Quadratic Equations | 二次方程

📚 Quadratic Equations | 二次方程

Quadratic equations are one of the most important topics in the Edexcel IGCSE Mathematics syllabus. They appear in algebra, graphs, geometry and problem-solving questions. Mastering them is essential for achieving a high grade.

二次方程是 Edexcel IGCSE 数学考纲中最重要的主题之一。它出现在代数、图像、几何和应用题中。掌握二次方程是取得高分的关键。

1. Standard Form and Identifying Coefficients | 标准形式与识别系数

A quadratic equation is any equation that can be written in the general form:

ax² + bx + c = 0

where a, b and c are real numbers, and a ≠ 0. The coefficient a is the coefficient of x², b is the coefficient of x, and c is the constant term.

二次方程是可以写成一般形式 ax² + bx + c = 0 的方程，其中 a、b、c 是实数，且 a ≠ 0。系数 a 是 x² 的系数，b 是 x 的系数，c 是常数项。

For example, in the equation 2x² − 3x + 5 = 0, we have a = 2, b = −3, and c = 5.

例如，在方程 2x² − 3x + 5 = 0 中，a = 2，b = −3，c = 5。

Sometimes you must expand or rearrange an expression before you can identify these coefficients. For example, (x + 1)² = 3x − 2 must be expanded first to give x² + 2x + 1 = 3x − 2, which then becomes x² − x + 3 = 0.

有时你需要先展开或整理表达式，才能识别这些系数。例如，(x + 1)² = 3x − 2 必须先展开为 x² + 2x + 1 = 3x − 2，然后整理为 x² − x + 3 = 0。

2. Solving by Factorisation | 用因式分解求解

If a quadratic expression can be factorised into a product of two linear factors, then the equation can be solved easily. The principle is: if the product of two expressions is zero, then at least one of them must be zero.

如果一个二次表达式可以分解成两个一次因式的乘积，那么方程就可以轻松求解。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必须为零。

Consider the equation x² − 5x + 6 = 0. We look for two numbers that multiply to give 6 and add to give −5. Those numbers are −2 and −3.

考虑方程 x² − 5x + 6 = 0。我们需要找到两个数，乘积为 6，和为 −5。这两个数是 −2 和 −3。

(x − 2)(x − 3) = 0

Therefore x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3.

因此 x − 2 = 0 或 x − 3 = 0，所以 x = 2 或 x = 3。

For equations where the coefficient of x² is not 1, you may need to use a method such as splitting the middle term. For example:

当 x² 的系数不为 1 时，你可能需要使用“拆中项”等方法。例如：

2x² + 7x + 3 = (2x + 1)(x + 3)

Check: (2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3. Always expand mentally to check your factorisation.

检验：(2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3。务必心里展开来检查因式分解是否正确。

3. Solving by Completing the Square | 用配方法求解

Completing the square rewrites a quadratic expression in the form a(x + p)² + q. This is useful when factorisation is not easy or when you need to find turning points.

配方法将二次表达式改写为 a(x + p)² + q 的形式。当因式分解困难或需要求顶点坐标时，这种方法很有用。

To solve x² + 6x + 2 = 0 by completing the square, first take the coefficient of x (which is 6), halve it to get 3, and write:

要用配方法求解 x² + 6x + 2 = 0，首先取 x 的系数 6，一半为 3，然后写：

(x + 3)² − 9 + 2 = 0

Simplify to get:

化简得到：

(x + 3)² − 7 = 0

Then (x + 3)² = 7, so x + 3 = ±√7, giving x = −3 ± √7.

然后 (x + 3)² = 7，所以 x + 3 = ±√7，因此 x = −3 ± √7。

If the coefficient of x² is not 1, first factor it out. For example, 2x² − 8x + 1 = 0 becomes 2(x² − 4x) + 1 = 0, then complete the square inside the bracket.

如果 x² 的系数不为 1，先提出来。例如，2x² − 8x + 1 = 0 可写成 2(x² − 4x) + 1 = 0，然后在括号内配方。

4. The Quadratic Formula | 求根公式

The quadratic formula can solve any quadratic equation, even those that cannot be factorised. For ax² + bx + c = 0, the solutions are given by:

求根公式可以解任何二次方程，包括无法因式分解的方程。对于 ax² + bx + c = 0，解为：

x = [−b ± √(b² − 4ac)] / (2a)

This formula is provided in the Edexcel IGCSE formula booklet, but you must know how to substitute values accurately.

