Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are a central topic in the Edexcel IGCSE Mathematics syllabus. This revision guide explains the key methods, useful formulas and common pitfalls.

二次方程是 Edexcel IGCSE 数学大纲的核心内容。本复习指南讲解主要方法、常用公式和常见陷阱。

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

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of x is 2.

二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b、c 是常数，且 a ≠ 0。x 的最高次数是 2。

If a = 0, the equation becomes linear, so the condition a ≠ 0 is essential.

如果 a = 0，方程就变成一次方程，因此条件 a ≠ 0 至关重要。

Examples of quadratic equations: x² − 5x + 6 = 0, 2x² + 3x − 2 = 0.

二次方程的例子：x² − 5x + 6 = 0，2x² + 3x − 2 = 0。
Non-examples: x + 3 = 0, x³ − 1 = 0.

反例：x + 3 = 0，x³ − 1 = 0。

2. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method when the quadratic can be written as a product of two linear factors.

当二次式可以写成两个一次因式相乘时，因式分解通常是最快的方法。

Step 1: Write the equation in the form ax² + bx + c = 0.

第一步：将方程写成 ax² + bx + c = 0 的形式。

Step 2: Factorise the left-hand side into two brackets.

第二步：将左边分解为两个括号的乘积。

Step 3: Set each bracket equal to zero and solve.

第三步：令每个括号等于零并求解。

Example: x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 or x = 3.

示例：x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 或 x = 3。

Always check whether the coefficient a has a common factor first.

务必先检查系数 a 是否有公因数。

3. Solving by Completing the Square | 配方法

Completing the square rewrites x² + bx as (x + b/2)² − (b/2)². For the full quadratic, we include the constant term.

配方法将 x² + bx 改写为 (x + b/2)² − (b/2)²。对于完整二次式，我们还要加入常数项。

x² + bx + c = (x + b/2)² − (b/2)² + c.

x² + bx + c = (x + b/2)² − (b/2)² + c。

If a ≠ 1, divide the whole equation by a first, or factor a out of the x terms.

如果 a ≠ 1，先将整个方程除以 a，或者从含 x 的项中提出 a。

Example: x² + 6x + 4 = 0 → (x + 3)² − 5 = 0 → (x + 3)² = 5 → x = −3 ± √5.

示例：x² + 6x + 4 = 0 → (x + 3)² − 5 = 0 → (x + 3)² = 5 → x = −3 ± √5。

This method is particularly useful for solving equations with irrational roots and for finding the turning point of a parabola.

该方法特别适用于求解根为无理数的方程，以及求抛物线的顶点。

4. The Quadratic Formula | 求根公式

The quadratic formula solves any quadratic equation. For ax² + bx + c = 0:

求根公式可以解任意二次方程。对于 ax² + bx + c = 0：

x = (−b ± √(b² − 4ac)) / (2a)

Substitute the values of a, b and c carefully, paying attention to signs.

小心代入 a、b、c 的值，特别注意符号。

Write the two roots separately if required: x = (−b + √Δ)/(2a) and x = (−b − √Δ)/(2a).

如果需要，分别写出两个根：x = (−b + √Δ)/(2a) 和 x = (−b − √Δ)/(2a)。

5. The Discriminant | 判别式

The discriminant is the expression under the square root: Δ = b² − 4ac.

判别式是根号下的表达式：Δ = b² − 4ac。

If Δ > 0, there are two distinct real roots.

如果 Δ > 0，方程有两个不相等的实数根。
If Δ = 0, there is exactly one repeated real root.

如果 Δ = 0，方程有一个二重实数根。
If Δ < 0, there are no real roots; the roots are complex.

如果 Δ < 0，方程没有实数根；根为复数。

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

例如，x² + 2x + 5 = 0 的 Δ = 4 − 20 = −16，因此没有实数根。

6. Roots and Coefficients | 根与系数的关系

If α and β are the roots of ax² + bx + c = 0, then:

如果 α 和 β 是 ax² + bx + c = 0 的两个根，则：

α + β = −b/a, αβ = c/a

α + β = −b/a，αβ = c/a

These relationships help you check your answers or form a quadratic from given roots.

这些关系可帮助你检验答案，或由已知根构造二次方程。

For example, if the roots are 2 and 5, then the sum is 7 and the product is 10, so the equation is x² − 7x + 10 = 0.

例如，若两根为 2 和 5，则和为 7，积为 10，所对应的方程为 x² − 7x + 10 = 0。

7. Equations Involving Fractions | 含分数的二次方程

Some equations look like fractions but reduce to quadratics after multiplying by the common denominator.

有些方程看似含分数，但乘以公分母后可化为二次方程。

Example: 1/x + 1/(x + 2) = 1

示例：1/x + 1/(x + 2) = 1

Multiply every term by x(x + 2): (x + 2) + x = x(x + 2).

每一项乘以 x(x + 2)：(x + 2) + x = x(x + 2)。

Simplify to x² = 2x + 2, or x² − 2x − 2 = 0. Then solve using the formula.

化简得 x² = 2x + 2，即 x² − 2x − 2 = 0。然后使用公式求解。

Always check that your solutions do not make any denominator zero.

务必检查解是否会使分母为零。

8. Quadratic Graphs | 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.

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

The x-intercepts are the real roots of ax² + bx + c = 0. The y-intercept is c.

与 x 轴的交点就是 ax² + bx + c = 0 的实数根。与 y 轴的交点是 c。

The axis of symmetry is x = −b/(2a), and the vertex lies on this line.

对称轴是 x = −b/(2a)，顶点就在这条直线上。

Using completing the square, y = a(x − h)² + k, the vertex is (h, k).

利用配方法，y = a(x − h)² + k，顶点为 (h, k)。

9. Setting Up Quadratic Equations from Word Problems | 由实际问题建立二次方程

Many exam questions describe a real-life situation and ask you to form and solve a quadratic equation.

许多考试题目会描述一个实际问题，要求你列出并求解二次方程。

Read the problem carefully and define a variable, for example x for the unknown number.

仔细阅读题目并定义变量，例如用 x 表示未知数。

Translate the words into an equation. Common contexts include area, product of consecutive numbers, and projectile motion.

将文字转化为方程。常见情境包括面积、连续整数之积、抛体运动。

Example: The product of two consecutive positive odd integers is 99. Find them.

示例：两个连续正奇数的乘积是 99，求这两个数。

Let the smaller integer be n, then n(n + 2) = 99, so n² + 2n − 99 = 0.

设较小奇数为 n，则 n(n + 2) = 99，所以 n² + 2n − 99 = 0。

Factorising: (n + 11)(n − 9) = 0, so n = 9 (positive). The integers are 9 and 11.

因式分解

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Solving Quadratic Equations | 解二次方程