Solving Quadratic Equations | 二次方程求解

📚 Solving Quadratic Equations | 二次方程求解

Quadratic equations appear frequently in IGCSE Mathematics. Mastering their solution techniques is essential for both Paper 1 and Paper 2 of the Edexcel syllabus.

二次方程是 IGCSE 数学中的高频考点。掌握其求解方法对于 Edexcel 考试局的 Paper 1 和 Paper 2 都至关重要。

1. Standard Form & Key Terms | 标准形式与关键术语

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b and c are real numbers and a ≠ 0.

二次方程是指可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为实数，且 a ≠ 0。

Here, ‘a’ is the coefficient of x², ‘b’ is the coefficient of x, and ‘c’ is the constant term. The highest power of x is 2, which is why it is called ‘quadratic’.

其中，a 是 x² 的系数，b 是 x 的系数，c 是常数项。未知数 x 的最高次数为 2，因此称为“二次”。

Example: 2x² + 3x – 5 = 0 is a quadratic equation.

例如：2x² + 3x – 5 = 0 就是一个二次方程。

2. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method when the quadratic has simple integer roots. The idea is to rewrite the quadratic as a product of two linear expressions.

因式分解法通常在方程具有简单整数根时最为快捷。其思路是将二次表达式改写为两个一次表达式的乘积。

To solve x² + 5x + 6 = 0, we look for two numbers that multiply to 6 and add to 5. These numbers are 2 and 3.

解 x² + 5x + 6 = 0 时，我们寻找两个数，它们相乘得 6，相加得 5。这两个数是 2 和 3。

x² + 5x + 6 = (x + 2)(x + 3) = 0

Then, by the zero product property, either x + 2 = 0 or x + 3 = 0. Hence x = -2 or x = -3.

接着根据零乘积性质，要么 x + 2 = 0，要么 x + 3 = 0，因此 x = -2 或 x = -3。

For equations where the coefficient a ≠ 1, such as 2x² + 7x + 3 = 0, you may need to use the ‘ac method’ or factor by grouping.

当 a ≠ 1 时，例如 2x² + 7x + 3 = 0，可能需要使用“ac 方法”或分组分解。

3. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x + p)² + q. This is useful for finding turning points and solving equations that do not factor nicely.

配方法将二次式改写为 a(x + p)² + q 的形式。它常用于求函数图像的顶点，也用于解不易因式分解的方程。

Consider x² + 6x – 7 = 0. Write the first two terms as (x + 3)² – 9, because (x + 3)² = x² + 6x + 9.

考虑 x² + 6x – 7 = 0。将前两项写成 (x + 3)² – 9，因为 (x + 3)² = x² + 6x + 9。

(x + 3)² – 9 – 7 = 0

So (x + 3)² = 16. Taking square roots gives x + 3 = ±4, therefore x = 1 or x = -7.

于是 (x + 3)² = 16。开平方得 x + 3 = ±4，所以 x = 1 或 x = -7。

Remember to add the square of half the coefficient of x, then subtract it to keep the expression balanced.

请记住，要加上 x 系数一半的平方，再减去同一数值以保持等式平衡。

4. The Quadratic Formula | 求根公式

The quadratic formula works for all quadratic equations, including those that are difficult to factorise. For ax² + bx + c = 0, the solutions are given by:

求根公式适用于所有二次方程，包括难以因式分解的类型。对于 ax² + bx + c = 0，其解由下式给出：

x = (-b ± √(b² – 4ac)) / (2a)

For example, solve 2x² – 4x – 3 = 0. Here a = 2, b = -4, and c = -3.

例如，解 2x² – 4x – 3 = 0。这里 a = 2，b = -4，c = -3。

x = (4 ± √((-4)² – 4×2×(-3))) / (2×2) = (4 ± √40) / 4

Simplifying gives x = (2 ± √10) / 2.

化简得 x = (2 ± √10) / 2。

Always write the formula down first, then substitute carefully to avoid sign errors.

务必先写出公式，再代入数值，以避免符号错误。

5. The Discriminant | 判别式

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

求根公式中的 b² – 4ac 称为判别式。它无需解方程即可判断根的性质。

Discriminant Δ	Nature of roots	Graph intersection with x-axis
Δ > 0	Two distinct real roots	Crosses at two points
Δ = 0	One repeated real root	Touches at one point
Δ < 0	No real roots	Does not touch x-axis

For example, 3x² + 2x + 1 = 0 has Δ = 4 – 12 = -8, so it has no real solutions.

例如，3x² + 2x + 1 = 0 的判别式 Δ = 4 – 12 = -8，因此没有实数解。

6. Solving Word Problems | 二次方程应用题

Many exam problems require forming a quadratic equation from a real-world situation. Read the question carefully, define the unknown, and translate the conditions into an equation.

许多考试题目需要从实际情境构建二次方程。仔细读题，设定未知数，并将条件转化为方程。

Example: A rectangle has length (x + 3) cm and width (x – 2) cm. Its area is 50 cm². Find x.

