Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

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

二次方程是 Edexcel IGCSE 数学大纲中最重要的主题之一。它出现在代数、图像、几何甚至应用题中。掌握这一主题对于取得高分至关重要。

1. What Is a Quadratic Equation? | 什么是二次方程？

A quadratic equation is an equation that can be written in the standard form:

二次方程是可以写成标准形式的方程：

ax² + bx + c = 0

where a, b and c are constants, and a ≠ 0. The highest power of the variable x is 2, which is why it is called “quadratic” (from the Latin word quadratus, meaning square).

其中 a、b 和 c 是常数，且 a ≠ 0。变量 x 的最高次数是 2，因此称为“二次”（源自拉丁语 quadratus，意为平方）。

x² − 3x + 2 = 0 is a quadratic equation.
x² − 3x + 2 = 0 是一个二次方程。
2x² + 4x − 6 = 0 is also quadratic.
2x² + 4x − 6 = 0 也是二次方程。
x³ − 2x + 1 = 0 is not quadratic.
x³ − 2x + 1 = 0 不是二次方程。

2. Solving by Factorisation | 因式分解法

The first method you should try is factorisation. This means writing the quadratic as a product of two brackets. For example:

你应该首先尝试的方法是因式分解。这意味着将二次方程写成两个括号的乘积。例如：

x² − 5x + 6 = (x − 2)(x − 3)

To solve the equation x² − 5x + 6 = 0, we use the fact that if the product of two numbers is zero, then at least one of them must be zero.

要解方程 x² − 5x + 6 = 0，我们利用一个事实：如果两个数的乘积为零，那么其中至少有一个为零。

(x − 2)(x − 3) = 0

So either x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.

因此要么 x − 2 = 0，要么 x − 3 = 0，得到 x = 2 或 x = 3。

For equations with a leading coefficient ≠ 1, such as 2x² + 5x − 3 = 0, you may need to factorise by grouping:

对于首项系数不等于 1 的方程，例如 2x² + 5x − 3 = 0，你可能需要分组分解：

2x² + 5x − 3 = (2x − 1)(x + 3)

Then the solutions are x = ½ and x = −3.

那么解为 x = ½ 和 x = −3。

3. Solving by Completing the Square | 配方法

Completing the square is a powerful algebraic technique. It rewrites a quadratic in the form:

配方法是一种强大的代数技巧。它将二次方程改写为以下形式：

a(x + p)² + q = 0

For example, take x² + 6x + 1 = 0. First, halve the coefficient of x (which is 6) to get 3, then write:

例如，取 x² + 6x + 1 = 0。首先，将 x 的系数（即 6）减半得到 3，然后写出：

(x + 3)² − 9 + 1 = 0

Simplify to get (x + 3)² − 8 = 0. Then:

化简得到 (x + 3)² − 8 = 0。然后：

(x + 3)² = 8

Taking square roots gives x + 3 = ±√8, so x = −3 ± 2√2.

开平方得 x + 3 = ±√8，所以 x = −3 ± 2√2。

Completing the square is especially useful when the equation cannot be factorised easily.

当方程不易因式分解时，配方法特别有用。

4. The Quadratic Formula | 求根公式

The quadratic formula works for every quadratic equation. For ax² + bx + c = 0:

求根公式适用于所有二次方程。对于 ax² + bx + c = 0：

x = (−b ± √(b² − 4ac)) / (2a)

This formula is derived from completing the square, and it is given in the Edexcel IGCSE formula booklet. You should memorise it anyway.

该公式由配方法推导而来，在 Edexcel IGCSE 公式手册中给出。你应该无论如何都要记住它。

Example: Solve 3x² − 5x − 2 = 0.

示例：解 3x² − 5x − 2 = 0。

Here a = 3, b = −5, c = −2.

这里 a = 3，b = −5，c = −2。

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

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

So x = 12/6 = 2 or x = −2/6 = −⅓.

所以 x = 12/6 = 2 或 x = −2/6 = −⅓。

5. The Discriminant | 判别式

The expression b² − 4ac inside the quadratic formula is called the discriminant. It tells us how many real roots the equation has:

求根公式中的式子 b² − 4ac 称为判别式。它告诉我们方程有多少个实数根：

Discriminant	Number of real roots
b² − 4ac > 0	Two distinct real roots
b² − 4ac = 0	One repeated real root
b² − 4ac < 0	No real roots (two complex roots)

For example, x² + 2x + 5 = 0 has discriminant 2² − 4×1×5 = 4 − 20 = −16, so it has no real roots.

例如，x² + 2x + 5 = 0 的判别式为 2² − 4×1×5 = 4 − 20 = −16，因此没有实数根。

The discriminant also tells us whether the graph crosses the x-axis, touches it, or does not meet it.

判别式还告诉我们图像是否与 x 轴相交、相切或不相交。

6. Roots and Coefficients (Vieta’s Formulas) | 根与系数的关系（韦达定理）

For a quadratic equation ax² + bx + c = 0 with roots α and β, the sum and product of the roots are related to the coefficients:

对于根为 α 和 β 的二次方程 ax² + bx + c = 0，根的和与积与系数有关：

α + β = −b/a

αβ = c/a

These are not always in the IGCSE syllabus, but they are useful for quick checks and for solving certain problems without factorising.

