📚 Solving Quadratic Equations | 解一元二次方程
A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in algebra, graphs, geometry and problem-solving questions. Mastering this topic gives you a strong foundation for higher-level mathematics.
一元二次方程是 IGCSE 数学中最重要的主题之一。它出现在代数、图像、几何和解题类问题中。掌握这一主题能为你学习更高层次的数学打下坚实基础。
1. Standard Form of a Quadratic Equation | 一元二次方程的标准形式
The general form is written as ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.
一元二次方程是变量最高次数为 2 的方程。其一般形式写作 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。
erwise the equation becomes linear. For example, 2x² + 3x − 5 = 0 is a quadratic equation.
这里 x 是未知数。系数 a 不能为零，否则方程就变成了一次方程。例如，2x² + 3x − 5 = 0 就是一个一元二次方程。
− 9 = 0. Always check if the equation can be rearranged into standard form before solving.
像 x² − 9 = 0 这样的方程也是一元二次方程，因为它可以写成 1x² + 0x − 9 = 0。在解题前，务必检查方程是否能整理成标准形式。
2. Solving by Factorising | 因式分解法解方程
Factorising is often the fastest method when the quadratic can be written as a product of two linear brackets. The principle is that if the product of two expressions is zero, then at least one of them must be zero.
当二次式可以写成两个一次括号相乘时，因式分解通常是最快的方法。其原理是：如果两个表达式的乘积为零，那么其中至少有一个表达式必须为零。
hat multiply to 6 and add to 5. Those numbers are 2 and 3, so we write (x + 2)(x + 3) = 0.
例如，解 x² + 5x + 6 = 0。我们要找两个数，它们相乘等于 6，相加等于 5。这两个数是 2 和 3，因此写成 (x + 2)(x + 3) = 0。
= 0. This gives x = −2 or x = −3. Always expand your factorised form to check your answer.
然后令每个括号等于零：x + 2 = 0 或 x + 3 = 0。解得 x = −2 或 x = −3。始终将因式分解后的形式展开，以检查答案是否正确。
x + 3 = 0, you may need to use the method of splitting the middle term or trial and error.
对于首项系数大于 1 的方程，例如 2x² + 7x + 3 = 0，你可能需要使用拆中项法或试错法。
3. Solving by Completing the Square | 配方法解方程
Completing the square rewrites a quadratic in the form a(x + p)² + q. This is useful when factorising is difficult and it also helps in finding the turning point of a quadratic graph.
配方法将二次式改写为 a(x + p)² + q 的形式。当因式分解困难时，这种方法很有用，同时也有助于找到二次图像的顶点。
ct this value. For example, x² + 6x + 2 can be written as (x + 3)² − 9 + 2 = (x + 3)² − 7.
对于 x² + bx + c，配平方时取 b 的一半，平方它，然后加上并减去这个值。例如，x² + 6x + 2 可写成 (x + 3)² − 9 + 2 = (x + 3)² − 7。
to (x + 3)² = 7. Then take the square root of both sides, remembering the ± sign: x + 3 = √7 or x + 3 = −√7.
要解 (x + 3)² − 7 = 0，整理为 (x + 3)² = 7。然后两边开平方，记住 ± 号：x + 3 = √7 或 x + 3 = −√7。
Finally, x = −3 + √7 or x = −3 − √7. This method always works, even when the roots are irrational.
最后得到 x = −3 + √7 或 x = −3 − √7。这种方法始终有效，即使根是无理数。
4. Solving by the Quadratic Formula | 公式法解方程
universal method for solving any quadratic equation. For ax² + bx + c = 0, the formula is:
公式法是解任意一元二次方程的通用方法。对于 ax² + bx + c = 0，公式为：
x = (−b ± √b² − 4ac) / 2a
To use the formula, substitute the values of a, b and c into the formula. Be careful with negative signs and fractions.
使用公式时，将 a、b、c 的值代入公式。注意负号和分数，避免符号错误。
le, solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, c = −3. Substitute these into the formula.
例如，解 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。代入公式。
x = (4 ± √16 + 24) / 4 = (4 ± √40) / 4
Simplify √40 = 2√10, so x = (4 ± 2√10) / 4, which can also be written as x = 1 ± √10 / 2. Always simplify surds where possible.
