Maths 204: Quadratic Equations | 数学204：二次方程

📚 Maths 204: Quadratic Equations | 数学204：二次方程

In the Edexcel IGCSE Mathematics syllabus, quadratic equations form a central part of Algebra. Understanding how to solve them is essential for higher-level topics such as functions, graphs, and problem solving.

在 Edexcel IGCSE 数学考纲中，二次方程是代数部分的核心内容。掌握其解法对于学习函数、图像以及解决实际问题等更高阶主题至关重要。

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

A quadratic equation is a polynomial equation of degree 2. It can be written in the standard form:

二次方程是最高次数为2的多项式方程，其标准形式为：

ax² + bx + c = 0, 其中 a ≠ 0

Here x is the unknown, a, b and c are constants, and a is not 0 because otherwise the equation would be linear, not quadratic.

其中 x 是未知数，a、b、c 是常数，且 a 不等于 0。若 a = 0，方程就会变成一次方程，而不是二次方程。

The values of x that make the equation true are called the roots, or solutions, of the quadratic equation.

使方程成立的 x 值称为这个二次方程的根，也叫解。

2. Solving by Factorising | 因式分解法

Factorising is often the fastest method when the quadratic expression can be written as a product of two binomial factors.

当二次表达式可以写成两个二项式的乘积时，因式分解法往往是最快捷的解法。

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

例如，解方程 x² − 5x + 6 = 0。我们要找两个相乘得 6、相加得 −5 的数，它们是 −2 和 −3，因此：

(x − 2)(x − 3) = 0

Using the zero product property, either x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3.

根据零乘积性质，要么 x − 2 = 0，要么 x − 3 = 0，所以 x = 2 或 x = 3。

Always expand your factors to check your answer.

记得把因式展开验证你的答案。

3. The Quadratic Formula | 求根公式

When factorising is difficult or impossible, the quadratic formula works for any quadratic equation.

当因式分解困难或无法进行时，求根公式适用于任何二次方程。

For ax² + bx + c = 0, the formula is:

对于 ax² + bx + c = 0，求根公式为：

x = (−b ± √(b² − 4ac)) / 2a

To use the formula, substitute the values of a, b and c into the expression, then simplify carefully.

使用公式时，将 a、b、c 的值代入表达式，然后仔细化简。

For instance, solve 2x² + 3x − 2 = 0. Here a = 2, b = 3, c = −2.

例如，解 2x² + 3x − 2 = 0。这里 a = 2，b = 3，c = −2。

x = ( −3 ± √(3² − 4×2×(−2)) ) / (2×2)

This gives x = 0.5 or x = −2.

得到 x = 0.5 或 x = −2。

4. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form p(x + q)² + r. This is especially useful for finding the vertex of a parabola.

配方法把二次式改写为 p(x + q)² + r 的形式。这特别有助于找到抛物线的顶点。

Start by halving the coefficient of x. For x² + 6x + 2 = 0, half of 6 is 3, so we write:

先取 x 系数的一半。对于 x² + 6x + 2 = 0，6 的一半是 3，于是写成：

x² + 6x = (x + 3)² − 9

Then the equation becomes (x + 3)² − 9 + 2 = 0, so (x + 3)² = 7.

于是原方程变为 (x + 3)² − 9 + 2 = 0，即 (x + 3)² = 7。

Taking square roots gives x + 3 = ±√7, therefore x = −3 ± √7.

开平方得 x + 3 = ±√7，所以 x = −3 ± √7。

5. The Discriminant | 判别式

The discriminant is the part of the quadratic formula under the square root: Δ = b² − 4ac.

判别式是求根公式中根号下的部分：Δ = b² − 4ac。

It tells us how many real roots a quadratic equation has.

它告诉我们二次方程有几个实数根。

Discriminant	Number of real roots
Δ > 0	Two distinct real roots
Δ = 0	One repeated real root
Δ < 0	No real roots

If Δ is a perfect square and a, b, c are rational, the quadratic can also be factorised easily.

如果 Δ 是完全平方数且 a、b、c 为有理数，那么二次式通常也能轻松因式分解。

6. Sum and Product of Roots | 根的和与积

For a quadratic equation ax² + bx + c = 0 with roots α and β, the sum and product of the roots have simple formulas.

对于根为 α 和 β 的二次方程 ax² + bx + c = 0，根的和与积有简洁的公式。

α + β = −b/a, αβ = c/a

These relationships allow you to find the roots without solving the full equation in some problems.

利用这些关系，在某些问题中无需解整个方程就能得出根。

For example, for x² − 7x + 12 = 0, sum = 7 and product = 12, so the roots are 3 and 4.

例如，对于 x² − 7x + 12 = 0，根和为 7，积为 12，因此根是 3 和 4。

7. Quadratic Graphs | 二次函数图像

The graph of y = ax² + bx + c is a parabola. Its shape depends on the sign of a.

y = ax² + bx + c 的图像是一条抛物线。其开口方向取决于 a 的符号。

If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point.

若 a > 0，抛物线开口向上，有最小值点；若 a < 0，开口向下，有最大值点。

The x-coordinate of the vertex is given by x = −b / (2a). Substituting this value into the equation gives the y-coordinate.

顶点的 x 坐标为 x = −b / (2a)。将 x 代入方程即可得到 y 坐标。

The roots of the equation ax² + bx + c = 0 correspond to the x-intercepts of the graph.

方程 ax² + bx + c = 0 的根对应图像与 x 轴的交点。

8. Word Problems | 应用题

Quadratic equations often arise in geometry and physics problems.

二次方程经常出现在几何与物理问题中。

Example: The area of a rectangle is 40 cm² and its length is 3 cm more than its width. Let width = x, then length = x + 3.

例如：一个矩形的面积为 40 平方厘米，长比宽多 3 厘米。设宽为 x，则长为 x + 3。

x(x + 3) = 40 → x² + 3x − 40 = 0

Factorising gives (x + 8)(x − 5) = 0, so x = −8 or x = 5. Since length cannot be negative, the width is 5 cm and the length is 8 cm.

因式分解得 (x + 8)(x − 5) = 0，所以 x = −8 或 x = 5。由于长度不能为负，宽为 5 厘米，长为 8 厘米。

Always interpret your answers in the context of the problem and reject impossible values.

务必结合题目背景解释答案，并排除不合实际的值。

9. Common Pitfalls | 常见错误

Students often forget to set the equation to zero before factorising.

同学们经常忘记在因式分解前把方程一边化为零。

Another common mistake is missing the ± sign when taking square roots.

另一个常见错误是开平方时漏掉 ± 符号。

When using the quadratic formula, be careful with negative values of b and c.

使用求根公式时，要特别注意 b 和 c 为负数的情况。

Always check whether the discriminant is negative before deciding that there are no real solutions.

在判断“无实数解”之前，一定要先计算判别式是否为负数。

10. Revision Strategy | 复习建议

Start by mastering factorising, then practise completing the square, and finally use the quadratic formula.

先从因式分解入手，再练习配方法，最后掌握求根公式。

Draw graphs for different quadratics to connect algebraic solutions with visual understanding.

画出不同二次函数的图像，把代数解法和直观理解联系起来。

Review the discriminant for every exercise to predict the number of roots.

每道练习都先通过判别式预测根的个数。
Try past paper questions on quadratic equations.

尝试往年试卷中的二次方程题目。
Write down the formula and steps clearly to avoid arithmetic errors.

把公式和步骤写清楚，避免计算错误。

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Maths 204: Quadratic Equations | 数学204：二次方程