Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are a fundamental part of the IGCSE mathematics syllabus. They appear in algebra, geometry, trigonometry, and many real-world problems. Mastering the different methods of solving them is essential for your Edexcel exams.

二次方程是 IGCSE 数学课程的基础部分。它们出现在代数、几何、三角学以及许多现实问题中。掌握不同求解方法对你的 Edexcel 考试至关重要。

1. Standard Form | 标准形式

A quadratic equation is an equation of the second degree, meaning the highest power of the variable is 2. The general form is written as:

二次方程是二次方程，即变量的最高次幂为 2。其一般形式写为：

ax² + bx + c = 0

where a, b and c are constants, and a ≠ 0. For example, x² – 5x + 6 = 0 is a quadratic equation with a = 1, b = -5, c = 6.

其中 a、b 和 c 是常数，且 a ≠ 0。例如，x² – 5x + 6 = 0 是一个二次方程，其中 a = 1，b = -5，c = 6。

2. Solving by Factorisation | 用因式分解求解

If a quadratic expression can be written as the product of two linear factors, we can use the zero-product property. This states that if A × B = 0, then either A = 0 or B = 0.

如果二次表达式可以写成两个一次因式的乘积，我们可以利用零积性质。该性质说明：若 A × B = 0，则 A = 0 或 B = 0。

For example, solve x² – 5x + 6 = 0. We look for two numbers that multiply to 6 and add to -5, which are -2 and -3:

例如，解 x² – 5x + 6 = 0。我们需要找到两个相乘为 6、相加为 -5 的数，它们是 -2 和 -3：

x² – 5x + 6 = (x – 2)(x – 3) = 0

Thus 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。

Always check if the coefficient of x² is not 1. For example, solve 2x² + 5x – 3 = 0. We split the middle term or use factor pairs:

当 x² 的系数不是 1 时，要特别小心。例如，解 2x² + 5x – 3 = 0。我们可以使用十字相乘法或因式分组：

2x² + 5x – 3 = (2x – 1)(x + 3) = 0

So x = ½ or x = -3.

所以 x = ½ 或 x = -3。

3. Solving by the Quadratic Formula | 用求根公式求解

When factorisation is difficult or impossible, the quadratic formula can always be used. For ax² + bx + c = 0, the solutions are given by:

当因式分解困难或不可能时，求根公式总是可以使用。对于 ax² + bx + c = 0，解为：

x = (-b ± √(b² – 4ac)) / (2a)

This formula works for all quadratic equations, provided a ≠ 0. You must be careful with signs and substitute correctly.

此公式适用于所有二次方程，前提是 a ≠ 0。使用时要特别注意符号并正确代入。

For example, solve 2x² – 4x – 3 = 0. Here a = 2, b = -4, c = -3. Substituting into the formula gives:

例如，解 2x² – 4x – 3 = 0。这里 a = 2，b = -4，c = -3。代入公式得：

x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4

Simplify √40 = 2√10, so x = (4 ± 2√10) / 4 = 1 ± √10 / 2. Therefore the two solutions are x ≈ 2.58 and x ≈ -1.58.

化简 √40 = 2√10，所以 x = (4 ± 2√10) / 4 = 1 ± √10 / 2。因此两个解为 x ≈ 2.58 和 x ≈ -1.58。

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

Completing the square rewrites a quadratic in the form a(x + p)² + q. This method is especially useful for finding turning points of graphs and for deriving the quadratic formula itself.

配方法将二次式改写为 a(x + p)² + q 的形式。此方法在求图像拐点以及推导求根公式时特别有用。

The basic steps for x² + bx + c = 0 are:

对于 x² + bx + c = 0 的基本步骤如下：

Halve the coefficient of x, b/2, and write (x + b/2)².
抽取 x 系数的一半 b/2，写成 (x + b/2)²。
Subtract (b/2)² to keep the expression equivalent.
减去 (b/2)² 以保持等价。

For instance, solve x² + 6x + 2 = 0. Complete the square:

例如，解 x² + 6x + 2 = 0。用配方法：

(x + 3)² – 9 + 2 = (x + 3)² – 7 = 0

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

于是 (x + 3)² = 7，所以 x + 3 = ±√7，得 x = -3 ± √7。

5. Solving by Graphical Method | 用图像法求解

The roots of a quadratic equation ax² + bx + c = 0 are the x-intercepts of the graph y = ax² + bx + c. Plotting the curve gives approximate solutions.