化简 √40 = 2√10，所以 x = (4 ± 2√10) / 4，也可写成 x = 1 ± √10 / 2。如果可能，始终化简根式。
5. The Discriminant | 判别式
The expression b² − 4ac inside the quadratic formula is called the discriminant. It tells us the number and type of roots without solving the equation fully.
公式中根号内的表达式 b² − 4ac 称为判别式。它可以在不完全解方程的情况下告诉我们根的数量和类型。
= 0, it has exactly one real root (repeated root). If b² − 4ac < 0, it has no real roots.
如果 b² − 4ac > 0，方程有两个不同的实数根。如果 b² − 4ac = 0，它只有一个实数根（重根）。如果 b² − 4ac < 0，则没有实数根。
4x + 4 = 0, the discriminant is (−4)² − 4×1×4 = 16 − 16 = 0, so it has one repeated root.
例如，对于 x² − 4x + 4 = 0，判别式为 (−4)² − 4×1×4 = 16 − 16 = 0，所以它有一个重根。
The discriminant also tells us about the graph: positive discriminant means the graph crosses the x-axis twice; zero means it touches once; negative means it never touches the x-axis.
判别式也告诉我们图像的信息：判别式为正，图像与 x 轴相交两次；为零，图像与 x 轴相切一次；为负，图像与 x 轴没有交点。
6. Roots, Solutions and x-intercepts | 根、解与 x 轴交点
solutions. Graphically, these roots are the x-intercepts of the parabola y = ax² + bx + c.
一元二次方程的根是满足方程的 x 值，也称为解。从图像上看，这些根就是抛物线 y = ax² + bx + c 与 x 轴的交点。
If the quadratic equation is given in factorised form, each factor gives one root. If it is in completed square form, the turning point gives direct information about the location of the roots.
如果一元二次方程以因式分解形式给出，每个因式给出一个根。如果以配方法形式给出，顶点直接提供根的位置信息。
These are the x-intercepts of the graph, and they match the solutions of x² − 5x + 6 = 0.
例如，y = x² − 5x + 6 的图形在 x = 2 和 x = 3 处与 x 轴相交。这些是图的 x 轴交点，与方程 x² − 5x + 6 = 0 的解一致。
7. Quadratic Graphs and the Turning Point | 二次函数图像与顶点
The graph of a quadratic function is a parabola. If a > 0, the parabola opens upwards and has a minimum turning point. If a < 0, it opens downwards and has a maximum turning point.
二次函数的图像是一条抛物线。如果 a > 0，抛物线开口向上，有最小值顶点。如果 a < 0，抛物线开口向下，有最大值顶点。
p)² + q. The vertex is at (−p, q). For example, y = (x − 3)² + 2 has its vertex at (3, 2).
顶点可以通过配方法形式 a(x + p)² + q 求得。顶点坐标为 (−p, q)。例如，y = (x − 3)² + 2 的顶点在 (3, 2)。
point, with equation x = −p. This line divides the parabola into two mirror-image halves.
对称轴是过顶点的一条竖直线，其方程为 x = −p。这条线将抛物线分成两个镜像对称的部分。
To sketch a quadratic graph, find the roots, the y-intercept and the turning point. Then draw a smooth curve through these key points.
要绘制二次函数草图，需要找到根、y 轴截距和顶点，然后用平滑曲线穿过这些关键点。
8. Forming a Quadratic Equation from Given Roots | 由已知根建立一元二次方程
the equation can be written as (x − α)(x − β) = 0. Expanding gives x² − (α + β)x + αβ = 0.
如果一元二次方程的根是 α 和 β，那么方程可以写成 (x − α)(x − β) = 0。展开后得到 x² − (α + β)x + αβ = 0。
So the sum of the roots is α + β and the product of the roots is αβ. This is a quick way to check or construct equations.
因此，两根之和为 α + β，两根之积为 αβ。这是检查或构建方程的快捷方法。
with roots 3 and −4 is x² − (3 + (−4))x + 3×(−4) = 0, which simplifies to x² + x − 12 = 0.
例如，根为 3 和 −4 的一元二次方程是 x² − (3 + (−4))x + 3×(−4) = 0，化简得 x² + x − 12 = 0。
This method is especially useful in problems where reverse engineering from roots is required, such as in coordinate geometry.
