分解、配方法和求根公式三种解法，可以为你应对核心卷和拓展卷提供可靠的工具。本文逐步讲解每种方法，并配有例题、常见错误和考试技巧。
1. What Is a Quadratic Equation? | 什么是二次方程？
A quadratic equation in one variable is any equation that can be written in the general form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0 tion becomes linear, not quadratic. The highest power of the variable x must be exactly 2.
一元二次方程是可以写成一般形式 ax² + bx + c = 0 的方程，其中 a、b、c 是常数，且 a ≠ 0。如果 a = 0，方程就变成一次方程，而不是二次方程。变量 x 的最高次数必须正好是 2。
the highest power of x is 2. Equations such as x³ + x² = 0 or x + 4 = 0 are not quadratic.
例如，2x² - 5x + 3 = 0、x² - 9 = 0 和 3x² + 2x = 0 都是二次方程，因为 x 的最高次数是 2。像 x³ + x² = 0 或 x + 4 = 0 这样的方程就不是二次方程。
- 2)(x + 5) = 0 and x² = 7x - 12 are both quadratic equations once expanded or rearranged.
二次方程在整理前可能形式不同。例如，(x - 2)(x + 5) = 0 和 x² = 7x - 12 在展开或移项后都是二次方程。
2. Standard Form and Key Terms | 标准形式与关键术语
Before solving, always rearrange the equation into standard form ax² + bx + c = 0 s a, b and c correctly. For example, in 4x² - 7x + 2 = 0, we have a = 4, b = -7 and c = 2.
在求解之前，要先将方程整理成标准形式 ax² + bx + c = 0。这样才能正确识别系数 a、b 和 c。例如，在 4x² - 7x + 2 = 0 中，a = 4，b = -7，c = 2。
adratic as long as a ≠ 0. For example, 5x² - 20 = 0 has b = 0, and 2x² + 3x = 0 has c = 0.
项 ax² 称为二次项，bx 称为一次项，c 称为常数项。当 b 或 c 为零时，只要 a ≠ 0，方程仍然是二次方程。例如，5x² - 20 = 0 中 b = 0，2x² + 3x = 0 中 c = 0。
Writing the equation in standard form also helps you compare coefficients when using the quadratic formula or completing the square. Never start solving until all terms are on one side and zero is on the other side.
将方程写成标准形式还有助于在使用求根公式或配方法时比较系数。在开始求解之前，一定要将所有项移到一边，使另一边为零。
3. Solving by Factorising | 因式分解法
Factorising works well when the quadratic expression can be written as a product of two linear factors. Since mn = 0 r equal to zero and solve. This method is often the quickest when the coefficients are small integers.
当二次表达式可以写成两个一次因式的乘积时，因式分解法非常有效。因为 mn = 0 意味着 m = 0 或 n = 0，所以我们可以令每个因式等于零并求解。当系数是较小的整数时，这种方法通常最快。
x² + 5x + 6 = 0 → (x + 2)(x + 3) = 0 → x = -2 or x = -3
例如，x² + 5x + 6 = 0 可分解为 (x + 2)(x + 3) = 0，因此 x = -2 或 x = -3。
x = -1/2 or x = -3. Always expand your factors to check they give the original expression.
对于 a ≠ 1 的二次式，可用观察法或拆中项法。例如，2x² + 7x + 3 = 0 可分解为 (2x + 1)(x + 3) = 0，解得 x = -1/2 或 x = -3。务必把因式展开，检查是否得到原表达式。
Special forms such as the difference of two squares can be factorised directly: x² - 16 = 0 becomes (x + 4)(x - 4) = 0, so x = 4 or x = -4. Recognising these patterns saves time in exams.
特殊形式如平方差公式可以直接因式分解：x² - 16 = 0 变为 (x + 4)(x - 4) = 0，所以 x = 4 或 x = -4。在考试中识别这些模式可以节省时间。
4. Solving by Completing the Square | 配方法
Completing the square transforms ax² + bx + c into the form a(x + p)² + q f x, square it, and add and subtract that value. This method is especially useful when the quadratic does not factorise easily.
