📚 Mastering Quadratic Equations for IGCSE Mathematics | 掌握 IGCSE 数学中的二次方程
Quadratic equations are one of the most important algebraic topics in IGCSE Mathematics. They appear in pure algebra, coordinate geometry, measurement and real-life modelling, so a reliable method for solving them is essential. This revision guide covers the standard form, factorisation, completing the square, the quadratic formula, the discriminant and exam-style applications.
二次方程是 IGCSE 数学中最重要的代数主题之一。它们在纯代数、坐标几何、测量和现实建模中都会出现，因此掌握可靠的求解方法至关重要。本复习指南涵盖标准形式、因式分解法、配方法、求根公式、判别式以及考试风格的应用题。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is a polynomial equation in which the highest power of the variable is 2. It can always be written in the form ax² + bx + c = 0, where a ≠ 0.
二次方程是最高次数为 2 的多项式方程。它总可以写成 ax² + bx + c = 0 的形式，其中 a ≠ 0。
The condition a ≠ 0 is essential. If a = 0, the equation becomes linear bx + c = 0, not quadratic.
条件 a ≠ 0 至关重要。如果 a = 0，方程就变成一元一次方程 bx + c = 0，而不是二次方程。
Quadratic equations can have two real solutions, one repeated solution or no real solutions. The correct method depends on the form of the equation and the question requirements.
二次方程可以有两个实数解、一个重复解或没有实数解。选择正确的求解方法取决于方程的形式和题目要求。
2. Standard Form and Coefficients | 标准形式与系数
In the standard form ax² + bx + c = 0, a is the quadratic coefficient, b is the linear coefficient, and c is the constant term.
在标准形式 ax² + bx + c = 0 中，a 是二次项系数，b 是一次项系数，c 是常数项。
Before solving, always rearrange the equation so that one side is equal to 0. For example, 3x² = 5x − 2 should be written as 3x² − 5x + 2 = 0.
求解前，始终要把方程整理成一边等于 0 的形式。例如，3x² = 5x − 2 应写成 3x² − 5x + 2 = 0。
Take care with negative signs when identifying b and c. In x² − 4x + 3 = 0, we have a = 1, b = −4 and c = 3.
在确定 b 和 c 时要注意负号。在 x² − 4x + 3 = 0 中，a = 1，b = −4，c = 3。
3. Solving by Factorisation | 因式分解法
If the quadratic expression ax² + bx + c can be factorised into two linear brackets, then each bracket can be set to zero. This uses the zero-product property: if pq = 0, then p = 0 or q = 0.
如果二次式 ax² + bx + c 可以因式分解成两个一次括号，那么每个括号都可以设为零。这利用了零乘积性质：若 pq = 0，则 p = 0 或 q = 0。
For example, solve x² − 5x + 6 = 0. Factorising gives (x − 2)(x − 3) = 0, so x = 2 or x = 3.
例如，解 x² − 5x + 6 = 0。因式分解得 (x − 2)(x − 3) = 0，所以 x = 2 或 x = 3。
When a ≠ 1, use splitting the middle term or trial and error. For 2x² + 7x + 3 = 0, we get (2x + 1)(x + 3) = 0, giving x = −1/2 or x = −3.
当 a ≠ 1 时，可使用十字相乘法或拆分中项。对于 2x² + 7x + 3 = 0，得到 (2x + 1)(x + 3) = 0，解得 x = −1/2 或 x = −3。
Difference of two squares is a special case: x² − 9 = 0 factorises as (x − 3)(x + 3) = 0, so x = 3 or x = −3.
平方差公式是一种特殊情况：x² − 9 = 0 可因式分解为 (x − 3)(x + 3) = 0，所以 x = 3 或 x = −3。
4. Solving by Completing the Square | 配方法
Completing the square rewrites a quadratic in the form (x + p)² = q. This is especially useful when factorisation is not straightforward.
配方法将二次式改写成 (x + p)² = q 的形式。这在因式分解不直接时特别有用。
For x² + 6x + 2 = 0, first rewrite as x² + 6x = −2. Half of 6 is 3, and 3² = 9, so add 9 to both sides: (x + 3)² = 7. Then x = −3 ± √7.
对于 x² + 6x + 2 = 0，先写成 x² + 6x = −2。6 的一半是 3，且 3² = 9，因此两边同时加 9：(x + 3)² = 7。于是 x = −3 ± √7。
If the leading coefficient is not 1, divide first. For 2x² + 8x + 5 = 0, divide by 2 to get x² + 4x + 5/2 = 0, then complete the square.
如果首项系数不是 1，应先除以该系数。对于 2x² + 8x + 5 = 0，先除以 2 得 x² + 4x + 5/2 = 0，再配方。
The completed-square form is also written as a(x − h)² + k, where (h, k) is the vertex of the parabola. This connects algebraic method with graph sketching.
配方法的完成式也可以写成 a(x − h)² + k 的形式，其中 (h, k) 是抛物线的顶点。这就把代数方法与画图联系了起来。
5. The Quadratic Formula | 求根公式
The quadratic formula solves any quadratic equation ax² + bx + c = 0:
求根公式可以解任何二次方程 ax² + bx + c = 0：
x = (−b ± √(b² − 4ac)) ÷ 2a
Write down a, b and c carefully after rearranging to standard form. Substitute them into the formula, then simplify the square root and the fraction.