这个公式在 Edexcel IGCSE 公式册中会给出，但你必须知道如何准确代入数值。

For example, solve 3x² + 5x − 2 = 0. Here a = 3, b = 5, c = −2.

例如，解 3x² + 5x − 2 = 0。这里 a = 3，b = 5，c = −2。

x = [−5 ± √(25 − 4 × 3 × (−2))] / (2 × 3)

x = [−5 ± √(25 + 24)] / 6 = [−5 ± √49] / 6 = [−5 ± 7] / 6

So x = (−5 + 7)/6 = 2/6 = 1/3, or x = (−5 − 7)/6 = −12/6 = −2.

所以 x = (−5 + 7)/6 = 2/6 = 1/3，或 x = (−5 − 7)/6 = −12/6 = −2。

5. The Discriminant | 判别式

The part of the quadratic formula under the square root, b² − 4ac, is called the discriminant. It tells us how many real roots a quadratic equation has.

求根公式中根号下的部分 b² − 4ac 称为判别式。它告诉我们二次方程有几个实数根。

If b² − 4ac > 0, the equation has two distinct real roots.

如果 b² − 4ac > 0，方程有两个不相等的实数根。
If b² − 4ac = 0, the equation has exactly one real root (repeated root).

如果 b² − 4ac = 0，方程恰好有一个实数根（重根）。
If b² − 4ac < 0, the equation has no real roots.

如果 b² − 4ac < 0，方程没有实数根。

For example, x² − 4x + 4 = 0 has discriminant (−4)² − 4 × 1 × 4 = 16 − 16 = 0, so it has one repeated root x = 2.

例如，x² − 4x + 4 = 0 的判别式为 (−4)² − 4 × 1 × 4 = 16 − 16 = 0，因此它有一个重根 x = 2。

6. Solving by Drawing Graphs | 用图像求解

A quadratic equation ax² + bx + c = 0 can also be solved graphically. The real roots are the x-coordinates where the curve y = ax² + bx + c crosses the x-axis.

二次方程 ax² + bx + c = 0 也可以通过图像求解。实数根是曲线 y = ax² + bx + c 与 x 轴交点的横坐标。

To use this method, plot the graph of y = ax² + bx + c and read off the x-intercepts. If the graph does not cross the x-axis, there are no real roots.

使用这种方法时，先画出 y = ax² + bx + c 的图像，然后读出与 x 轴的交点。如果图像不与 x 轴相交，则没有实数根。

The graph of a quadratic function is a parabola. When a > 0 it opens upwards and has a minimum point; when a < 0 it opens downwards and has a maximum point.

二次函数的图像是抛物线。当 a > 0 时开口向上，有最低点；当 a < 0 时开口向下，有最高点。

7. Finding the Turning Point | 求顶点坐标

The turning point of a quadratic graph can be found by using the completed square form y = a(x + p)² + q. The vertex is at (−p, q).

二次函数图像的顶点可以通过配方法得到的形式 y = a(x + p)² + q 来求。顶点坐标为 (−p, q)。

For example, for y = (x − 3)² + 2, the vertex is (3, 2). The line of symmetry is x = 3.

例如，对于 y = (x − 3)² + 2，顶点是 (3, 2)。对称轴是 x = 3。

If you start from y = ax² + bx + c, you can also use the formula x = −b/(2a) to find the x-coordinate of the turning point, then substitute back to find y.

如果从 y = ax² + bx + c 出发，也可以用公式 x = −b/(2a) 求顶点的横坐标，再代入求纵坐标。

x_vertex = −b / (2a)

8. Quadratic Word Problems | 二次方程应用题

Many real-world problems can be modelled by quadratic equations. For example, the area of a rectangle, the height of a projectile, or the product of two consecutive integers.

许多实际问题都可以用二次方程建模。例如矩形的面积、抛射体的高度、两个连续整数的乘积等。

Example: The length of a rectangle is 3 cm more than its width, and its area is 40 cm². Let the width be x cm. Then the length is x + 3 cm, so x(x + 3) = 40.

