Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations appear throughout the Edexcel IGCSE Mathematics syllabus. Mastering the methods for solving them is essential for success in both Paper 1 and Paper 2. In this guide, we will break down everything you need to know: from the standard form to factorising, completing the square, the quadratic formula, and graph interpretation.

二次方程贯穿爱德思 IGCSE 数学大纲。掌握其解法对 Paper 1 和 Paper 2 的成功至关重要。在本指南中，我们将详细分解你需要了解的全部内容：从标准形式到因式分解、配方法、求根公式以及图像解读。

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

A quadratic equation is an equation of degree 2, meaning the highest power of the variable is 2. In algebra, it can be written in the general form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. If a = 0, the equation becomes linear, not quadratic.

二次方程是次数为 2 的方程，意味着变量的最高次幂是 2。在代数中，它可写成一般形式 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。若 a = 0，则方程变为线性方程，而不是二次方程。

The values of x that satisfy the equation are called roots or solutions. A quadratic equation can have at most two distinct roots.

满足方程的 x 值称为根或解。二次方程最多有两个不同的根。

ax² + bx + c = 0

2. Standard Form and Key Terms | 标准形式与关键术语

Before solving, always rearrange the equation into the standard form ax² + bx + c = 0. The coefficient ‘a’ is the quadratic coefficient, ‘b’ is the linear coefficient, and ‘c’ is the constant term.

在求解之前，始终将方程重排为标准形式 ax² + bx + c = 0。系数 a 称为二次项系数，b 称为一次项系数，c 称为常数项。

For example, the equation 2x² – 3x + 5 = 0 has a = 2, b = –3 and c = 5. Remember to include the signs.

例如，方程 2x² – 3x + 5 = 0 中 a = 2，b = –3，c = 5。注意符号。

A monic quadratic has a = 1, such as x² – 5x + 6 = 0.

首项系数为 1 的二次方程称为首一二次方程，例如 x² – 5x + 6 = 0。
A non-monic quadratic has a ≠ 1, such as 2x² + 7x + 3 = 0.

首项系数不为 1 的二次方程称为非首一二次方程，例如 2x² + 7x + 3 = 0。

Recognising the form helps you choose the fastest solving method.

识别方程形式有助于你选择最快的求解方法。

3. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method when the equation can be written as a product of two binomials. Once factorised, you use the zero product property: if AB = 0, then A = 0 or B = 0.

当方程能写成两个二项式的乘积时，因式分解通常是最快的方法。分解后，利用零乘积性质：若 AB = 0，则 A = 0 或 B = 0。

For a monic quadratic x² + bx + c = 0, look for two numbers that multiply to give c and add to give b.

对于首一二次方程 x² + bx + c = 0，找到两个数，它们的乘积为 c，和为 b。

x² + 5x + 6 = 0 → (x + 2)(x + 3) = 0

Then set each bracket to zero: x + 2 = 0 gives x = –2, and x + 3 = 0 gives x = –3.

然后令每个括号为零：x + 2 = 0 得到 x = –2，x + 3 = 0 得到 x = –3。

For non-monic quadratics, try factorising by grouping or use trial and error with factors of a.

对于非首一二次方程，可尝试分组分解，或利用 a 的因数进行试错。

Always expand your brackets to check your answer.

始终展开括号以检查答案。
If the equation is not in standard form, rearrange it first.

如果方程不是标准形式，先重排。

4. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x + p)² + q. This method is especially useful when the equation cannot be factorised 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，此方法尤为有用。

For x² + bx + c = 0, take half of b, square it, and add and subtract it inside the expression.

对于 x² + bx + c = 0，取 b 的一半，平方，然后在表达式中加上并减去该值。

x² + 6x + 2 = 0 → (x + 3)² – 9 + 2 = 0

Simplify to (x + 3)² = 7, then take square roots: x + 3 = ±√7.

化简得 (x + 3)² = 7，然后开平方：x + 3 = ±√7。

So x = –3 + √7 or x = –3 – √7. Do not forget the ± sign.

因此 x = –3 + √7 或 x = –3 – √7。不要忘记 ± 号。

When a ≠ 1, factor out a from the x² and x terms before completing the square.

当 a ≠ 1 时，先从 x² 和 x 项中提出 a，再进行配方。

5. Solving by the Quadratic Formula | 公式法

The quadratic formula works for all quadratic equations. You should memorise it:

求根公式适用于所有二次方程。你应该记住它：

x = (−b ± √(b² − 4ac)) / (2a)

Given ax² + bx + c = 0, substitute the values of a, b and c into the formula. Be careful with negative signs.

已知 ax² + bx + c = 0，将 a、b、c 的值代入公式。注意负号。

Example: Solve 2x² + 3x – 2 = 0 using the formula.

示例：用公式法解 2x² + 3x – 2 = 0。

Here a = 2, b = 3, c = –2. Then b² – 4ac = 9 – 4(2)(–2) = 9 + 16 = 25, so x = (–3 ± √25) / 4 = (–3 ± 5) / 4.

这里 a = 2，b = 3，c = –2。则 b² – 4ac = 9 – 4(2)(–2) = 9 + 16 = 25，所以 x = (–3 ± √25) / 4 = (–3 ± 5) / 4。

Thus x = (2)/4 = 0.5 or x = (–8)/4 = –2.

因此 x = (2)/4 = 0.5 或 x = (–8)/4 = –2。

Use the quadratic formula when factorisation is difficult or impossible.

当因式分解困难或不可能时，使用求根公式。

6. The Discriminant | 判别式

The discriminant is the part of the formula inside the square root: Δ = b² – 4ac. It tells us how many real roots a quadratic equation has without solving it.

