📚 Mastering Quadratic Equations | 掌握二次方程
a ≠ 0. Understanding how to solve these equations is essential for higher-level algebra, coordinate geometry and problem-solving questions.
二次方程贯穿 IGCSE 数学课程的始终，从基础的因式分解到曲线作图以及实际建模。二次方程是指任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数且 a ≠ 0。掌握二次方程的解法对于高等代数、坐标几何和应用题都至关重要。
1. What is a Quadratic Equation? | 什么是二次方程？
Quadratic equations can have two real solutions, one repeated real solution, or no real solutions depending on the value of the discriminant.
二次方程是一个二次多项式方程。变量 x 的最高次数为 2，这意味着 y = ax² + bx + c 的图像是一条抛物线。根据判别式的值，二次方程可能有两个实数解、一个重复实数解或没有实数解。
The general form is:
一般形式为：
ax² + bx + c = 0, where a ≠ 0
- a is the coefficient of x² and cannot be zero.
- b is the coefficient of x.
- c is the constant term.
- a 是 x² 的系数，不能为零。
- b 是 x 的系数。
- c 是常数项。
2. Standard Form and Coefficients | 标准形式与系数
iplying by -1. Keeping the equation in standard form helps you identify a, b and c correctly.
在求解二次方程之前，应始终将其整理为标准形式。例如，3x - x² = 4 应改写为 -x² + 3x - 4 = 0，或者乘以 -1 后等价地写成 x² - 3x + 4 = 0。保持标准形式有助于正确识别 a、b 和 c。
For the equation 2x² - 5x + 3 = 0:
对于方程 2x² - 5x + 3 = 0：
| Coefficient | Value |
| a | 2 |
| b | -5 |
| c | 3 |
A common mistake is to forget the sign of b or c when substituting into the quadratic formula. Always include the negative sign if it is part of the coefficient.
一个常见错误是在代入二次公式时忘记 b 或 c 的符号。如果系数本身带负号，一定要包含负号。
3. Solving by Factorisation | 因式分解法
to 6 and add to 5. These numbers are 2 and 3, so the equation becomes (x + 2)(x + 3) = 0.
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。如果方程是 x² + 5x + 6 = 0，你需要找到两个数，乘积为 6，和为 5。这两个数是 2 和 3，因此方程变为 (x + 2)(x + 3) = 0。
at least one factor must be zero. This gives x + 2 = 0 or x + 3 = 0, so x = -2 or x = -3.
然后使用零乘积性质：如果乘积等于零，那么至少有一个因式为零。由此得到 x + 2 = 0 或 x + 3 = 0，所以 x = -2 或 x = -3。
se trial and error. The factorised form is (2x + 1)(x + 3) = 0, giving x = -1/2 or x = -3.
对于非首一二次式，如 2x² + 7x + 3 = 0，需要拆分中项或使用试验法。因式分解形式为 (2x + 1)(x + 3) = 0，得到 x = -1/2 或 x = -3。
4. Solving by Completing the Square | 配方法
Completing the square is a powerful technique because it works for every quadratic equation and also helps find the vertex of a parabola. The key idea is to rewrite x² + bx as (x + b/2)² - (b/2)². This creates a perfect square plus a constant.
配方法是一种非常有效的方法，因为它适用于所有二次方程，同时也有助于求抛物线的顶点。关键思想是将 x² + bx 改写为 (x + b/2)² - (b/2)²。这样就产生了一个完全平方加上一个常数。
(x + 3)² - 9. So the equation becomes (x + 3)² - 9 = 7, which simplifies to (x + 3)² = 16.
例如，解 x² + 6x - 7 = 0。首先写成 x² + 6x = 7。然后对左边进行配方：x² + 6x = (x + 3)² - 9。因此方程变为 (x + 3)² - 9 = 7，化简为 (x + 3)² = 16。
ng the square is especially useful when the quadratic formula is not allowed or when exact surd form is required.
两边开平方得到 x + 3 = ±4，所以 x = 1 或 x = -7。当不允许使用二次公式或需要保留根式精确形式时，配方法尤其有用。
5. The Quadratic Formula | 二次公式法
It is derived by completing the square on the general quadratic equation. The formula is:
二次公式可以直接给出 ax² + bx + c = 0 的解。它是对一般二次方程配方后得到的。公式为：
x = ( -b ± √b² - 4ac ) ÷ 2a
them into a calculator. For 2x² - 3x - 5 = 0, a = 2, b = -3 and c = -5. The formula gives:
代入 a、b、c 的值时要仔细，使用计算器时应加上括号。对于 2x² - 3x - 5 = 0，a = 2，b = -3，c = -5。公式给出：
x = ( 3 ± √9 + 40 ) ÷ 4 = ( 3 ± √49 ) ÷ 4 = ( 3 ± 7 ) ÷ 4
1. The quadratic formula is reliable and should be used when factorisation is not obvious.
因此 x = 10/4 = 2.5 或 x = -4/4 = -1。二次公式非常可靠，当因式分解不明显时应使用它。
6. The Discriminant | 判别式
² - 4ac. It tells you how many real solutions a quadratic equation has without solving it.
