📚 Quadratic Equations: Factorising, Completing the Square and the Formula | IGCSE数学二次方程：因式分解、配方法与公式法
Quadratic equations lie at the heart of IGCSE algebra. They appear both as stand-alone solving questions and inside wider problems on graphs, areas, sequences and modelling. This revision guide explains the three main algebraic methods: factorising, completing the square, and using the quadratic formula, together with the discriminant and exam tips.
二次方程是IGCSE代数的核心内容。它不仅作为独立求解题出现，还广泛出现在图像、面积、数列和建模等综合问题中。本复习指南讲解三种主要代数方法：因式分解法、配方法和公式法，同时介绍判别式以及考试技巧。
1. What is a Quadratic Equation? | 什么是二次方程？
d c are constants and a ≠ 0. The highest power of the unknown x is 2, which gives the name 'quadratic'. If a were zero, the equation would become linear, so the condition a ≠ 0 is essential.
二次方程是可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 是常数且 a ≠ 0。未知数 x 的最高次数为 2，因此称为“二次”。如果 a 为零，方程就变成一次方程，所以 a ≠ 0 这一条件至关重要。
ns. Some equations may need rearranging before they look like the standard form, but as long as the highest power of x is 2 after simplification, they are quadratic.
例如，x² - 3x + 2 = 0、2x² + 5x - 7 = 0 和 x² - 4 = 0 都是二次方程。有些方程在整理成标准形式之前可能看起来不像，但只要化简后 x 的最高次数为 2，它们就是二次方程。
2. Standard Form and Key Terms | 标准形式与关键术语
e a is the coefficient of x², b is the coefficient of x, and c is the constant term. The term 'root' or 'solution' ation true. A quadratic equation can have two distinct real roots, one repeated real root, or no real roots.
标准形式为 ax² + bx + c = 0。其中 a 是 x² 的系数，b 是 x 的系数，c 是常数项。“根”或“解”是指使方程成立的 x 值。二次方程可能有两个不同的实根、一个重根，或者没有实根。
-2 and c = 1. Getting these values right is the first step in avoiding sign errors later.
在使用求根公式之前，先识别出 a、b 和 c 会很有帮助。对于 3x² - 2x + 1 = 0，我们有 a = 3、b = -2 和 c = 1。正确找出这些值是避免后面出现符号错误的第一步。
3. Solving by Factorising | 因式分解法求解
each bracket equal to zero. For example, x² + 5x + 6 = 0 factorises to (x + 2)(x + 3) = 0.
当二次式可以整齐地因式分解时，这通常是最快的方法。将表达式写成两个一次因式的乘积，然后令每个因式分别等于零。例如，x² + 5x + 6 = 0 可分解为 (x + 2)(x + 3) = 0。
= -2 or x = -3. Always check by substituting both values back into the original equation.
一旦完成因式分解，方程就变成两个独立的一次方程。由 (x + 2)(x + 3) = 0，我们解 x + 2 = 0 或 x + 3 = 0，得到 x = -2 或 x = -3。务必把两个值代回原方程进行检验。
4. The Zero Product Property | 零乘积性质
ackets equals zero, at least one bracket must equal zero. This splits the quadratic into two simple linear equations.
因式分解法成立的依据是零乘积性质：如果 p × q = 0，那么 p = 0 或 q = 0。由于两个因式的乘积为零，至少有一个因式必须为零。这样就把二次方程拆成两个简单的一次方程。
2 = 6 or x + 3 = 6. You must first rearrange the equation until the right-hand side is 0.
这个性质只有在方程一边为零时才成立。如果有 (x + 2)(x + 3) = 6，不能得出 x + 2 = 6 或 x + 3 = 6。必须先把方程重新整理，使右边等于 0。
5. Completing the Square | 配方法
Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This method is useful when factorising is difficult and is also needed for finding turning points of quadratic graphs. To complete the square for x² + bx + c, add and subtract (b ÷ 2)².
配方法将 ax² + bx + c 改写为 a(x + p)² + q 的形式。当因式分解较困难时，该方法非常有用，而且求二次图像顶点也需要它。对于 x² + bx + c，配方时加上并减去 (b ÷ 2)²。
6x + 5 = (x + 3)² - 9 + 5 = (x + 3)² - 4. This form reveals the turning point at (-3, -4).
