Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are a fundamental part of the Edexcel IGCSE Mathematics syllabus. In this revision guide, we will explore multiple methods for solving them, understand the discriminant, and apply these skills to real-world problems.

二次方程是 Edexcel IGCSE 数学大纲的基础内容。在本复习指南中，我们将探讨求解二次方程的多种方法，理解判别式，并将这些技能应用于实际问题。

1. Expanding and Factorising Quadratics | 展开与因式分解

Before solving a quadratic equation, you must be comfortable with expanding brackets and factorising expressions. A quadratic expression in the form \(ax^2 + bx + c\) can often be written as a product of two binomials.

在求解二次方程之前，你必须熟练掌握展开括号和将表达式因式分解。形如 \(ax^2 + bx + c\) 的二次表达式通常可以写成两个二项式的乘积。

Expanding: \((x + 3)(x – 2) = x^2 + x – 6\)

展开：\((x + 3)(x – 2) = x^2 + x – 6\)
Factorising: \(x^2 + 5x + 6 = (x + 2)(x + 3)\)

因式分解：\(x^2 + 5x + 6 = (x + 2)(x + 3)\)

When the coefficient of \(x^2\) is greater than 1, use methods such as grouping or the “ac” method. Practice is key to speed and accuracy.

当 \(x^2\) 的系数大于 1 时，可以使用分组法或 “ac” 法。熟能生巧，练习是提高速度和准确性的关键。

2. Solving by Factorisation | 用因式分解法求解

A quadratic equation written as \(ax^2 + bx + c = 0\) can be solved if the expression factorises. The principle is: if the product of two factors is zero, then at least one factor must be zero.

如果二次方程写成 \(ax^2 + bx + c = 0\) 且表达式能因式分解，即可求解。其原理是：如果两个因式的乘积为零，则至少有一个因式为零。

If \(pq = 0\), then \(p = 0\) or \(q = 0\).

若 \(pq = 0\)，则 \(p = 0\) 或 \(q = 0\)。

Example: Solve \(x^2 – 7x + 10 = 0\). Factorise to \((x – 5)(x – 2) = 0\). Thus \(x = 5\) or \(x = 2\).

例如：解方程 \(x^2 – 7x + 10 = 0\)。因式分解得 \((x – 5)(x – 2) = 0\)。所以 \(x = 5\) 或 \(x = 2\)。

Always check your solutions by substituting them back into the original equation.

始终将解代入原方程进行检验。

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

Completing the square rewrites the quadratic in the form \(a(x + p)^2 + q\). This method is useful when the equation does not factorise easily.

配方法将二次式改写为 \(a(x + p)^2 + q\) 的形式。当方程不易因式分解时，这种方法很有用。

For a quadratic \(x^2 + bx + c = 0\), add and subtract \(\left(\frac{b}{2}\right)^2\).

对于二次方程 \(x^2 + bx + c = 0\)，加上并减去 \(\left(\frac{b}{2}\right)^2\)。

\(x^2 + 6x + 2 = (x + 3)^2 – 9 + 2 = (x + 3)^2 – 7\)

Then solve by rearranging: \((x + 3)^2 = 7\), so \(x + 3 = ±√7\), giving \(x = -3 ± √7\).

然后通过移项求解：\((x + 3)^2 = 7\)，所以 \(x + 3 = ±√7\)，即 \(x = -3 ± √7\)。

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

The quadratic formula works for any quadratic equation \(ax^2 + bx + c = 0\), even when factorisation is impossible.

求根公式适用于任何二次方程 \(ax^2 + bx + c = 0\)，即使无法因式分解也能求解。

\(x = \frac{-b ± \sqrt{b^2 – 4ac}}{2a}\)

You are given this formula in the Edexcel IGCSE formula booklet, but you must know how to substitute values correctly.

Edexcel IGCSE 公式手册中会给出这个公式，但你必须知道如何正确代入数值。

Example: Solve \(2x^2 – 3x – 2 = 0\). Here \(a = 2\), \(b = -3\), \(c = -2\). Substitute into the formula:

例如：解 \(2x^2 – 3x – 2 = 0\)。其中 \(a = 2\)，\(b = -3\)，\(c = -2\)。代入公式：

\(x = \frac{3 ± \sqrt{9 + 16}}{4} = \frac{3 ± 5}{4}\)

Thus \(x = 2\) or \(x = -\frac{1}{2}\).

因此 \(x = 2\) 或 \(x = -\frac{1}{2}\)。

5. The Discriminant | 判别式

The expression \(b^2 – 4ac\) inside the quadratic formula is called the discriminant. It tells us the number and type of roots.

求根公式中的 \(b^2 – 4ac\) 称为判别式。它告诉我们根的数量和类型。

Discriminant \(Δ = b^2 – 4ac\)	Number of Real Roots	Real 根的数量
\(Δ > 0\)	Two distinct real roots	两个不同的实根
\(Δ = 0\)	One repeated real root	一个重根（两个相等实根）
\(Δ < 0\)	No real roots	没有实根

Example: For \(3x^2 – 5x + 2 = 0\), \(Δ = 25 – 24 = 1 > 0\), so there are two distinct real roots.

例如：对于 \(3x^2 – 5x + 2 = 0\)，\(Δ = 25 – 24 = 1 > 0\)，所以有两个不同的实根。

The discriminant also helps determine whether the graph of a quadratic intersects the x-axis, touches it, or does not meet it.

