📚 Solving Quadratic Equations: Factorising, Completing the Square and the Formula | 二次方程：因式分解、配方法与求根公式
raic solution methods: factorising, completing the square and the quadratic formula. It also covers the discriminant, graphs and common exam mistakes.
二次方程是 IGCSE 数学的核心主题。它们出现在纯代数题、坐标几何、测量问题和现实建模中。本复习指南解释标准形式 ax² + bx + c = 0，并比较三种代数解法：因式分解、配方法和求根公式。它还涵盖判别式、图像和常见考试错误。
1. What is a Quadratic Equation? | 什么是二次方程？
b and c are constants and a ≠ 0. The highest power of the variable x is 2, so the equation has degree 2.
在 IGCSE 数学中，二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 是常数，且 a ≠ 0。变量 x 的最高次数是 2，因此该方程的次数为 2。
which is a linear equation, not a quadratic one. That is why the condition a ≠ 0 is essential.
如果 a = 0，方程就变成 bx + c = 0，这是一元一次方程，而不是二次方程。这就是为什么 a ≠ 0 这个条件必不可少。
e two real solutions, one repeated solution, or no real solutions depending on the discriminant.
典型例子包括 x² − 5x + 6 = 0、2x² + 3x − 2 = 0 和 x² − 4 = 0。根据判别式的不同，每个方程可能有两个实数解、一个重复实数解或没有实数解。
2. Standard Form and Coefficients | 标准形式与系数
rm ax² + bx + c = 0. This makes it easy to identify the coefficients a, b and c correctly.
在解二次方程之前，你应始终将其整理成标准形式 ax² + bx + c = 0。这样可以方便地正确识别系数 a、b 和 c。
quation x² = 4x − 3 should be rearranged as x² − 4x + 3 = 0. Here a = 1, b = −4 and c = 3.
例如，方程 x² = 4x − 3 应整理为 x² − 4x + 3 = 0。这里 a = 1，b = −4，c = 3。
Many errors in the quadratic formula or factorising happen because students identify the wrong values of a, b or c. Always move every term to one side so that the other side is zero.
许多使用求根公式或因式分解时的错误，都是因为学生认错了 a、b、c 的值。始终把所有项移到等号一边，使另一边为零。
ax² + bx + c = 0, a ≠ 0
3. Solving by Factorising | 因式分解法
integer roots. It relies on the zero product principle: if p × q = 0, then p = 0 or q = 0.
当二次式的根为简单整数时，因式分解通常是最快的方法。它依赖于零乘积原理：如果 p × q = 0，那么 p = 0 或 q = 0。
The steps are: write the equation in standard form, factorise the left side into two brackets, set each bracket equal to zero, then solve the resulting linear equations.
步骤是：将方程写成标准形式，把左边分解成两个括号，令每个括号分别等于零，然后解所得的一次方程。
6 = 0 factorises as (x − 2)(x − 3) = 0. Setting each bracket to zero gives x = 2 or x = 3.
例如，x² − 5x + 6 = 0 可分解为 (x − 2)(x − 3) = 0。令每个括号分别等于零，得到 x = 2 或 x = 3。
Factorising works quickly when the roots are rational, but it is less convenient for surds or decimals. You should always expand your brackets as a mental check.
当根为有理数时，因式分解非常快，但当根含有根式或小数时就不太方便。你应总是把括号展开作为心里检查。
4. Completing the Square | 配方法
Completing the square transforms a quadratic expression into the form (x + p)² + q. This method is especially useful for finding the turning point of a parabola and for deriving the quadratic formula.
配方法将二次式转化为 (x + p)² + q 的形式。这种方法对于求抛物线的顶点以及推导求根公式特别有用。
f the coefficient of x, square it, and adjust the constant: x² + bx = (x + b/2)² − (b/2)².
要对 x² + bx 进行配方，取 x 系数的一半，将其平方，并调整常数项：x² + bx = (x + b/2)² − (b/2)²。
9 + 5 = 0, so (x + 3)² = 4. Taking square roots gives x + 3 = ±2, hence x = −1 or x = −5.
例如，x² + 6x + 5 = 0 可以写成 (x + 3)² − 9 + 5 = 0，所以 (x + 3)² = 4。两边开平方得到 x + 3 = ±2，因此 x = −1 或 x = −5。
This method always works, but it can be slower when a is not 1. In that case, factor out a first before completing the square.
这种方法总是有效，但当 a 不等于 1 时可能较慢。在这种情况下，应先把 a 提取出来，再进行配方。
5. The Quadratic Formula | 求根公式
ard form. It is derived by completing the square on the general equation ax² + bx + c = 0.
求根公式可以解任何标准形式的二次方程。它由对一般方程 ax² + bx + c = 0 进行配方推导得出。
x = (−b ± √b² − 4ac) ÷ 2a
e discriminant. The ± sign means you must calculate two values: one with + and one with −.
在这个公式中，根号下的表达式 b² − 4ac 称为判别式。± 号表示你必须计算两个值：一个取 +，一个取 −。
For 2x² + 3x − 2 = 0, we have a = 2, b = 3 and c = −2. Substituting gives x = (−3 ± √9 + 16) ÷ 4 = (−3 ± 5) ÷ 4, so x = 1/2 or x = −2.