判别式是公式中根号内的部分：Δ = b² – 4ac。它无需解方程就能告诉我们二次方程有多少个实数根。

Value of Δ	Number of Real Roots	图形含义
Δ > 0	Two distinct real roots	与 x 轴交于两点
Δ = 0	One repeated real root	与 x 轴相切于一点
Δ < 0	No real roots	不与 x 轴相交

If the discriminant is a perfect square and a, b, c are rational, the equation can be factorised.

如果判别式是完全平方数，且 a、b、c 为有理数，则方程可以因式分解。

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

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

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 are the x-coordinates where the curve crosses the x-axis.

方程 ax² + bx + c = 0 的根就是曲线与 x 轴交点的 x 坐标。

The y-intercept is the constant term c, since at x = 0, y = c.

y 截距是常数项 c，因为当 x = 0 时，y = c。

Sketching the graph helps you understand the relationship between roots, turning point and the equation.

画出图像有助于你理解根、顶点与方程之间的关系。

8. Roots and Turning Points | 根与顶点

The turning point (or vertex) of a parabola is the point where the curve changes direction. For y = ax² + bx + c, its x-coordinate is given by x = –b/(2a).

抛物线的顶点（或转向点）是曲线改变方向的点。对于 y = ax² + bx + c，其 x 坐标为 x = –b/(2a)。

Substitute this x value back into the equation to find the y-coordinate.

将此 x 值代回方程，求出 y 坐标。

Alternatively, if the quadratic is written in completed square form y = a(x – p)² + q, the vertex is (p, q).

或者，如果二次式写成完成平方形式 y = a(x – p)² + q，则顶点为 (p, q)。

The line of symmetry is the vertical line x = –b/(2a). It passes through the vertex.

对称轴是竖直线 x = –b/(2a)，它经过顶点。

Knowing the vertex helps you sketch the graph accurately and solve optimisation problems.

知道顶点有助于你准确画出图像并解决最优化问题。

9. Solving Word Problems | 应用题

Many quadratic problems are presented in context. Start by defining a variable, then form a quadratic equation from the given information.

许多二次问题以实际情景呈现。首先定义变量，然后根据已知信息建立二次方程。

Example: A rectangle has length 4 cm longer than its width. Its area is 21 cm². Find the width.

示例：一个矩形的长比宽多 4 cm，面积为 21 cm²。求宽。

Let width = x, length = x + 4. Area = x(x + 4) = 21 → x² + 4x – 21 = 0.

设宽为 x，长为 x + 4。面积 = x(x + 4) = 21 → x² + 4x – 21 = 0。

Factorise: (x + 7)(x – 3) = 0, so x = –7 or x = 3. Since a width cannot be negative, x = 3 cm.

因式分解：(x + 7)(x – 3) = 0，因此 x = –7 或 x = 3。由于宽度不能为负，所以 x = 3 cm。

Always check that your answer makes sense in the context.

始终检查你的答案在情境中是否合理。

10. Common Exam Mistakes | 常见考试错误

Here are typical errors that lose marks in Edexcel IGCSE exams.

以下是在爱德思 IGCSE 考试中常见的失分错误。

Forgetting to set the equation equal to zero before factorising.

在因式分解前忘记将方程设为等于零。
Omitting the ± sign when taking the square root.

开平方时漏掉 ± 号。
Misidentifying a, b and c when using the quadratic formula.

使用求根公式时错误识别 a、b、c。
Losing negative signs during substitution.

代入过程中丢失负号。
Not giving the answer to the required degree of accuracy.

未按要求的精度给出答案。

Practicing these points will prevent calculator slips and arithmetic mistakes.

练习这些要点可以避免计算器失误和算术错误。

11. Practice Questions | 练习题

Try these questions on your own, then check against the answers below.

请独立尝试以下问题，然后核对下面的答案。

Solve x² – 7x + 10 = 0.

解方程 x² – 7x + 10 = 0。
Solve 2x² + 5x – 3 = 0 by factorisation.

用因式分解法解 2x² + 5x – 3 = 0。
Use the quadratic formula to solve x² + 2x – 4 = 0, giving answers to 3 significant figures.

用求根公式解 x² + 2x – 4 = 0，答案保留 3 位有效数字。
Find the discriminant of 3x² – 6x + 2 = 0 and state the number of real roots.

求 3x² – 6x + 2 = 0 的判别式，并说明实数根的个数。

Answers: 1. x = 2 or 5. 2. x = 0.5 or –3. 3. x = –1 ± √5 ≈ 1.24 or –3.24. 4. Δ = 36 – 24 = 12 > 0, two distinct real roots.

答案：1. x = 2 或 5。2. x = 0.5 或 –3。3. x = –1 ± √5 ≈ 1.24 或 –3.24。4. Δ = 36 – 24 = 12 > 0，有两个不同的实数根。

12. Summary | 总结

To solve a quadratic equation efficiently, first rearrange it into standard form. Then choose a method: factorisation for simple cases, completing the square for vertex work, and the quadratic formula for any quadratic.

要高效地解二次方程，首先将其重排为标准形式。然后选择方法：简单情况用因式分解，涉及顶点用配方法，任何二次方程都可使用求根公式。

Remember the discriminant tells you the nature of the roots. Combine this with graphical understanding to check your solutions.

记住判别式告诉你根的性质。结合图像理解来检查你的解。

With regular practice, quadratic equations will become one of the most reliable topics in your IGCSE Maths exam.

通过经常练习，二次方程将成为你 IGCSE 数学考试中最可靠的考点之一。

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Solving Quadratic Equations | 解二次方程