在整理为标准形式后，仔细写出 a、b 和 c。将它们代入公式，然后化简根号和分数。
Example: solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, c = −3. Then x = (4 ± √(16 + 24)) ÷ 4 = (4 ± √40) ÷ 4 = (4 ± 2√10) ÷ 4 = (2 ± √10) ÷ 2.
例：解 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。则 x = (4 ± √(16 + 24)) ÷ 4 = (4 ± √40) ÷ 4 = (4 ± 2√10) ÷ 4 = (2 ± √10) ÷ 2。
6. Discriminant and Nature of Roots | 判别式与根的性质
The discriminant is the expression inside the square root in the quadratic formula: Δ = b² − 4ac.
判别式是求根公式中根号内的表达式：Δ = b² − 4ac。
It tells us the nature of the roots without solving the equation.
它无需解方程就能告诉我们根的性质。
If b² − 4ac > 0, there are two distinct real roots.
若 b² − 4ac > 0，有两个不相等的实根。
If b² − 4ac = 0, there is one repeated real root.
若 b² − 4ac = 0，有一个重实根。
If b² − 4ac < 0, there are no real roots.
若 b² − 4ac < 0，没有实根。
Example: for x² + 2x + 5 = 0, Δ = 2² − 4(1)(5) = 4 − 20 = −16 < 0, so there are no real roots.
例：对于 x² + 2x + 5 = 0，Δ = 2² − 4(1)(5) = 4 − 20 = −16 < 0，因此没有实根。
7. Graphical Interpretation | 图像解释
The graph of y = ax² + bx + c is a parabola. The solutions of ax² + bx + c = 0 are the x-intercepts of the graph where y = 0.
y = ax² + bx + c 的图像是一条抛物线。ax² + bx + c = 0 的解就是图像上 y = 0 时的 x 轴截距。
If the discriminant is positive, the parabola crosses the x-axis twice. If it is zero, the vertex touches the x-axis once. If it is negative, the parabola does not meet the x-axis.
若判别式为正，抛物线与 x 轴相交两次。若为零，顶点恰好与 x 轴接触一次。若为负，抛物线与 x 轴不相交。
The axis of symmetry is x = −b ÷ 2a, and the vertex lies on this line. This helps sketch the graph quickly.
对称轴为 x = −b ÷ 2a，顶点就在这条线上。这有助于快速画出草图。
If a > 0, the parabola opens upwards, so the vertex is a minimum. If a < 0, it opens downwards, so the vertex is a maximum.
若 a > 0，抛物线开口向上，顶点是最低点。若 a < 0，抛物线开口向下，顶点是最高点。
8. Choosing the Best Method | 选择最佳方法
Use factorisation when the quadratic has simple integer roots and is easy to factor. It is usually the fastest method.
当二次方程有简单的整数根且易于因式分解时，使用因式分解法。它通常是最快的方法。
Use the quadratic formula when factorisation is difficult, when coefficients are large or when answers are required in surd form. Completing the square is useful for deriving the formula and for finding the vertex.
当因式分解困难、系数较大或答案需要用根式表示时，使用求根公式。配方法在推导公式和求顶点时很有用。
In the exam, always check the question wording. If it says ‘give your answer in the form a ± √b’, the quadratic formula or completing the square is likely appropriate.
考试时，一定要看清题目要求。如果题目说“答案写成 a ± √b 的形式”，那么求根公式或配方法可能更合适。
For word problems, translate the scenario into an equation first, then decide the method. Drawing a diagram can help identify the unknown.
对于文字题，先把情景转化为方程，然后决定方法。画图有助于找出未知量。
9. Common Mistakes | 常见错误
A common mistake is forgetting to set the equation to zero before factorising. Factorising 2x² + x = 6 without rearranging gives an incorrect result.
一个常见错误是在因式分解前忘记将方程设为零。不整理 2x² + x = 6 就直接因式分解会得到错误结果。
Another error is dividing by an expression containing the variable. For example, dividing x² = 3x by x loses the solution x = 0.
另一个错误是除以含有未知数的表达式。例如，将 x² = 3x 两边除以 x 会丢失解 x = 0。
Also, when using the formula, be careful with negative coefficients. For b = −3, b² = 9, not −9.
此外，使用公式时要小心负系数。当 b = −3 时，b² = 9，而不是 −9。
When completing the square, do not forget to add the same value to both sides. Balance is essential in every algebraic step.
配方时，不要忘记在等式两边同时加上相同的值。每一步代数变形都要保持平衡。
10. Exam Tips | 考试技巧
Show every step: rearranging, substituting into the formula, simplifying. Method marks can be earned even if the final answer is wrong.
写出每一步：整理、代入公式、化简。即使最终答案错误，也能获得方法分。
When an equation has no real roots, do not write a fake root. State clearly ‘no real roots’ or ‘no real solution’ and justify using the discriminant if needed.
当方程没有实根时，不要编造根。清楚地写出“没有实根”或“没有实数解”，必要时用判别式说明理由。
Practise mixed questions where you must decide the method. This builds speed and confidence for the non-calculator and calculator papers.
练习混合题型，训练自己选择方法。这能提高速度和信心，应对非计算器和计算器试卷。
Always substitute your answers back into the original equation to check, especially when the question asks for exact values.
始终将答案代回原方程检验，尤其是当题目要求精确值时。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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