例：一个矩形的长为 (x + 3) cm，宽为 (x – 2) cm，面积为 50 cm²。求 x。

(x + 3)(x – 2) = 50

Expanding gives x² + x – 6 = 50, so x² + x – 56 = 0. Factoring gives (x + 8)(x – 7) = 0, hence x = 7 (reject x = -8 because length cannot be negative).

展开得 x² + x – 6 = 50，即 x² + x – 56 = 0。因式分解得 (x + 8)(x – 7) = 0，故 x = 7（舍去 x = -8，因为长度不能为负）。

Always check whether your answers make sense in the original context, especially when negative values are involved.

务必检查答案在原始情境中是否合理，尤其是涉及负值时。

7. Graphs of Quadratic Functions | 二次函数图像

The graph of y = ax² + bx + c is a parabola. If a > 0 it opens upwards; if a < 0 it opens downwards.

函数 y = ax² + bx + c 的图像是抛物线。当 a > 0 时开口向上；当 a < 0 时开口向下。

The roots of the equation ax² + bx + c = 0 correspond to the x-intercepts of the parabola. The vertex is the maximum or minimum point.

方程 ax² + bx + c = 0 的根对应抛物线与 x 轴的交点。顶点是最大值或最小值点。

For y = x² – 4x + 3, completing the square gives y = (x – 2)² – 1, so the vertex is at (2, -1) and the roots are x = 1 and x = 3.

对于 y = x² – 4x + 3，配方得 y = (x – 2)² – 1，因此顶点为 (2, -1)，根为 x = 1 和 x = 3。

Sketching the graph quickly helps you visualise the solutions and verify your algebraic results.

快速画出草图有助于直观理解解的情况，并验证代数结果。

8. Simultaneous Equations with a Quadratic | 含二次项的联立方程

In IGCSE, you may be asked to solve a linear equation and a quadratic equation simultaneously. This often involves substitution.

在 IGCSE 考试中，你可能会遇到一个线性方程与一个二次方程联立求解的问题。通常使用代入法。

Example: Solve y = x + 1 and y = x² – 3x + 2.

例：解方程组 y = x + 1 和 y = x² – 3x + 2。

Since y = x + 1, substitute into the quadratic: x + 1 = x² – 3x + 2.

因为 y = x + 1，将其代入二次方程：x + 1 = x² – 3x + 2。

x² – 4x + 1 = 0

Using the quadratic formula with a = 1, b = -4, c = 1 gives x = 2 ± √3. Then substitute back to find y.

使用求根公式，a = 1，b = -4，c = 1，得 x = 2 ± √3。再代回求 y 值。

Always substitute both x-values into the linear equation to get the corresponding y-values.

始终将两个 x 值代入线性方程，以求得对应的 y 值。

9. Common Mistakes & Exam Tips | 常见错误与考试技巧

Do not forget the ± sign when taking square roots of both sides.
Do not cancel a factor of x unless x = 0 is already accounted for.
When expanding brackets, check the signs carefully, especially with subtraction.
Always rearrange the equation into standard form before factorising or using the formula.
If a quadratic has no obvious factors, use the quadratic formula rather than guessing.

请不要在两边开平方时忘记 ± 符号。

不可随意约去因式 x，除非已考虑 x = 0 情况。

展开括号时仔细检查符号，尤其是减法时。

在因式分解或使用公式前，务必将方程整理为标准形式。

如果二次式不易因式分解，使用求根公式，不要猜测。

10. Practice Questions | 练习题目

Try these questions to test your understanding:

尝试以下题目以检验你的理解：

Solve x² – 9x + 14 = 0 by factorisation.
Solve 4x² + 12x + 9 = 0 by completing the square.
Given f(x) = 2x² – 3x + 5, find the discriminant and state the number of real roots.
A right-angled triangle has legs of length x and (x + 2), and hypotenuse of length 10. Form and solve a quadratic equation to find x.

Answers: 1) x = 2 or x = 7. 2) x = -1.5 (repeated root). 3) Δ = 9 – 40 = -31, no real roots. 4) x² + 2x – 48 = 0, so x = 6 (reject x = -8).

答案：1) x = 2 或 x = 7。2) x = -1.5（重根）。3) Δ = 9 – 40 = -31，无实根。4) x² + 2x – 48 = 0，所以 x = 6（舍去 x = -8）。

11. Summary | 总结

You have learned the standard form of a quadratic equation, three main solving methods (factorisation, completing the square, and the quadratic formula), and how to use the discriminant.

你已经学习了二次方程的标准形式、三种主要解法（因式分解、配方、求根公式）以及判别式的应用。

Remember to choose the most efficient method for each problem. Factorising is quick, completing the square helps with graphs, and the formula always works.

请记住针对不同题目选择最有效的方法。因式分解快捷，配方有助于作图，而公式法则始终有效。

With regular practice, solving quadratic equations will become a confident skill in your exam toolkit.

通过经常练习，求解二次方程将成为你考试技能库中的自信本领。

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Solving Quadratic Equations | 二次方程求解