判别式是二次公式中平方根里面的表达式：Δ = b² - 4ac。它可以在不求解的情况下告诉你二次方程有多少个实数解。
The three cases are:
三种情况如下：
- If Δ > 0, there are two distinct real solutions.
- If Δ = 0, there is exactly one repeated real solution.
- If Δ < 0, there are no real solutions, but two complex solutions.
- 如果 Δ > 0，有两个不同的实数解。
- 如果 Δ = 0，恰好有一个重复实数解。
- 如果 Δ < 0，没有实数解，但有两个复数解。
ample, for x² - 4x + 4 = 0, Δ = (-4)² - 4(1)(4) = 0, so there is one repeated root, x = 2.
在 IGCSE 中，题目经常要求判断根的性质。例如，对于 x² - 4x + 4 = 0，Δ = (-4)² - 4(1)(4) = 0，因此有一个重复根 x = 2。
7. Graphical Interpretation | 图像意义
x-intercepts of the graph, that is, the points where the curve crosses the x-axis. If the curve does not cross the x-axis, the equation has no real solutions.
y = ax² + bx + c 的图像是一条抛物线。ax² + bx + c = 0 的解就是图像与 x 轴的交点，也就是曲线穿过 x 轴的点。如果曲线不与 x 轴相交，则方程没有实数解。
If a is positive, the parabola opens upwards and has a minimum point. If a is negative, it opens downwards and has a maximum point. The vertex has x-coordinate -b ÷ 2a, which is also the axis of symmetry.
如果 a 为正，抛物线开口向上，有一个最低点。如果 a 为负，抛物线开口向下，有一个最高点。顶点的 x 坐标为 -b ÷ 2a，这也是对称轴。
Sketching the graph is quicker if you find the y-intercept at (0, c), the x-intercepts by solving the equation, and the vertex. These three features give a clear shape for most IGCSE questions.
如果找到 y 轴截距 (0, c)、通过解方程找到 x 轴截距以及顶点，画图像会更快。这三个特征足以应对大多数 IGCSE 题目中的图像形状。
8. Quadratic Inequalities | 二次不等式
s zero. Factorising gives (x - 3)(x + 2) > 0, so the critical values are x = 3 and x = -2.
二次不等式，如 x² - x - 6 > 0，可以通过先找到表达式等于零的临界值来求解。因式分解得到 (x - 3)(x + 2) > 0，因此临界值为 x = 3 和 x = -2。
These values divide the number line into three intervals: x < -2, -2 < x < 3, and x > 3. Test a value from each interval to determine where the inequality is true. The solution is x < -2 or x > 3.
这些值将数轴分为三个区间：x < -2、-2 < x < 3 和 x > 3。从每个区间中选取一个值进行检验，以确定不等式在哪些区间成立。解为 x < -2 或 x > 3。
points: -2 ≤ x ≤ 3. Always pay attention to whether the inequality is strict or inclusive.
如果不等式是 x² - x - 6 ≤ 0，解将包含端点：-2 ≤ x ≤ 3。始终要注意不等式是严格不等还是包含等号。
9. Word Problems and Applications | 应用题与实际应用
Many IGCSE questions embed quadratic equations in real-life contexts, such as projectile motion, area problems, or profit maximisation. The key is to define a variable, form a quadratic equation, solve it, and then check that the solution makes sense in the original context.
许多 IGCSE 题目将二次方程嵌入实际情境中，例如抛体运动、面积问题或利润最大化。关键是要设出变量，列二次方程，求解，然后检验解在原始情境中是否有意义。
40 = 0. Factorising gives (x + 8)(x - 5) = 0, so x = 5 because a width cannot be negative.
例如，一个矩形的长比宽多 3 米，面积为 40 平方米。设宽为 x，则长为 x + 3。方程为 x(x + 3) = 40，即 x² + 3x - 40 = 0。因式分解为 (x + 8)(x - 5) = 0，所以 x = 5，因为宽度不能为负。
Always reject solutions that are impossible in context, such as negative lengths or times. This final check is often required to gain full marks in IGCSE word problems.
一定要舍去在情境中不合理的解，例如负长度或负时间。这个最后的检验通常是 IGCSE 应用题获得满分所必需的。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
One common error is forgetting to rearrange the equation into standard form before identifying a, b and c. Another is mishandling negative signs when substituting into the quadratic formula, especially when b is negative or c is negative.
一个常见错误是在识别 a、b、c 之前忘记将方程整理为标准形式。另一个常见错误是在代入二次公式时处理负号不当，尤其是当 b 或 c 为负数时。
When factorising, always expand your factors back to check they produce the original quadratic. In completing the square, remember to add the same value to both sides of the equation. In inequalities, sketching a quick sign table can prevent interval mistakes.
因式分解后，一定要将因式乘回去检查是否得到原来的二次式。配方法时要记住在方程两边同时加上相同的值。解不等式时，快速画出符号表可以避免区间错误。
- Always write the equation in standard form first.
- Check your solutions by substituting them back into the original equation.
- Use the discriminant to verify the number of real roots.
- Show all steps clearly because method marks are awarded even if the final answer is wrong.
- 始终先把方程写成标准形式。
- 将解代回原方程进行检验。
- 利用判别式验证实数根的个数。
- 清晰写出所有步骤，因为即使最终答案错误，方法分仍然可以获得。
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