📚 Mastering IGCSE Quadratic Equations | 掌握 IGCSE 二次方程
e written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.
二次方程是 IGCSE 数学中最重要的代数技能之一。它们出现在纯代数题、图像草图以及实际建模中。二次方程总可以写成标准形式 ax² + bx + c = 0，其中 a、b、c 为常数且 a ≠ 0。
1. Recognising Quadratic Equations | 识别二次方程
A quadratic equation contains an x² term as the highest power of x. If an equation has x³, √x, or 1/x, it is not quadratic in standard form.
二次方程含有 x² 项，并且它是 x 的最高次幂。如果方程含有 x³、√x 或 1/x，它就不是标准形式的二次方程。
c = 0, where a ≠ 0. The coefficient a can be positive or negative, but it cannot be zero.
一般形式为 ax² + bx + c = 0，其中 a ≠ 0。系数 a 可正可负，但不能为零。
2. Solving by Factorising | 因式分解法
in the form ax² + bx + c = 0, then factorise the left-hand side into two linear brackets.
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。首先将方程改写为 ax² + bx + c = 0 的形式，然后将左边分解为两个一次因式。
0 becomes (x + 2)(x + 3) = 0. Then solve x + 2 = 0 or x + 3 = 0, giving x = −2 or x = −3.
例如，x² + 5x + 6 = 0 可化为 (x + 2)(x + 3) = 0。然后解 x + 2 = 0 或 x + 3 = 0，得到 x = −2 或 x = −3。
Always check that the equation equals zero before factorising. If it equals another number, move all terms to one side first.
因式分解前务必检查方程是否等于零。如果等于其他数，先把所有项移到一边。
3. The Zero Product Property | 零乘积性质
zero, then at least one factor must be zero. Symbolically, if mn = 0, then m = 0 or n = 0.
零乘积性质指出：如果两个因式相乘为零，那么至少有一个因式为零。用符号表示为：若 mn = 0，则 m = 0 或 n = 0。
atic is written as (px + q)(rx + s) = 0, each bracket can be set equal to zero separately.
这就是因式分解法成立的原因。一旦二次方程写成 (px + q)(rx + s) = 0，每个括号就可以分别设为零来求解。
4. Completing the Square | 配方法
Completing the square transforms ax² + bx + c into the form a(x + p)² + q. This is useful for finding the vertex of a parabola and for solving equations that do not factorise neatly.
配方法将 ax² + bx + c 转化为 a(x + p)² + q 的形式。这对求抛物线的顶点以及解不易因式分解的方程非常有用。
monic quadratic x² + bx, add and subtract (b/2)². That is, x² + bx = (x + b/2)² − (b/2)².
对于首项系数为 1 的二次式 x² + bx，加上并减去 (b/2)²。也就是 x² + bx = (x + b/2)² − (b/2)²。
Example: x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7. This gives the vertex (−3, −7).
例如：x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7。由此可得顶点为 (−3, −7)。
5. The Quadratic Formula | 求根公式
on, even when factorising is difficult or impossible. For ax² + bx + c = 0, the roots are:
求根公式可以解任何二次方程，即使因式分解很难或无法进行。对于 ax² + bx + c = 0，根为：
x = (−b ± √b² − 4ac) ÷ 2a
Substitute the values of a, b and c carefully, including negative signs. Then simplify the square root and divide by 2a.
代入 a、b、c 的值时要小心，包括负号。然后化简平方根并除以 2a。
This formula is provided on some IGCSE formula sheets, but you must know how to use it accurately under time pressure.
该公式在某些 IGCSE 公式表中会提供，但你必须在时间压力下准确使用它。
6. The Discriminant and Number of Roots | 判别式与根的个数
ide the square root: Δ = b² − 4ac. It tells you how many real solutions the quadratic has.
判别式是平方根内的表达式：Δ = b² − 4ac。它告诉你二次方程有多少个实数解。
If Δ > 0, there are two distinct real roots.
如果 Δ > 0，则有两个不同的实数根。
If Δ = 0, there is one repeated real root, so the graph touches the x-axis.
如果 Δ = 0，则有一个重复的实数根，图像与 x 轴相切。
If Δ < 0, there are no real roots, so the graph does not cross the x-axis.
如果 Δ < 0，则没有实数根，图像不与 x 轴相交。
Exam questions often ask you to use the discriminant to determine the number of solutions without actually solving the equation.
考试题经常要求你使用判别式来判断解的个数，而无需真正解方程。
7. Choosing the Best Method | 选择最佳方法
If the quadratic has small integer coefficients and factorises quickly, use factorising. If you need the vertex or turning point, completing the square is the best choice.
如果二次方程系数较小且能快速因式分解，就使用因式分解法。如果需要求顶点或转折点，配方法是最佳选择。
Use the quadratic formula when the coefficients are large, negative, or non-integer, or when factorising is not obvious. The formula always works.
当系数较大、为负数或非整数，或因式分解不明显时，使用求根公式。该公式始终有效。
In exams, check the wording: 'solve' allows any method; 'give your answer to 2 decimal places' often signals the formula or completing the square.
考试时注意题目措辞：'求解'允许使用任何方法；'答案保留两位小数'通常提示使用公式或配方法。
8. Quadratic Graphs and Key Features | 二次函数图像与关键特征
. If a > 0, it opens upwards with a minimum point. If a < 0, it opens downwards with a maximum point.
y = ax² + bx + c 的图像是一条抛物线。若 a > 0，开口向上，有最低点；若 a < 0，开口向下，有最高点。
ertex lies on this line, and its y-coordinate is found by substituting this x-value back into the equation.
对称轴为 x = −b ÷ 2a。顶点位于该直线上，其 y 坐标通过将该 x 值代回方程求得。
of the graph y = ax² + bx + c. If the discriminant is negative, there are no x-intercepts.
方程 ax² + bx + c = 0 的根是图像 y = ax² + bx + c 与 x 轴的交点横坐标。如果判别式为负，则没有 x 轴截距。
9. Real-World Applications | 实际应用
Quadratic models appear in projectile motion, area problems, and profit optimisation. A typical question gives a situation and asks you to form a quadratic equation, solve it, and interpret the roots.
二次模型出现在抛体运动、面积问题和利润优化中。典型题目会给出情境，要求你建立二次方程、求解并解释根的意义。
its area is 40, then x(x + 3) = 40. Expand and rearrange to x² + 3x − 40 = 0, then solve.
例如，一个矩形的宽为 x，长为 x + 3，面积为 40，则 x(x + 3) = 40。展开并移项得 x² + 3x − 40 = 0，然后求解。
Always check whether a negative root makes sense in context. Lengths, times and quantities are usually non-negative, so discard impossible values.
一定要检查负根在情境中是否有意义。长度、时间和数量通常为非负，因此舍弃不合理的值。
10. Common Exam Mistakes | 常见考试错误
re factorising. For example, x² + 5x + 6 = 12 must be rearranged to x² + 5x − 6 = 0 first.
许多学生忘记在因式分解前
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