这些关系并不总是在 IGCSE 大纲中，但对于快速检验以及解决某些无需因式分解的问题很有用。

Example: For 2x² − 8x + 3 = 0, the sum of roots is −(−8)/2 = 4 and the product is 3/2.

示例：对于 2x² − 8x + 3 = 0，根的和为 −(−8)/2 = 4，积为 3/2。

7. Graphing Quadratic Functions | 二次函数的图像

The graph of y = ax² + bx + c is a parabola. If a > 0, it opens upwards (U-shaped). If a < 0, it opens downwards (n-shaped).

函数 y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，开口向上（U 形）。如果 a < 0，开口向下（n 形）。

The solutions of ax² + bx + c = 0 are the x-coordinates where the graph crosses the x-axis.

方程 ax² + bx + c = 0 的解就是图像与 x 轴交点的 x 坐标。

The turning point (vertex) can be found by completing the square. For y = (x + p)² + q, the vertex is at (−p, q).

顶点（转向点）可以通过配方找到。对于 y = (x + p)² + q，顶点在 (−p, q)。

Example: y = x² − 4x + 1 = (x − 2)² − 3. The vertex is at (2, −3).

示例：y = x² − 4x + 1 = (x − 2)² − 3。顶点在 (2, −3)。

8. Solving Quadratic Inequalities | 解二次不等式

Quadratics also appear in inequalities. For example, to solve x² − 5x + 6 > 0, first factorise:

二次式也出现在不等式中。例如，要解 x² − 5x + 6 > 0，先因式分解：

(x − 2)(x − 3) > 0

The roots are 2 and 3. Test intervals:

根为 2 和 3。测试区间：

x < 2: both factors negative → product positive
x < 2：两个因子均为负 → 乘积为正
2 < x < 3: one negative, one positive → product negative
2 < x < 3：一负一正 → 乘积为负
x > 3: both positive → product positive
x > 3：两个因子均为正 → 乘积为正

So the solution is x < 2 or x > 3.

因此解为 x < 2 或 x > 3。

Remember to use ≤ or ≥ when the inequality includes equality.

记住当不等式包含等号时要用 ≤ 或 ≥。

9. Applications in Geometry | 在几何中的应用

Quadratic equations often arise in geometry problems. For example, finding the side length of a square when its area is given.

二次方程经常出现在几何问题中。例如，已知正方形面积求边长。

Suppose a rectangle has length (x + 3) cm and width x cm. Its area is 40 cm².

假设一个矩形的长为 (x + 3) cm，宽为 x cm，面积为 40 cm²。

x(x + 3) = 40

x² + 3x − 40 = 0

Factorising: (x + 8)(x − 5) = 0, so x = 5 (since length cannot be negative).

因式分解：(x + 8)(x − 5) = 0，所以 x = 5（因为长度不能为负）。

Always check your answers in word problems — discard any negative lengths or times.

在应用题中始终检查你的答案——舍弃任何负的长度或时间。

10. Common Mistakes | 常见错误

Here are some frequent errors students make:

以下是一些学生常犯的错误：

Forgetting to set the equation to zero before factorising.
在因式分解之前忘记将方程化为零。
Dividing both sides by x when x could be zero — you lose roots.
当 x 可能为零时两边同时除以 x —— 你会丢失根。
Forgetting the ± sign when taking square roots.
开平方时忘记 ± 号。
Using the quadratic formula with an error in signs.
使用求根公式时符号出错。
Confusing a, b, c when the equation is not in standard form.
当方程不是标准形式时混淆 a、b、c。

Always double-check by substituting your answers back into the original equation.

始终通过将答案代入原方程来复查。

11. Practice Questions | 练习题目

Try these questions to test your understanding.

试试下面这些题来检验你的理解。

Solve x² − 7x + 12 = 0 by factorisation.
用因式分解法解 x² − 7x + 12 = 0。
Solve 2x² + 3x − 2 = 0 using the quadratic formula.
用求根公式解 2x² + 3x − 2 = 0。
Find the range of values of k for which x² + kx + 4 = 0 has real roots.
求使得 x² + kx + 4 = 0 有实数根时 k 的取值范围。

Answers: 1) x = 3 or 4. 2) x = ½ or −2. 3) Discriminant ≥ 0 → k² − 16 ≥ 0 → k ≤ −4 or k ≥ 4.

答案：1) x = 3 或 4。2) x = ½ 或 −2。3) 判别式 ≥ 0 → k² − 16 ≥ 0 → k ≤ −4 或 k ≥ 4。

12. Summary | 总结

Quadratic equations can be solved by factorisation, completing the square, or using the quadratic formula. The discriminant tells you about the nature of the roots. Graphs of quadratics are parabolas, and their roots correspond to x-intercepts.

二次方程可以通过因式分解、配方法或求根公式来解。判别式告诉你根的性质。二次函数的图像是抛物线，其根对应于 x 轴交点。

Practise all three methods and know when to use each one. For Edexcel IGCSE, factorisation is often quickest, but the quadratic formula always works as a safety net.

练习所有三种方法，并知道何时使用哪一种。对于 Edexcel IGCSE，因式分解通常最快，但求根公式始终是安全网。

Good luck with your revision!

祝复习顺利！

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Solving Quadratic Equations | 解二次方程