📚 IGCSE Mathematics G-2: Quadratic Equations and Graphs | IGCSE 数学 G-2：二次方程与图像
A quadratic equation is one of the most important topics in the IGCSE Mathematics syllabus. It appears in algebra, graphs, problem solving and even in real-life modelling. This article covers the key methods you need to solve quadratic equations, understand their graphs and score well in the exam.
二次方程是 IGCSE 数学大纲中最重要的主题之一。它出现在代数、图像、问题求解甚至实际生活建模中。本文涵盖求解二次方程所需的关键方法、理解其图像以及在考试中取得好成绩的要点。
1. What is a Quadratic Equation? | 什么是二次方程？
onstants and a ≠ 0. If a were equal to 0, the equation would become linear, not quadratic.
二次方程是一个次数为 2 的方程。其标准形式为 ax² + bx + c = 0，其中 a、b 和 c 是常数，且 a ≠ 0。如果 a 等于 0，方程就变成了一次方程，而不是二次方程。
The highest power of the variable x is 2, which means a quadratic equation can have up to two real solutions. These solutions are also called roots or x-intercepts when the equation is linked to a graph.
变量 x 的最高次数为 2，这意味着二次方程最多可以有两个实数解。这些解也称为根，当方程与图像联系时，它们也叫作 x 轴截距。
ax² + bx + c = 0 (a ≠ 0)
2. Solving by Factorising | 因式分解法求解
uct of two brackets. For example, x² + 5x + 6 = 0 can be factorised as (x + 2)(x + 3) = 0.
当二次式可以写成两个括号的乘积时，因式分解通常是最快的方法。例如，x² + 5x + 6 = 0 可以因式分解为 (x + 2)(x + 3) = 0。
ust equal zero. This gives x + 2 = 0 or x + 3 = 0, so the solutions are x = −2 and x = −3.
如果两个因式的乘积等于零，那么至少有一个因式必须等于零。因此得到 x + 2 = 0 或 x + 3 = 0，所以解为 x = −2 和 x = −3。
- Factorise first: write the quadratic as a product of brackets.
- 先因式分解：将二次式写成括号的乘积。
- Set each bracket to zero: solve the resulting linear equations.
- 令每个括号等于零：求解得到的一次方程。
Not every quadratic can be factorised using integers. When the numbers are not simple, you should use the quadratic formula or completing the square.
并非每个二次式都能用整数进行因式分解。当数字不简单时，你应当使用求根公式或配方法。
3. Solving by the Quadratic Formula | 公式法求解
ratic equation, even when factorising is difficult or impossible. It is given by x = (−b ± √b² − 4ac) ÷ 2a.
求根公式适用于每一个二次方程，即使因式分解很困难或无法进行。公式为 x = (−b ± √b² − 4ac) ÷ 2a。
+ bx + c = 0. Then substitute these values carefully, paying attention to negative signs.
使用公式时，首先从 ax² + bx + c = 0 中确定 a、b 和 c。然后仔细代入这些值，特别注意负号。
x = (−b ± √b² − 4ac) ÷ 2a
xample: solve 2x² − 4x − 6 = 0. Here a = 2, b = −4 and c = −6. Substituting gives x = (4 ± √16 + 48) ÷ 4 = (4 ± √64) ÷ 4, so x = (4 ± 8) ÷ 4. The solutions are x = 3 and x = −1.
示例：求解 2x² − 4x − 6 = 0。这里 a = 2，b = −4，c = −6。代入得 x = (4 ± √16 + 48) ÷ 4 = (4 ± √64) ÷ 4，所以 x = (4 ± 8) ÷ 4。解为 x = 3 和 x = −1。
4. Completing the Square | 配方法
Completing the square transforms a quadratic expression into the form a(x + p)² + q. This method is especially useful for finding the turning point of a graph and for deriving the quadratic formula.
配方法将二次式转化为 a(x + p)² + q 的形式。这种方法特别适用于求图像的顶点，以及推导求根公式。
f 6, which is 3, and square it to get 9. So x² + 6x + 1 = (x + 3)² − 9 + 1 = (x + 3)² − 8.
对于 x² + 6x + 1，取 6 的一半为 3，再平方得 9。因此 x² + 6x + 1 = (x + 3)² − 9 + 1 = (x + 3)² − 8。
To solve x² + 6x + 1 = 0, write (x + 3)² − 8 = 0. Then (x + 3)² = 8, so x + 3 = ±√8. This gives x = −3 ± √8 = −3 ± 2√2.
要求解 x² + 6x + 1 = 0，写成 (x + 3)² − 8 = 0。于是 (x + 3)² = 8，所以 x + 3 = ±√8。得到 x = −3 ± √8 = −3 ± 2√2。
5. The Discriminant | 判别式
tells you how many real solutions a quadratic equation has without solving it completely.
判别式是求根公式中平方根号下的部分：Δ = b² − 4ac。它无需完整求解就能告诉你二次方程有多少个实数解。
| Discriminant Δ = b² − 4ac | Number of real roots |
| Δ > 0 | Two distinct real roots |
| Δ = 0 | One repeated real root |
| Δ < 0 | No real roots |
When Δ is positive, the graph crosses the x-axis twice. When Δ is zero, the graph touches the x-axis at one point. When Δ is negative, the graph does not meet the x-axis at all.