这种方法在需要从根反推方程的问题中特别有用，例如在坐标几何问题中。
9. Word Problems with Quadratic Equations | 一元二次方程应用题
Many real-world problems lead to quadratic equations. Common topics include area, projectile motion, number puzzles and sales or profit calculations.
许多现实问题会引出一元二次方程。常见主题包括面积、抛体运动、数字谜题以及销售或利润计算。
and its area is 40 cm². Let width = x, then length = x + 3. The equation is x(x + 3) = 40.
例如，一个矩形的长比宽多 3 cm，面积为 40 cm²。设宽为 x，则长为 x + 3。方程为 x(x + 3) = 40。
(x + 8)(x − 5) = 0, so x = −8 or x = 5. Since a length cannot be negative, we take x = 5.
展开得 x² + 3x − 40 = 0。因式分解：(x + 8)(x − 5) = 0，所以 x = −8 或 x = 5。由于长度不能为负，我们取 x = 5。
Always check whether your solutions make sense in the context of the problem. Reject any negative or unrealistic answers, even if they are mathematically valid.
始终检查你的解是否符合问题的实际背景。即使数学上有效，也要舍去负数或不现实的答案。
10. Common Mistakes and How to Avoid Them | 常见错误与避免方法
dard form before applying the formula or factorising. Always write ax² + bx + c = 0 first.
一个常见错误是忘记在使用公式或因式分解前将方程整理成标准形式。务必先写成 ax² + bx + c = 0。
is losing the ± sign when taking square roots. Remember that x² = k has two solutions: x = √k and x = −√k, unless k is zero.
另一个错误是在开平方时丢掉 ± 号。记住 x² = k 有两个解：x = √k 和 x = −√k，除非 k 为零。
s x = 2 or x = 3. The roots should be the numbers that make each bracket zero: x = −2, −3.
因式分解时，学生有时会写出 (x + 2)(x + 3) = 0，然后错误地解得 x = 2 或 x = 3。根应使每个括号为零，即 x = −2、−3。
Check your answers by substituting them back into the original equation. This simple step catches most careless errors.
将所有解代回原方程进行检验。这个简单的步骤可以捕捉到大多数粗心错误。
11. Choosing the Best Method | 选择最佳解题方法
In the exam, you should choose the method that is fastest for the given equation. Factorising is best when the equation has simple integer roots.
考试中，应根据给定方程选择最快的方法。当方程具有简单整数根时，因式分解最好。
rm like (x + p)² = q, or when the coefficient of x is even and you need the turning point.
当方程已经接近 (x + p)² = q 的形式，或者 x 项系数为偶数且需要求顶点时，配方法很有用。
The quadratic formula is always reliable, especially for equations with irrational or decimal roots. It may take longer, but it works for every quadratic.
公式法始终可靠，尤其适用于根为无理数或小数的方程。它可能花费更多时间，但适用于所有一元二次方程。
Use the discriminant to check whether the roots are real before you start solving. This can save time and prevent frustration.
在开始求解前，使用判别式检查根是否真实存在。这样可以节省时间并避免不必要的麻烦。
12. Summary of Key Formulas | 关键公式总结
a quadratic equation ax² + bx + c = 0, the following are the key tools you must remember.
对于一元二次方程 ax² + bx + c = 0，以下是必须记住的关键工具。
| Name / 名称 | Formula / 公式 |
|---|---|
| Quadratic formula / 求根公式 | x = (−b ± √b² − 4ac) / 2a |
| Discriminant / 判别式 | Δ = b² − 4ac |
| Sum of roots / 两根之和 | −b / a |
| Product of roots / 两根之积 | c / a |
| Completed square form / 配方形式 | a(x + p)² + q |
Mastering these formulas and knowing when to apply them will help you solve any quadratic equation efficiently. Practice different types of questions to build confidence.
掌握这些公式并知道何时应用，可以帮助你高效地解任何一元二次方程。练习不同类型的题目以建立信心。
Remember to always write your working clearly. In an exam, show every step so that you can earn method marks even if you make a small arithmetic error.
记住始终清晰地写出你的步骤。在考试中，展示每一步，即使出现小的算术错误，也能获得方法分。
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