📚 Mastering Quadratic Equations | 掌握一元二次方程
Quadratic equations are one of the most important algebra topics in IGCSE Mathematics. They appear in algebraic manipulation, graph sketching, and real-world problems. Mastering the methods of solving them will give you confidence in both Core and Extended papers.
二次方程是 IGCSE 数学代数中最重要的主题之一。它们会出现在代数运算、图像绘制和实际应用问题中。掌握二次方程的解法能让你在 Core 和 Extended 试卷中更有把握。
This guide walks through every method you are expected to know - factorising, completing the square and the quadratic formula - and connects them to the discriminant, to graph work, and to the word problems that examiners love. Work through the sections in order, then use the final checklist to test yourself.
本指南会依次讲解你需要掌握的全部方法——因式分解法、配方法和求根公式，并把它们与判别式、图像题以及考官偏爱的应用题联系起来。建议按顺序阅读，最后用文末的复习清单自测。
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 standard form below, where a, b and c are constants and a ≠ 0.
一元二次方程是可以写成下列标准形式的方程，其中 a、b、c 为常数且 a ≠ 0。
ax² + bx + c = 0, a ≠ 0
erm, and c is the constant term. If a = 0, the equation becomes linear, not quadratic, so the condition a ≠ 0 is essential.
x 的最高次数是 2，这正是它被称为二次方程而不是一次方程的原因。ax² 称为二次项，bx 称为一次项，c 称为常数项。如果 a = 0，方程就变成了一次方程，不再是二次方程，因此 a ≠ 0 这个条件必不可少。
Here are some typical examples, with the values of a, b and c identified:
下面是几个典型例子，并标出了 a、b、c 的取值：
- x² - 5x + 6 = 0, so a = 1, b = -5, c = 6.
x² - 5x + 6 = 0，因此 a = 1，b = -5，c = 6。 - 3x² + 2x - 8 = 0, so a = 3, b = 2, c = -8.
3x² + 2x - 8 = 0，因此 a = 3，b = 2，c = -8。 - x² = 9 must first be rearranged to x² - 9 = 0.
x² = 9 必须先整理为 x² - 9 = 0。 - (x + 2)(x - 3) = 2x must first be expanded and rearranged to x² - 3x - 6 = 0.
(x + 2)(x - 3) = 2x 必须先展开并整理为 x² - 3x - 6 = 0。
linear and the other is cubic. Before solving anything, always rearrange so that one side is exactly zero.
注意像 2x + 1 = 0 或 x³ - x = 0 这样的式子并不是二次方程：前者是一次方程，后者是三次方程。在求解之前，一定要先整理，使等式一边恰好为 0。
2. The Zero Product Property | 零乘积性质
then at least one of the factors must be zero. In symbols, if ab = 0 then a = 0 or b = 0.
零乘积性质是指：如果两个或多个因式的乘积等于零，那么其中至少有一个因式等于零。用符号表示就是：若 ab = 0，则 a = 0 或 b = 0。
if ab = 0, then a = 0 or b = 0
that multiply to give 6. That is why the very first step in solving a quadratic is always to move every term to one side so the other side is 0.
这个性质只对零成立。如果 ab = 6，你不能得出 a = 6 或 b = 6，因为乘积为 6 的数对有无穷多组。这正是解二次方程时第一步永远是把所有项移到一边、使另一边为 0 的原因。
7. Both values are valid answers, and an exam answer that gives only one of them loses marks.
例如，(x - 5)(x + 7) = 0 立刻可以推出 x - 5 = 0 或 x + 7 = 0，因此 x = 5 或 x = -7。两个值都是正确答案，考试中只写出其中一个会丢分。
3. Solving by Factorising | 因式分解法
x - p)(x - q) = 0 and then set each bracket equal to zero using the zero product property.
当二次式可以分解为两个一次因式时，这是最快的解法。把式子写成 (x - p)(x - q) = 0，然后利用零乘积性质，令每个括号等于零。
and add to -7, namely -3 and -4. Factorising gives (x - 3)(x - 4) = 0, so x = 3 or x = 4.
例如：解 x² - 7x + 12 = 0。我们需要两个数，它们的乘积是 12、和是 -7，即 -3 和 -4。因式分解得 (x - 3)(x - 4) = 0，因此 x = 3 或 x = 4。
The same four steps apply to every factorising question:
所有因式分解题都遵循同样的四个步骤：
- Step 1: Rearrange the equation into the form ax² + bx + c = 0.
第一步：把方程整理成 ax² + bx + c = 0 的形式。 - Step 2: Factorise the left-hand side completely.
第二步：把左边彻底因式分解。 - Step 3: Set each factor equal to zero.
第三步：令每个因式等于零。 - Step 4: Solve each linear equation and state both answers.
第四步：解出每个一次方程并写出两个答案。
econd, the difference of two squares: x² - 25 = 0 becomes (x - 5)(x + 5) = 0, so x = 5 or x = -5. Third, when a ≠ 1: 6x² + 11x - 35 = 0 factorises as (2x + 7)(3x - 5) = 0, giving x = -3.5 or x = 5/3.
要注意三种重要变化。第一，有公因式的情形：2x² - 8x = 0 化为 2x(x - 4) = 0，因此 x = 0 或 x = 4——如果两边同时除以 x，就会丢掉 x = 0 这个根。第二，平方差情形：x² - 25 = 0 化为 (x - 5)(x + 5) = 0，因此 x = 5 或 x = -5。第三，a ≠ 1 的情形：6x² + 11x - 35 = 0 分解为 (2x + 7)(3x - 5) = 0，得到 x = -3.5 或 x = 5/3。
gives 6x² - 10x + 21x - 35 = 6x² + 11x - 35, which confirms the factorisation is correct.