例如，对 x² + 6x + 5 配方：取 6 的一半为 3，再平方得到 9。然后写成 x² + 6x + 5 = (x + 3)² - 9 + 5 = (x + 3)² - 4。这种形式直接显示顶点为 (-3, -4)。
x² + 6x + 5 = (x + 3)² - 4
6. The Quadratic Formula | 求根公式
c formula solves any quadratic equation in standard form. The roots are given by x = (-b ± √b² - 4ac) ÷ 2a. It is derived from completing the square and is a reliable method when factorising is not obvious.
求根公式可以求解任何标准形式的二次方程。根由公式 x = (-b ± √b² - 4ac) ÷ 2a 给出。该公式由配方法推导而来，在因式分解不明显时是一种可靠方法。
x = (-b ± √b² - 4ac) ÷ 2a
carefully. For 2x² + 3x - 2 = 0, we substitute a = 2, b = 3 and c = -2 to obtain x = (-3 ± √9 + 16) ÷ 4 = (-3 ± 5) ÷ 4, giving x = ½ or x = -2.
使用公式时，先将方程写成 ax² + bx + c = 0。然后小心代入 a、b、c 的值。对于 2x² + 3x - 2 = 0，代入 a = 2、b = 3、c = -2，得到 x = (-3 ± √9 + 16) ÷ 4 = (-3 ± 5) ÷ 4，从而 x = ½ 或 x = -2。
7. The Discriminant | 判别式
0 there is one repeated real root; if Δ < 0 there are no real roots but two complex roots.
判别式是求根公式中平方根号下的部分：Δ = b² - 4ac。它告诉我们根的数量和类型。如果 Δ > 0，有两个不同的实根；如果 Δ = 0，有一个重根；如果 Δ < 0，没有实根但有两个复数根。
| Discriminant | Nature of roots |
| Δ > 0 | Two distinct real roots |
| Δ = 0 | One repeated real root |
| Δ < 0 | No real roots |
1 > 0, so it has two distinct real roots. The discriminant is especially useful in IGCSE questions that ask how many roots an equation has without requiring the actual roots.
例如，x² - 5x + 6 = 0 的判别式为 Δ = 25 - 24 = 1 > 0，所以它有两个不同的实根。在 IGCSE 题目中，当只要求判断方程根的个数而不需要求出根时，判别式特别有用。
8. Graphical Interpretation | 图像解释
e graph crosses the x-axis. The turning point can be found from completed square form; its x-coordinate is -b ÷ 2a.
y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的实根就是图像与 x 轴交点的横坐标。顶点坐标可从配方式求得；其横坐标为 -b ÷ 2a。
If a is positive, the parabola opens upwards and the turning point is a minimum. If a is negative, the parabola opens downwards and the turning point is a maximum. The number of x-axis intersections matches the number of real roots.
如果 a 为正，抛物线开口向上，顶点是最低点；如果 a 为负，抛物线开口向下，顶点是最高点。图像与 x 轴交点的个数与实根个数一致。
9. Choosing the Best Method | 选择最佳方法
In an exam, first try factorising if the coefficient of x² is 1 and the constant term has small factors. If the quadratic does not factorise easily, use the quadratic formula or completing the square. The formula is usually safest when coefficients are not integers or the discriminant is not a perfect square.
考试中，如果 x² 的系数为 1 且常数项的因数较小，首先尝试因式分解。如果二次式不容易分解，则使用求根公式或配方法。当系数不是整数或判别式不是完全平方数时，公式法通常最稳妥。
Completing the square is the best choice when you also need to find the turning point or sketch the graph. Factorising is quick for simple quadratics, while the formula always works but requires more steps and careful substitution.
当还需要求顶点或画图像时，配方法是最佳选择。因式分解对简单二次式更快捷，而公式法始终有效，但步骤更多，代入时需更加仔细。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Common mistakes include forgetting to set the equation equal to zero before factorising, losing the negative sign when substituting into the formula, and writing the wrong sign for c. Always rearrange into standard form first, use brackets when substituting, and check your roots by substitution.
常见错误包括：因式分解前忘记把方程化为等于零的形式；代入公式时丢失负号；以及写错 c 的符号。务必先把方程整理成标准形式，代入时使用括号，并通过代回原方程检验根。
In IGCSE exams, write down the formula before substituting values. Show each step clearly, because method marks are awarded even if a small arithmetic error occurs. Finally, decide whether the answer is reasonable by comparing it with the graph or discriminant.
在 IGCSE 考试中，应先写出公式再代入数值。每一步都要清晰展示，因为即使出现小的计算错误，方法分仍然可以获得。最后，可以通过与图像或判别式比较，判断答案是否合理。
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
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