例子：一个矩形的长比宽多 3 cm，面积为 40 cm²。设宽为 x cm，则长为 x + 3 cm，所以 x(x + 3) = 40。

x² + 3x − 40 = 0

Factorise: (x + 8)(x − 5) = 0. So x = −8 or x = 5. Since a width cannot be negative, x = 5 cm.

因式分解：(x + 8)(x − 5) = 0。所以 x = −8 或 x = 5。由于宽度不能为负数，所以 x = 5 cm。

9. The Sum and Product of Roots | 根的和与积

If α and β are the roots of the equation ax² + bx + c = 0, then we have the following relationships:

如果 α 和 β 是方程 ax² + bx + c = 0 的两个根，则有如下关系：

α + β = −b / a

αβ = c / a

These formulas are very useful for forming quadratic equations. If you are given the sum S and product P of the roots, the equation is x² − Sx + P = 0.

这些公式在构造二次方程时非常有用。如果已知根的和 S 与积 P，则方程为 x² − Sx + P = 0。

For example, if a quadratic equation has roots 2 and −5, then S = −3 and P = −10, so the equation is x² + 3x − 10 = 0.

例如，如果一个二次方程的根为 2 和 −5，则 S = −3，P = −10，所以方程为 x² + 3x − 10 = 0。

10. Quadratic Inequalities | 二次不等式

Quadratic inequalities often appear in exam questions. To solve them, first find the roots of the corresponding equation, then sketch the graph to decide which regions satisfy the inequality.

二次不等式经常出现在考试题中。要解它们，先求对应方程的根，然后画出草图以判断哪些区间满足不等式。

For example, solve x² − x − 6 > 0. Factorise to get (x − 3)(x + 2) > 0. The roots are x = 3 and x = −2.

例如，解不等式 x² − x − 6 > 0。因式分解得到 (x − 3)(x + 2) > 0。根为 x = 3 和 x = −2。

Since the graph of y = x² − x − 6 opens upwards, it is positive outside the roots. Therefore the solution is x < −2 or x > 3.

因为 y = x² − x − 6 的图像开口向上，所以在两根之外函数值为正。因此解为 x < −2 或 x > 3。

11. Common Mistakes and Tips | 常见错误与技巧

Many students lose marks on quadratic questions because of small errors. Here are some important tips to avoid them.

许多学生因为小错误在二次方程题目上失分。以下是一些避免错误的要点。

Always rearrange the equation into the form ax² + bx + c = 0 before solving.

求解前务必把方程整理成 ax² + bx + c = 0 的形式。
Do not forget to change the signs when bringing terms to the other side.

移项时不要忘记变号。
When using the quadratic formula, be careful with negative signs: −b and b² − 4ac must be handled correctly.

使用求根公式时，注意负号：−b 和 b² − 4ac 必须正确处理。
Check your solutions by substituting them back into the original equation.

将解代回原方程进行检验。
In word problems, reject any negative or impossible answers based on the context.

在应用题中，根据实际情况舍去负数或不可能的答案。

12. Exam Practice Summary | 考试练习总结

To score well in quadratic questions, you must be confident with all the methods: factorisation, completing the square, the quadratic formula, and using the discriminant. Practice past-paper style questions until the steps become automatic.

要在二次方程题目中拿高分，你必须熟练掌握所有方法：因式分解、配方法、求根公式和判别式。反复练习真题风格的题目，直到步骤变得熟练自如。

Remember that a quadratic can have two roots, one root, or no real roots. The question may ask you to find the set of values of k for which the equation has two distinct roots, so use the discriminant carefully.

记住二次方程可能有两个根、一个根或没有实数根。题目可能会要求你求使方程有两个不等实根的 k 的取值范围，此时要谨慎使用判别式。

Finally, always write your working clearly. In Edexcel IGCSE Mathematics, method marks are awarded even if your final answer is wrong, so show every step.

最后，务必清晰写出解题过程。在 Edexcel IGCSE 数学中，即使最终答案错误，也会按步骤给分，所以每一步都要写清楚。

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Quadratic Equations | 二次方程