📚 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 central to the IGCSE Mathematics syllabus. A quadratic equation contains a variable raised to the power of two, such as x² terms. You need to be able to solve these equations by factorising, completing the square, and using the quadratic formula. You also need to interpret the discriminant and relate quadratic functions to their graphs. This revision article covers the key methods, common mistakes, and exam-style problem solving.
二次方程是 IGCSE 数学课程的核心内容。二次方程含有二次项，例如 x² 项。你需要掌握因式分解法、配方法和求根公式三种解法，同时理解判别式的意义，并能将二次函数与其图像联系起来。本复习文章涵盖主要方法、常见错误以及考试类型的解题思路。
1. Standard Form of a Quadratic Equation | 二次方程的标准形式
e constants and a ≠ 0. The coefficient a cannot be zero because then the equation would become linear rather than quadratic. Recognising the values of a, b and c is the first step for many solution methods.
二次方程的标准形式是 ax² + bx + c = 0，其中 a、b、c 是常数且 a ≠ 0。系数 a 不能为零，否则方程就变成一次方程而不是二次方程。识别 a、b、c 的值是许多解题方法的第一步。
riting the equation in standard form helps you use factoring, completing the square or the quadratic formula correctly.
例如，在方程 2x² − 7x + 3 = 0 中，a = 2，b = −7，c = 3。在 x² − 9 = 0 中，a = 1，b = 0，c = −9。将方程写成标准形式有助于正确使用因式分解、配方法或求根公式。
2. Solving by Factorising | 因式分解法
zero product property: if two factors multiply to zero, at least one factor must be zero.
当二次式可以写成两个一次因式的乘积时，因式分解通常是最快的方法。如果 ax² + bx + c = 0 可以因式分解为 (px + q)(rx + s) = 0，那么 px + q = 0 或 rx + s = 0。这利用了零乘积性质：如果两个因式相乘为零，则至少一个因式为零。
ore x² + 5x + 6 = (x + 2)(x + 3) = 0. Setting each bracket to zero gives x = −2 or x = −3.
要分解 x² + 5x + 6 = 0，需要找到两个数，它们相乘得 6，相加得 5。数字 2 和 3 符合条件，因为 2 × 3 = 6 且 2 + 3 = 5。因此 x² + 5x + 6 = (x + 2)(x + 3) = 0。令每个括号为零，得到 x = −2 或 x = −3。
(x − 3) = 0. Setting each factor to zero gives 2x − 1 = 0 so x = ½, or x − 3 = 0 so x = 3.
当 x² 的系数不是 1 时，例如 2x² − 7x + 3 = 0，可以使用拆分中间项或尝试括号法。因式分解后的形式为 (2x − 1)(x − 3) = 0。令每个因式为零，得到 2x − 1 = 0，所以 x = ½；或 x − 3 = 0，所以 x = 3。
Always expand your brackets to check the factorisation before solving. A sign error is very common when the constant term is negative.
解题前一定要展开括号检查因式分解是否正确。当常数项为负数时，符号错误非常常见。
3. Solving by Completing the Square | 配方法
Completing the square transforms a quadratic expression into the form (x + p)² + q. This is useful when the quadratic cannot be easily factorised, and it also helps find the vertex of a quadratic graph. The method works by taking half the coefficient of x, squaring it, and adjusting the constant term.
配方法将二次式转化为 (x + p)² + q 的形式。当二次式不容易因式分解时，该方法非常有用，同时它还能帮助求出二次函数图像的顶点。配方法的步骤是：取 x 项系数的一半，平方后调整常数项。
btract 9 to keep the expression equivalent: x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7.
对 x² + 6x + 2 进行配方，先关注 x² + 6x。6 的一半是 3，3² = 9。因此 x² + 6x + 9 = (x + 3)²。因为我们加上了 9，需减去 9 以保持等价：x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7。
ite (x + 3)² − 7 = 0. Then (x + 3)² = 7. Taking square roots of both sides gives x + 3 = ±√7, so x = −3 ± √7. These are the exact solutions; approximate values are x ≈ −0.354 or x ≈ −5.646.
要求解 x² + 6x + 2 = 0，写成 (x + 3)² − 7 = 0。然后 (x + 3)² = 7。两边开平方得到 x + 3 = ±√7，所以 x = −3 ± √7。这是精确解；近似值为 x ≈ −0.354 或 x ≈ −5.646。
+ 5 = 0 becomes x² + 4x + 2.5 = 0 after dividing by 2. Then complete the square normally.