当 Δ 为正时，图像与 x 轴相交两次。当 Δ 为零时，图像与 x 轴相切于一点。当 Δ 为负时，图像与 x 轴完全没有交点。
6. Quadratic Graphs: Parabolas | 二次函数图像：抛物线
. If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point.
y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，抛物线开口向上，并有一个最低点。如果 a < 0，抛物线开口向下，并有一个最高点。
rm c. The x-intercepts are the real roots of the equation ax² + bx + c = 0, if they exist.
图像的 y 轴截距是常数项 c。x 轴截距是方程 ax² + bx + c = 0 的实数根（如果存在）。
You should be able to sketch a parabola by identifying its shape, y-intercept, roots and turning point. Use symmetry to make the sketch accurate.
你应当能够通过确定抛物线的形状、y 轴截距、根和顶点来画出草图。利用对称性使草图更加准确。
7. Roots and Turning Points | 根与顶点
point is at (p, q). The sign of a tells you whether this point is a maximum or a minimum.
对于 y = a(x − p)² + q，顶点坐标为 (p, q)。a 的符号告诉你该点是最大值点还是最小值点。
)² − 5, then the turning point is (3, −5). Since a = 2 is positive, it is a minimum point.
如果方程以完全平方形式给出，例如 y = 2(x − 3)² − 5，则顶点为 (3, −5)。由于 a = 2 为正，所以它是最低点。
e passing through the turning point. For the example above, the axis of symmetry is x = 3.
抛物线的对称轴是一条经过顶点的竖直线。对于上述示例，对称轴为 x = 3。
8. Solving Quadratic Inequalities | 解二次不等式
ng the roots of the corresponding equation x² − 4x + 3 = 0. The roots are x = 1 and x = 3.
二次不等式如 x² − 4x + 3 < 0 可以通过先求对应方程 x² − 4x + 3 = 0 的根来求解。根为 x = 1 和 x = 3。
wards. It is below the x-axis between x = 1 and x = 3, so the solution is 1 < x < 3.
画出抛物线 y = x² − 4x + 3 的草图。由于 a = 1 为正，图像开口向上。在 x = 1 和 x = 3 之间图像位于 x 轴下方，因此解为 1 < x < 3。
he inequality includes equality. If it is x² − 4x + 3 ≤ 0, then the solution is 1 ≤ x ≤ 3.
务必检查不等式是否包含等号。如果是 x² − 4x + 3 ≤ 0，那么解为 1 ≤ x ≤ 3。
9. Real-life Problems | 实际应用问题
Quadratic equations often appear in problems involving area, projectile motion, profit and product design. The first step is to translate the word problem into an equation in x.
二次方程经常出现在涉及面积、抛体运动、利润和产品设计的问题中。第一步是将文字题转化为关于 x 的方程。
4 m², find its dimensions. Let the width be x, so the length is x + 3. Then x(x + 3) = 54.
示例：一个矩形花园的长度比宽度长 3 m。如果面积为 54 m²，求其尺寸。设宽度为 x，则长度为 x + 3。于是 x(x + 3) = 54。
− 6) = 0. Since length cannot be negative, x = 6. The width is 6 m and the length is 9 m.
得到 x² + 3x − 54 = 0。因式分解得 (x + 9)(x − 6) = 0。由于长度不能为负，所以 x = 6。宽度为 6 m，长度为 9 m。
10. Common Mistakes | 常见错误
le, x² + 3x = 4 should first become x² + 3x − 4 = 0, then factorise to (x + 4)(x − 1) = 0.
学生经常在因式分解前忘记将方程化为等于零的形式。例如，x² + 3x = 4 应首先变为 x² + 3x − 4 = 0，然后因式分解为 (x + 4)(x − 1) = 0。
Another common mistake is losing a solution when dividing both sides by x. Never divide by a variable unless you know it is not zero; always factor instead.
另一个常见错误是在两边同除以 x 时丢失一个解。除非你知道变量不为零，否则绝不要除以变量；应当始终进行因式分解。
Sign errors in the quadratic formula are also frequent. Write down a, b and c clearly, and use brackets when substituting negative values.
求根公式中的符号错误也很常见。清楚地写下 a、b 和 c，并在代入负值时使用括号。
11. Exam Tips | 考试技巧
In the IGCSE exam, always choose the most efficient method. If the quadratic factorises easily, factorise. If not, use the quadratic formula or completing the square.
在 IGCSE 考试中，始终选择最高效的方法。如果二次式容易因式分解，就因式分解。如果不方便，就使用求根公式或配方法。
Show all steps clearly, especially when substituting into the formula. Even if your final answer is wrong, you can gain method marks for correct working.
清晰展示所有步骤，尤其是在代入公式时。即使最终答案错误，正确的解题过程也能获得方法分。
Finally, check your solutions by substituting them back into the original equation. This takes only a few seconds and can prevent careless errors.
最后，将解代回原方程进行检验。这只需几秒钟，却能避免粗心错误。
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