📚 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 are a central topic in IGCSE Mathematics. They appear in algebra, coordinate geometry, mensuration and real-world modelling problems. This guide covers the standard form, factorising, completing the square, the quadratic formula, the discriminant and graphical interpretation, with worked examples and exam tips.
二次方程是 IGCSE 数学的核心主题。它们出现在代数、坐标几何、测量和实际问题建模中。本指南涵盖标准形式、因式分解、配方法、求根公式、判别式和图形解释，并配有例题和考试技巧。
1. What is a Quadratic Equation? | 什么是二次方程？
hat can be written in the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.
一元二次方程是可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数，且 a ≠ 0。
The word 'quadratic' comes from 'quadratus', meaning square, because the highest power of the variable x is 2.
“二次”一词源于“平方”，因为变量 x 的最高次数是 2。
a ≠ 0 is essential. For example, 3x² + 2x − 5 = 0 is quadratic, but 2x + 5 = 0 is linear.
如果 a = 0，方程就变为一次方程，因此 a ≠ 0 这一条件非常重要。例如，3x² + 2x − 5 = 0 是二次方程，而 2x + 5 = 0 是一次方程。
Quadratic equations can have two real roots, one repeated real root, or no real roots, depending on the value of the discriminant.
根据判别式的值，二次方程可以有两个实数根、一个重根或没有实数根。
2. Standard Form and Key Terms | 标准形式与关键术语
nge a quadratic into the standard form ax² + bx + c = 0 before choosing a solution method.
在选择解法之前，一定要把二次方程整理成标准形式 ax² + bx + c = 0。
- a is the coefficient of x², b is the coefficient of x, and c is the constant term.
a 是 x² 的系数，b 是 x 的系数，c 是常数项。
nd direction of the parabola, while c gives the y-intercept of the graph y = ax² + bx + c.
首项系数 a 决定抛物线的形状和开口方向，而 c 给出图像 y = ax² + bx + c 的 y 轴截距。
2. Identifying these values correctly is the first step when using the quadratic formula.
例如，在 4x² − 7x + 2 = 0 中，a = 4，b = −7，c = 2。正确识别这些值是使用求根公式的第一步。
Many mistakes come from copying the wrong sign for b or c, especially when negative numbers are involved.
许多错误来自抄错 b 或 c 的符号，尤其是在涉及负数时。
3. Solving by Factorising | 因式分解法
Factorising is usually the fastest method when the quadratic has simple integer roots.
当二次方程具有简单的整数根时，因式分解通常是最快的方法。
bx + c = (px + q)(rx + s), then the roots are found by setting each factor equal to zero.
如果 ax² + bx + c = (px + q)(rx + s)，则令每个因式等于零即可求出根。
e x² − 5x + 6 = 0, find two numbers that multiply to 6 and add to −5: these are −2 and −3.
当 a = 1 时，寻找两个数，使它们的乘积为 c，和为 b。例如，解 x² − 5x + 6 = 0 时，找到两个数，乘积为 6，和为 −5：它们是 −2 和 −3。
x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 or x = 3
x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 或 x = 3。
or or the ac method. For example, 2x² + 7x + 3 = (2x + 1)(x + 3), giving x = −½ or x = −3.
当 a 不等于 1 时，可以使用试错法或 ac 法。例如，2x² + 7x + 3 = (2x + 1)(x + 3)，得到 x = −½ 或 x = −3。
You must check that expanding the factors returns the original quadratic; sign errors are common.
必须检查展开因式是否能还原原方程；符号错误很常见。
4. Special Cases: Difference of Two Squares | 特殊情形：平方差
quadratic has no x term, it may be a difference of two squares: a² − b² = (a + b)(a − b).
当二次方程没有 x 项时，它可能是平方差：a² − b² = (a + b)(a − b)。
For example, x² − 9 = 0 factorises as (x + 3)(x − 3), giving x = −3 or x = 3.
例如，x² − 9 = 0 可因式分解为 (x + 3)(x − 3)，得到 x = −3 或 x = 3。
on case is 4x² − 25 = 0, which factorises as (2x + 5)(2x − 5), giving x = −5/2 or x = 5/2.
另一个常见例子是 4x² − 25 = 0，它可以因式分解为 (2x + 5)(2x − 5)，得到 x = −5/2 或 x = 5/2。
Do not try to apply this pattern to a sum of squares such as x² + 9, which has no real roots.
不要把这个公式用于平方和，例如 x² + 9，它没有实数根。
Recognising special forms saves time in the exam because you can skip several steps of trial factorising.