如果 x² 的系数不是 1，先对整个方程除以 a。例如，2x² + 8x + 5 = 0 除以 2 后变为 x² + 4x + 2.5 = 0。然后按常规方法配方。
4. The Quadratic Formula | 求根公式
the formula below. It is derived from completing the square on the general standard form.
求根公式适用于所有二次方程，包括无法因式分解的方程。对于 ax² + bx + c = 0，其解由以下公式给出。该公式由对一般标准形式进行配方推导而来。
x = (−b ± √b² − 4ac) ÷ 2a
c is negative. For 3x² − 5x − 2 = 0, a = 3, b = −5 and c = −2. Substitution gives x = (5 ± √25 + 24) ÷ 6 = (5 ± √49) ÷ 6 = (5 ± 7) ÷ 6. Therefore x = 12 ÷ 6 = 2 or x = −2 ÷ 6 = −⅓.
使用公式时，先确定 a、b、c 的值，然后仔细代入，尤其当 b 或 c 为负数时。对于 3x² − 5x − 2 = 0，a = 3，b = −5，c = −2。代入得到 x = (5 ± √25 + 24) ÷ 6 = (5 ± √49) ÷ 6 = (5 ± 7) ÷ 6。因此 x = 12 ÷ 6 = 2 或 x = −2 ÷ 6 = −⅓。
In the formula, the expression b² − 4ac is called the discriminant. It determines the number and type of solutions. Always calculate the discriminant before evaluating the whole formula to avoid arithmetic mistakes.
公式中的 b² − 4ac 称为判别式。它决定了方程解的个数和类型。在计算整个公式之前，先计算判别式可以避免算术错误。
5. The Discriminant | 判别式
real root. If D < 0, it has no real roots, but two complex roots are introduced in advanced courses.
判别式 D = b² − 4ac 可以在不求解的情况下判断二次方程有多少个实数根。如果 D > 0，方程有两个不同的实数根。如果 D = 0，方程恰好有一个重根。如果 D < 0，方程没有实数根，但在更高阶课程中会引入两个复数根。
the equation has one repeated root, and the factorised form is (x − 2)² = 0, giving x = 2.
对于 x² − 4x + 4 = 0，D = (−4)² − 4 × 1 × 4 = 16 − 16 = 0。因此方程有一个重根，因式分解为 (x − 2)² = 0，解得 x = 2。
31. Since D < 0, the equation has no real solutions. This is confirmed by the graph, which would not cross the x-axis.
对于 2x² + 3x + 5 = 0，D = 3² − 4 × 2 × 5 = 9 − 40 = −31。因为 D < 0，方程没有实数解。图像也证实了这一点，因为该二次函数图像不会与 x 轴相交。
6. Graphical Interpretation | 图像解释
solutions of ax² + bx + c = 0 are the x-coordinates where the parabola 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，具有最低点。如果 a < 0，抛物线开口向下，具有最高点。方程 ax² + bx + c = 0 的解就是抛物线与 x 轴交点的横坐标。
e parabola touches the x-axis at exactly one point. If D < 0, the parabola does not touch the x-axis at all.
这些 x 轴交点也称为方程的根。如果判别式为正，抛物线与 x 轴相交于两点。如果 D = 0，抛物线与 x 轴恰好相切于一点。如果 D < 0，抛物线与 x 轴没有交点。
)² + q, the vertex is (−p, q). This helps you sketch the graph accurately and answer questions about maximum or minimum values.
抛物线的顶点可以通过配方法求得。如果 y = (x + p)² + q，则顶点为 (−p, q)。这有助于准确画出草图，并解答关于最大值或最小值的问题。
7. Solving Word Problems | 解应用题
Many IGCSE exam questions embed quadratic equations in real-life problems such as area, projectile motion, or number puzzles. The key is to define a variable, form a quadratic equation, solve it, and then check whether the solution makes sense in the original context.
许多 IGCSE 考试题将二次方程嵌入到面积、抛体运动或数字谜题等实际情境中。关键是设出变量，列出二次方程，求解，然后检查解在原情境中是否合理。
= 0, so x = −12 or x = 5. Since width cannot be negative, x = 5 m and the length is 12 m.