一定要检查因式分解后的式子展开后是否能还原为原式。展开 (2x + 7)(3x - 5) 得 6x² - 10x + 21x - 35 = 6x² + 11x - 35，说明分解正确。
4. Completing the Square | 配方法
he method behind the quadratic formula, and it is also the standard way to find the vertex of a parabola.
配方法把二次式改写成 (x + p)² + q = 0 的形式。它是求根公式背后的推导方法，也是求抛物线顶点的标准手段。
The key identity is that x² + bx can be written as (x + b/2)² - (b/2)². In words: halve the coefficient of x, square it, and subtract that square.
关键恒等式是：x² + bx 可以写成 (x + b/2)² - (b/2)²。用语言说就是：把一次项系数取一半，再平方，然后减去这个平方。
x² + bx = (x + b/2)² - (b/2)²
on becomes (x + 3)² - 9 + 1 = 0, that is (x + 3)² = 8. Taking square roots gives x + 3 = ±√8 = ±2√2, so x = -3 + 2√2 ≈ -0.172 or x = -3 - 2√2 ≈ -5.828.
例如：解 x² + 6x + 1 = 0。6 的一半是 3，3² = 9，所以 x² + 6x = (x + 3)² - 9。方程变为 (x + 3)² - 9 + 1 = 0，即 (x + 3)² = 8。开平方得 x + 3 = ±√8 = ±2√2，因此 x = -3 + 2√2 ≈ -0.172 或 x = -3 - 2√2 ≈ -5.828。
orm tells you immediately that the curve y = 2x² + 8x + 3 has a minimum point at (-2, -5).
当 a ≠ 1 时，先把系数 a 提取出来。对于 2x² + 8x + 3：2x² + 8x + 3 = 2(x² + 4x) + 3 = 2[(x + 2)² - 4] + 3 = 2(x + 2)² - 5。这个顶点式立刻告诉你曲线 y = 2x² + 8x + 3 的最低点是 (-2, -5)。
5. The Quadratic Formula | 求根公式
When a quadratic does not factorise with integers, use the quadratic formula. It solves every quadratic equation, whatever the values of a, b and c.
当二次式无法用整数因式分解时，就使用求根公式。无论 a、b、c 取什么值，它都能解出方程。
x = (-b ± √b² - 4ac) ÷ 2a
th a = 2, b = -4, c = -3. Then b² - 4ac = (-4)² - 4 × 2 × (-3) = 16 + 24 = 40, so x = (4 ± √40) ÷ 4 = (2 ± √10) ÷ 2. This gives x ≈ 2.58 or x ≈ -0.58 to 3 significant figures.
先写成标准形式再代入。例如：解 2x² - 4x - 3 = 0，其中 a = 2，b = -4，c = -3。则 b² - 4ac = (-4)² - 4 × 2 × (-3) = 16 + 24 = 40，所以 x = (4 ± √40) ÷ 4 = (2 ± √10) ÷ 2。精确到 3 位有效数字得 x ≈ 2.58 或 x ≈ -0.58。
symbol, because a quadratic normally has two roots; and divide the whole numerator by 2a, not just the square-root part.
三个好习惯可以避免绝大多数公式错误：注意 b 的符号要一起代入，因为 -(-4) = +4；不要忘记 ± 号，因为二次方程通常有两个根；以及要用 2a 去除整个分子，而不是只除根号那一部分。
6. The Discriminant | 判别式
The expression under the square root, b² - 4ac, is called the discriminant and is usually written as Δ. It tells you how many real roots a quadratic has, without solving it.
根号下的表达式 b² - 4ac 称为判别式，通常记作 Δ。不用解方程，它就能告诉你二次方程有多少个实根。
Δ = b² - 4ac
| Value of Δ / Δ 的取值 | Nature of roots / 根的性质 | Graph / 图像 |
| Δ > 0 | Two distinct real roots / 两个不同的实根 | Crosses the x-axis twice / 与 x 轴交于两点 |
| Δ = 0 | One repeated real root / 一个重根 | Touches the x-axis once / 与 x 轴相切于一点 |
| Δ < 0 | No real roots / 没有实根 | Does not meet the x-axis / 与 x 轴不相交 |
oots. Equal roots mean Δ = 0, so (-4)² - 4 × 1 × k = 0, that is 16 - 4k = 0, giving k = 4.
典型考题：求使 x² - 4x + k = 0 有相等实根的 k 值。有相等实根意味着 Δ = 0，所以 (-4)² - 4 × 1 × k = 0，即 16 - 4k = 0，得 k = 4。
A useful extra fact: if a, b and c are integers and Δ is a perfect square, then the quadratic factorises neatly with integer or simple fraction factors. If Δ is not a perfect square, the roots are irrational and you should use the formula, giving answers to 3 significant figures.
一个有用的附加结论：如果 a、b、c 都是整数，且 Δ 是完全平方数，那么这个二次式可以用整数或简单分数因式分解；如果 Δ 不是完全平方数，根就是无理数，这时应使用求根公式，并把答案保留到 3 位有效数字。
7. Sketching Quadratic Graphs | 二次函数图像的绘制
The graph of y = ax² + bx + c is a parabola. If a &
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