识别特殊形式可以在考试中节省时间，因为你可以跳过试错因式分解的若干步骤。
5. Completing the Square | 配方法
Completing the square rewrites ax² + bx + c in the form a(x + p)² + q, which is useful when factorising is difficult.
配方法把 ax² + bx + c 写成 a(x + p)² + q 的形式，在因式分解困难时非常有用。
ract (b/2)² to create a perfect square. The key identity is x² + bx = (x + b/2)² − (b/2)².
对于 x² + bx，加上并减去 (b/2)² 以构造完全平方。关键恒等式为 x² + bx = (x + b/2)² − (b/2)²。
For example, solve x² + 6x + 1 = 0. Here b = 6, so (b/2)² = 9.
例如，解方程 x² + 6x + 1 = 0。这里 b = 6，所以 (b/2)² = 9。
x² + 6x + 1 = 0 → (x + 3)² − 9 + 1 = 0 → (x + 3)² = 8 → x = −3 ± √8
x² + 6x + 1 = 0 → (x + 3)² − 9 + 1 = 0 → (x + 3)² = 8 → x = −3 ± √8。
t first. For example, 2x² + 8x + 5 = 2(x² + 4x) + 5 = 2[(x + 2)² − 4] + 5 = 2(x + 2)² − 3.
当 x² 的系数不是 1 时，要先把它提出来。例如，2x² + 8x + 5 = 2(x² + 4x) + 5 = 2[(x + 2)² − 4] + 5 = 2(x + 2)² − 3。
This method also gives the coordinates of the turning point of a quadratic graph: the vertex is (−p, q) for a(x + p)² + q.
该方法还能给出二次函数图像顶点的坐标：对于 a(x + p)² + q，顶点为 (−p, q)。
6. Quadratic Formula | 求根公式
The quadratic formula solves any quadratic equation ax² + bx + c = 0, provided a ≠ 0.
求根公式可以解任意二次方程 ax² + bx + c = 0，前提是 a ≠ 0。
x = (−b ± √b² − 4ac) / 2a
x = (−b ± √b² − 4ac) / 2a。
It is especially useful when the quadratic does not factorise over integers, or when the coefficients are large or awkward.
当二次方程不能在整数范围内因式分解，或者系数较大或不易处理时，该公式尤其有用。
For example, solve 2x² + 3x − 2 = 0. Here a = 2, b = 3 and c = −2.
例如，解方程 2x² + 3x − 2 = 0。这里 a = 2，b = 3，c = −2。
2x² + 3x − 2 = 0 → x = (−3 ± √9 + 16) / 4 = (−3 ± 5) / 4 → x = ½ or x = −2
2x² + 3x − 2 = 0 → x = (−3 ± √9 + 16) / 4 = (−3 ± 5) / 4 → x = ½ 或 x = −2。
Always write the formula first, substitute carefully, and simplify the square root fully. If the square root is not a perfect square, leave your answer in surd form unless the question asks for a decimal.
一定要先写出公式，仔细代入，并完全化简平方根。如果平方根不是完全平方数，除非题目要求小数，否则保留根式形式。
7. Discriminant and Nature of Roots | 判别式与根的性质
scriminant is the expression D = b² − 4ac inside the square root of the quadratic formula.
判别式是求根公式中平方根内的表达式 D = b² − 4ac。
- If D > 0, there are two distinct real roots.
- If D = 0, there is one repeated real root.
- If D < 0, there are no real roots.
如果 D > 0，有两个不同的实数根；如果 D = 0，有一个重根；如果 D < 0，没有实数根。
x² − 6x + 9 = 0 has D = (−6)² − 4(1)(9) = 36 − 36 = 0, so it has one repeated root: x = 3.
例如，x² − 6x + 9 = 0 的判别式 D = (−6)² − 4(1)(9) = 36 − 36 = 0，所以它有一个重根：x = 3。
You can use the discriminant to decide how many times a parabola crosses the x-axis: two crossings, one touching point, or no crossing.
你可以用判别式判断抛物线与 x 轴相交的次数：两次相交、一个切点或没有相交。
u to find the value of k for which an equation has equal roots; set D = 0 and solve for k.
IGCSE 题目经常要求你求出方程有重根时 k 的值；令 D = 0 并解出 k。
8. Graphical Interpretation | 图形解释
= ax² + bx + c is a parabola. Its x-intercepts are the real solutions of ax² + bx + c = 0.
y = ax² + bx + c 的图像是一条抛物线。它与 x 轴的交点就是方程 ax² + bx + c = 0 的实数解。
ards; if a < 0, it opens downwards. The axis of symmetry is the vertical line x = −b/(2a).
如果 a > 0，抛物线开口向上；如果 a < 0，抛物线开口向下。对称轴是竖直直线 x = −b/(2a)。
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