示例：一个矩形花园的面积为 60 平方米，长比宽多 7 米。设宽为 x 米，则长为 x + 7 米。面积方程为 x(x + 7) = 60。展开得到 x² + 7x − 60 = 0。因式分解得 (x + 12)(x − 5) = 0，所以 x = −12 或 x = 5。因为宽不能为负数，所以 x = 5 米，长为 12 米。
0. Use the quadratic formula with a = 5, b = −20, c = −1 to find the positive value of t.
抛体问题通常使用方程 h = −5t² + vt + h₀。例如，一个球向上抛出，t 秒后的高度 h（米）为 h = −5t² + 20t + 1，求球何时落地。令 h = 0：−5t² + 20t + 1 = 0。乘以 −1 得 5t² − 20t − 1 = 0。使用求根公式，a = 5，b = −20，c = −1，求出 t 的正值。
8. Choosing the Best Method | 选择最佳方法
In an exam, choose the method that is fastest and least error-prone. If the quadratic factorises easily with small integer factors, use factorising. If you need the exact roots in surd form or the vertex, completing the square is helpful. If the coefficients are awkward or the equation does not factorise, use the quadratic formula.
在考试中，应选择最快且最不容易出错的方法。如果二次式容易因式分解且因数为小整数，就使用因式分解法。如果需要以根式形式给出精确解或求顶点，配方法很有帮助。如果系数复杂或方程不能因式分解，就使用求根公式。
The quadratic formula is the most general and works every time, but it requires careful substitution and simplification. Factorising is quick but only works when roots are rational. Completing the square is especially useful for graphing because it reveals the vertex directly.
求根公式最通用，任何时候都适用，但需要仔细代入和化简。因式分解法快速，但只适用于根为有理数的情况。配方法在绘制图像时特别有用，因为它能直接显示顶点。
In questions that ask for solutions to a specified degree of accuracy, use the quadratic formula and a calculator. In non-calculator papers, factorising or completing the square are often expected.
如果题目要求将解精确到指定精度，应使用求根公式和计算器。在不允许使用计算器的试卷中，通常要求使用因式分解法或配方法。
9. Common Mistakes to Avoid | 常见错误与避免方法
ing x = 6 or x + 5 = 6. First write x² + 5x − 6 = 0, then factorise to (x + 6)(x − 1) = 0.
一个常见错误是在因式分解前忘记方程必须等于零。不能把 x² + 5x = 6 写成 x(x + 5) = 6 后就直接认为 x = 6 或 x + 5 = 6。应先写成 x² + 5x − 6 = 0，再因式分解为 (x + 6)(x − 1) = 0。
s 7. Also, b² is always positive. Be careful when calculating b² − 4ac; if c is negative, −4ac becomes positive.
另一个常见错误是在求根公式中处理负号不当。例如，如果 b = −7，那么 −b 是 7。此外，b² 总是正的。计算 b² − 4ac 时要小心；如果 c 为负，−4ac 会变成正数。
When completing the square, do not forget to balance the added square term. If you add 9 inside the bracket to complete the square, you must also subtract 9 outside, or the expression changes value. Always expand to verify.
配方时，不要忘记平衡添加的平方项。如果在括号内加上 9 来配方，必须在外面也减去 9，否则表达式的值会改变。务必展开验证。
10. Exam Tips and Summary | 考试技巧与总结
In IGCSE exams, always show your working clearly, especially when substituting into the quadratic formula. Write down the values of a, b and c first. This earns method marks even if the final answer is incorrect.
在 IGCSE 考试中，始终清晰地展示解题步骤，尤其是在代入求根公式时。先写出 a、b、c 的值。即使最终答案错误，这也能获得方法分。
posed solutions, verify that the equation is satisfied. This is a quick way to catch sign and arithmetic errors.
通过将解代回原方程来检查答案。例如，如果 x = 2 和 x = −3 是所得解，应验证方程是否成立。这是快速发现符号和算术错误的方法。
, and the product of the roots is c ÷ a. These relationships are useful for checking factorisation and for solving problems about roots. In summary, practise all three methods and know when to apply each one.
记住关键关系：ax² + bx + c = 0 的根之和为 −b ÷ a，根之积为 c ÷ a。这些关系有助于检查因式分解以及解答与根有关的问题。总之，要多练习三种方法，并知道何时使用每一种。
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