配方法将 ax² + bx + c 转化为 a(x + p)² + q 的形式。当 a = 1 时，取 x 项系数的一半，平方后在式子中加上并减去该值。当二次式不容易因式分解时，这种方法尤其有用。
x² + 6x + 5 = 0 → x² + 6x + 9 - 9 + 5 = 0 → (x + 3)² - 4 = 0
例如，x² + 6x + 5 = 0 可写成 (x + 3)² - 4 = 0。
because both 2² and (-2)² equal 4. Remember to consider both positive and negative square roots.
然后求解：(x + 3)² = 4，因此 x + 3 = ±2，得到 x = -1 或 x = -5。出现 ± 符号是因为 2² 和 (-2)² 都等于 4。记住要考虑正负两个平方根。
hen complete the square inside the bracket: 2[(x + 2)² - 4] + 5 = 0, so 2(x + 2)² - 3 = 0.
如果 a ≠ 1，先对 x² 项和 x 项提取公因数 a。例如，2x² + 8x + 5 = 0 变为 2(x² + 4x) + 5 = 0，然后在括号内配方：2[(x + 2)² - 4] + 5 = 0，所以 2(x + 2)² - 3 = 0。
5. The Quadratic Formula | 求根公式
When factorising is difficult, use the quadratic formula. For ax² + bx + c = 0, the solutions are given by the formula below. This formula works for all quadratic equations, including those with irrational or complex roots.
当因式分解比较困难时，可以使用求根公式。对于 ax² + bx + c = 0，解由下面的公式给出。该公式适用于所有二次方程，包括根为无理数或复数的情况。
x = (-b ± √b² - 4ac) / (2a)
nd c carefully, paying attention to negative signs. Use brackets around negative numbers to avoid sign errors.
该公式是通过对一般二次式 ax² + bx + c = 0 配方推导出来的。代入 a、b、c 时一定要仔细，尤其是负号。负数要用括号括起来，以避免符号错误。
Example: Solve 3x² + 5x - 2 = 0. Here a = 3, b = 5, c = -2, so x = (-5 ± √25 + 24) / 6 = (-5 ± 7) / 6, giving x = 1/3 or x = -2.
例如，解 3x² + 5x - 2 = 0，这里 a = 3，b = 5，c = -2，所以 x = (-5 ± √25 + 24) / 6 = (-5 ± 7) / 6，得到 x = 1/3 或 x = -2。
Always simplify the fraction fully. If the square root simplifies to an integer, write the two answers separately. If the root is not exact, leave your answer in surd form unless the question asks for decimals.
一定要将分数完全化简。如果平方根可以化为整数，就把两个答案分开写。如果根号不能开尽，除非题目要求小数，否则保留根式形式。
6. Discriminant and Nature of Roots | 判别式与根的性质
The expression Δ = b² - 4ac is called the discriminant. It tells you the nature of the roots without solving the equation. This is a common short-answer topic in IGCSE exams.
表达式 Δ = b² - 4ac 称为判别式。它可以在不求解方程的情况下判断根的性质。这是 IGCSE 考试中常见的简答题考点。
- 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 one repeated real root.
- 如果 Δ = 0，方程有一个重根（两个相等的实根）。
- If Δ < 0, there are no real roots; the roots are complex in Further Pure Mathematics.
- 如果 Δ < 0，方程没有实数根；在进阶纯数中根为复数。
lutions. In x² - 6x + 9 = 0, Δ = 36 - 36 = 0, so the equation has one repeated root x = 3.
例如，在 x² - 4x + 5 = 0 中，Δ = (-4)² - 4(1)(5) = 16 - 20 = -4 < 0，所以该方程没有实数解。在 x² - 6x + 9 = 0 中，Δ = 36 - 36 = 0，所以方程有一个重根 x = 3。
often ask you to find the value of k for which a quadratic has equal roots. Set Δ = 0 and
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