二次方程 ax² + bx + c = 0 的根就是图像 y = ax² + bx + c 与 x 轴的交点。绘制曲线可得到近似解。

To sketch or plot, calculate the vertex and axis of symmetry. The axis is x = -b/(2a), and the vertex lies on this line.

绘制图形时，需计算顶点和对称轴。对称轴为 x = -b/(2a)，顶点在该直线上。

For example, y = x² – 2x – 3 has vertex at x = 1 and y = -4. Its roots are x = -1 and x = 3, as seen when the graph crosses the x-axis.

例如，y = x² – 2x – 3 的顶点在 x = 1，y = -4。当图像与 x 轴相交时可看出根为 x = -1 和 x = 3。

If the graph does not cross the x-axis, the equation has no real roots. This is linked to the discriminant.

如果图像不与 x 轴相交，则方程没有实数根。这与判别式有关。

6. The Discriminant | 判别式

The expression b² – 4ac inside the quadratic formula is called the discriminant. It determines the nature of the roots without fully solving the equation.

求根公式中的表达式 b² – 4ac 称为判别式。它无需完全求解方程即可判断根的性质。

If b² – 4ac > 0, there are two distinct real roots.
若 b² – 4ac > 0，则有两个不同的实数根。
If b² – 4ac = 0, there is exactly one repeated real root.
若 b² – 4ac = 0，则有一个相等的实数根。
If b² – 4ac < 0, there are no real roots (two complex roots).
若 b² – 4ac < 0，则没有实数根（有两个复数根）。

For example, for 3x² – 4x + 2 = 0, the discriminant is (-4)² – 4(3)(2) = 16 – 24 = -8, so there are no real roots.

例如，对于 3x² – 4x + 2 = 0，判别式为 (-4)² – 4(3)(2) = 16 – 24 = -8，因此没有实数根。

7. Word Problems | 应用题

Many exam questions present a quadratic equation from a real-life context. You must translate the problem into an equation, solve it, and check the validity of your answers.

许多考试题目会从实际情境中给出二次方程。你需要将问题转化为方程，求解，并检查答案的合理性。

Example: A rectangle has length 3 cm more than its width. Its area is 70 cm². Let width = x cm. Then length = x + 3, so x(x + 3) = 70.

例如：一个矩形的长比宽多 3 cm。面积为 70 cm²。设宽为 x cm。则长为 x + 3，所以 x(x + 3) = 70。

x² + 3x – 70 = 0

Factorising gives (x + 10)(x – 7) = 0, so x = -10 or x = 7. Since width cannot be negative, x = 7 cm, and length = 10 cm.

因式分解得 (x + 10)(x – 7) = 0，所以 x = -10 或 x = 7。由于宽不能为负，x = 7 cm，长为 10 cm。

8. Choosing the Best Method | 选择最佳方法

For the Edexcel IGCSE exam, you need to decide which method to use efficiently. Factorisation is quick when the roots are integers or simple fractions. Completing the square is good for finding turning points. The quadratic formula is a reliable backup for everything.

对于 Edexcel IGCSE 考试，你需要高效地决定使用哪种方法。当根为整数或简单分数时，因式分解很快。配方法适合求拐点。求根公式是万无一失的后备方法。

Method	Best used when
Factorisation	Simple integer or fractional roots
Completing the square	Need vertex form or irrational roots
Quadratic formula	Any quadratic, especially with messy coefficients

Always check whether the equation is already in standard form. If not, rearrange first. Remember that sometimes an equation can be solved by taking the square root directly, like (x – 4)² = 25.

始终检查方程是否已化为标准形式。如果没有，先移项整理。请记住，有些方程可以直接开平方求解，例如 (x – 4)² = 25。

9. Practice Tips | 练习建议

To master quadratic equations, practice a variety of problems. In the exam, always write down your working clearly, because method marks are awarded even if a small arithmetic error occurs.

为了掌握二次方程，请练习多种类型的问题。在考试中，一定要清晰写出步骤，因为即使有小的计算错误，方法分仍然会给。

Memorise the quadratic formula and the discriminant condition.
牢记求根公式和判别式的条件。
Practise factorising quadratics with leading coefficient not equal to 1.
练习首项系数不为 1 的二次式因式分解。
Use approximations to check your answers make sense.
使用近似值检查答案是否合理。

Consistent revision will build your confidence for the algebra sections of the IGCSE Edexcel exams.

持续复习将使你对 Edexcel IGCSE 考试中的代数部分充满信心。

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Solving Quadratic Equations | 解二次方程