判别式还可以帮助我们判断二次函数的图像与 x 轴相交、相切还是不相交。

6. Solving Word Problems with Quadratics | 二次方程应用题

Many real-world problems involve quadratic equations. Follow a clear strategy: define variables, set up the equation, solve, and check the reasonableness of the answer.

许多实际问题涉及二次方程。遵循清晰的策略：定义变量、建立方程、求解并检查答案是否合理。

Example: A rectangle has length \(x + 4\) cm and width \(x\) cm. Its area is 45 cm². Find \(x\).

例：一个长方形，长为 \(x + 4\) cm，宽为 \(x\) cm，面积为 45 cm²。求 \(x\)。

Set up: \(x(x + 4) = 45\) → \(x^2 + 4x – 45 = 0\). Factorise: \((x + 9)(x – 5) = 0\). Since length cannot be negative, \(x = 5\).

建立方程：\(x(x + 4) = 45\) → \(x^2 + 4x – 45 = 0\)。因式分解：\((x + 9)(x – 5) = 0\)。由于长度不能为负，所以 \(x = 5\)。

Always discard solutions that do not make sense in the context of the problem.

始终舍弃在问题情境中没有意义的解。

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

A quadratic function \(y = ax^2 + bx + c\) always produces a parabola. The sign of \(a\) determines whether it opens upwards (\(a > 0\)) or downwards (\(a < 0\)).

二次函数 \(y = ax^2 + bx + c\) 的图像总是抛物线。\(a\) 的符号决定开口方向：\(a > 0\) 向上开口，\(a < 0\) 向下开口。

The roots of the equation \(ax^2 + bx + c = 0\) are the x-intercepts of the graph. The vertex can be found by completing the square or using \(x = -\frac{b}{2a}\).

方程 \(ax^2 + bx + c = 0\) 的根就是图像与 x 轴交点的横坐标。顶点可以通过配方法或使用 \(x = -\frac{b}{2a}\) 来求得。

Vertex x-coordinate: \(x = -\frac{b}{2a}\)

顶点的 x 坐标：\(x = -\frac{b}{2a}\)

The y-intercept is simply \(c\). Sketching the graph requires knowing the roots, the vertex, and the y-intercept.

y 轴截距就是 \(c\)。绘制图像需要知道根、顶点和 y 轴截距。

8. Quadratic Simultaneous Equations | 二次联立方程

In IGCSE, you may be asked to solve a linear and a quadratic equation simultaneously. The usual method is substitution.

在 IGCSE 中，你可能会遇到一个线性方程与一个二次方程的联立求解。通常使用代入法。

Example: Solve \(y = 2x + 1\) and \(y = x^2 + x – 3\). Since both equal \(y\), set them equal: \(2x + 1 = x^2 + x – 3\).

例：解方程组 \(y = 2x + 1\) 和 \(y = x^2 + x – 3\)。因为都等于 \(y\)，所以令它们相等：\(2x + 1 = x^2 + x – 3\)。

Rearrange to \(x^2 – x – 4 = 0\). Solve using the quadratic formula: \(x = \frac{1 ± \sqrt{17}}{2}\). Then substitute back to find the corresponding \(y\) values.

整理得 \(x^2 – x – 4 = 0\)。用求根公式求解：\(x = \frac{1 ± \sqrt{17}}{2}\)。然后代回求对应的 \(y\) 值。

Geometrically, the solutions are the intersection points of a line and a parabola.

从几何上看，解就是直线与抛物线的交点。

9. Common Mistakes and Tips | 常见错误与提示

Students often lose marks due to avoidable errors. Here are the most common ones and how to avoid them.

学生常常因为可避免的错误而失分。以下是最常见的错误以及如何避免它们。

Forgetting to set the equation to zero before factorising – always rearrange to \(ax^2 + bx + c = 0\).

因式分解前忘记将方程化为零的形式——始终整理成 \(ax^2 + bx + c = 0\)。
Misapplying the quadratic formula signs – pay careful attention to negative values of \(b\).

使用求根公式时弄错符号——特别注意 \(b\) 的负值。
Dropping the ± sign when taking square roots in completing the square.

配方法中取平方根时漏掉 ± 号。
Not checking whether a solution is valid in word problems (e.g., negative lengths).

在应用题中未检查解是否有效（例如负长度）。

Tip: Always attempt to factorise first; if you cannot spot factors quickly, use the formula or complete the square.

提示：优先尝试因式分解；如果无法快速找到因式，就用公式法或配方法。

10. Summary and Final Revision | 总结与最后复习

Quadratic equations appear frequently across the IGCSE papers. Make sure you can:

二次方程在 IGCSE 试卷中频繁出现。确保你能：

Expand and factorise quadratics reliably.

可靠地展开和因式分解二次式。
Solve using all three methods: factorisation, completing the square, and the quadratic formula.

使用三种方法求解：因式分解法、配方法和求根公式。
Use the discriminant to describe the nature of roots.

使用判别式描述根的性质。
Sketch quadratic graphs from key features.

根据关键特征绘制二次函数图像。
Solve quadratic simultaneous equations using substitution.

使用代入法解二次联立方程。

Practise past-paper questions to build familiarity with the style and timing. Good luck!

通过练习往年真题来熟悉题型和考试时间分配。祝你好运！

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Solving Quadratic Equations | 解二次方程