对于 2x² + 3x − 2 = 0，我们有 a = 2，b = 3，c = −2。代入公式得到 x = (−3 ± √9 + 16) ÷ 4 = (−3 ± 5) ÷ 4，所以 x = 1/2 或 x = −2。
Use this method when factorising is difficult or when the roots are irrational. Remember to write the formula clearly before substituting values.
当因式分解困难或根为无理数时，使用这种方法。记住在代入数值前先把公式清楚地写出来。
6. The Discriminant | 判别式
the nature of the roots without solving the equation. It is one of the most useful tools for checking answers.
判别式 D = b² − 4ac 可以在不解方程的情况下告诉你根的性质。它是检查答案的最有用工具之一。
ne repeated real root. If D < 0, there are no real roots, only complex roots which are not required at IGCSE.
如果 D > 0，方程有两个不相等的实数根。如果 D = 0，恰好有一个重复的实数根。如果 D < 0，没有实数根，只有复数根，而 IGCSE 不要求复数根。
ne repeated root. The equation x² + x + 1 = 0 has D = 1 − 4 = −3, so it has no real roots.
例如，x² − 4x + 4 = 0 的判别式 D = 16 − 16 = 0，所以它有一个重复根。方程 x² + x + 1 = 0 的判别式 D = 1 − 4 = −3，所以它没有实数根。
= ax² + bx + c, the discriminant tells you how many times the parabola crosses the x-axis.
在 y = ax² + bx + c 的图像中，判别式告诉你抛物线与 x 轴相交的次数。
7. Graphical Interpretation | 图像解释
he solutions of ax² + bx + c = 0 are the x-coordinates where the graph crosses the x-axis.
y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，抛物线开口向上；如果 a < 0，抛物线开口向下。方程 ax² + bx + c = 0 的解就是图像与 x 轴交点的横坐标。
The completed square form (x + p)² + q gives the vertex of the parabola at (−p, q). This is useful for sketching the graph and solving maximum or minimum problems.
配方法得到的形式 (x + p)² + q 给出抛物线的顶点为 (−p, q)。这对画图像以及解最大值或最小值问题很有用。
. When D = 0, it touches the x-axis at one point. When D < 0, it never crosses the x-axis.
当 D > 0 时，抛物线与 x 轴相交两次。当 D = 0 时，它在一点与 x 轴相切。当 D < 0 时，它不与 x 轴相交。
8. Word Problems and Modelling | 应用题与建模
Quadratic equations often arise from real-life contexts such as area, product of numbers, projectile motion or profit. The key steps are: define the unknown, form an equation, solve it and interpret the answers.
二次方程经常来自现实情境，如面积、数的乘积、抛体运动或利润。关键步骤是：设未知数，建立方程，解方程并解释答案。
Always check that the solutions make sense in the original context. For example, a negative length or time is usually rejected even though it is a valid algebraic solution.
一定要检查解在原始情境中是否有意义。例如，负的长度或时间通常要舍去，尽管它在代数上是有效解。
ves (n + 8)(n − 7) = 0, so n = 7 because n must be positive. The two integers are 7 and 8.
例如，如果两个连续正整数的乘积是 56，设较小的整数为 n。那么 n(n + 1) = 56，所以 n² + n − 56 = 0。因式分解得到 (n + 8)(n − 7) = 0，因为 n 必须为正，所以 n = 7。两个整数为 7 和 8。
In area problems, drawing a diagram and labelling the sides in terms of one variable can help you form the correct quadratic equation.
在面积问题中，画图并用一个变量标记各边有助于你建立正确的二次方程。
9. Common Mistakes | 常见错误
Many marks are lost through avoidable errors. One common mistake is forgetting that a must not be zero when identifying a quadratic equation.
许多失分都来自可避免的错误。一个常见错误是在识别二次方程时忘记 a 不能为零。
Another frequent error is a sign mistake in the quadratic formula, especially when b or c is negative. Always use brackets around negative values when substituting.
另一个常见错误是求根公式中的符号错误，特别是当 b 或 c 为负数时。代入时始终用括号括住负值。
, which can remove the solution x = 0. Instead, bring all terms to one side and factorise.
学生有时会把二次方程两边同时除以 x，这可能会丢掉 x = 0 这个解。正确的做法是把所有项移到一边然后因式分解。
h roots from the ± sign and to reject unrealistic answers in word problems when necessary.
还要记住写出 ± 号对应的两个根，并在应用题中必要时舍去不符合实际的答案。
10. Exam Tips and Practice | 考试技巧与练习
In the exam, quote the quadratic formula before substituting values. This helps you avoid substitution errors and shows the examiner your method.
考试时，在代入数值前先写出求根公式。这有助于避免代入错误，也能向考官展示你的解题方法。
Always check factorised answers by expanding the brackets quickly. Use the discriminant to confirm the number of real roots when a question asks about the nature of roots.
总是通过快速展开括号来检查因式分解的答案。当题目询问根的性质时，使用判别式确认实数根的个数。
Practise past paper questions that mix factorising, completing the square, the quadratic formula and graph interpretation. Structured practice will help you choose the most efficient method under time pressure.
练习真题中混合因式分解、配方法、求根公式和图像解释的题目。有组织的练习能帮助你在时间压力下选择最高效的方法。
Finally, present your working clearly. Even if the final answer is wrong, correct substitution and method can still earn method marks in IGCSE Mathematics.
最后，清楚地呈现你的解题过程。即使最终答案错误，正确的代入和方法在 IGCSE 数学中仍可获得方法分。
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