Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations are a fundamental part of the IGCSE Mathematics syllabus. They appear in algebra, graphs and word problems. Mastering the methods for solving them is essential for exam success.

二次方程是 IGCSE 数学大纲中的基础部分。它们出现在代数、图像和应用题中。掌握求解这些方程的方法，对于在考试中取得成功至关重要。

1. Introduction to Quadratic Equations | 二次方程简介

A quadratic equation is a polynomial equation of degree 2. The highest power of the variable is 2, which means the equation has the general shape of a parabola when graphed. In this guide we focus on equations with one unknown, usually written as x.

二次方程是最高次数为 2 的多项式方程。变量的最高幂次是 2，这意味着它的图像是抛物线形状。在本指南中，我们专注于包含一个未知数（通常用 x 表示）的方程。

ax² + bx + c = 0, 其中 a ≠ 0

The letter a is called the quadratic coefficient, b is the linear coefficient, and c is the constant term. Any equation that can be rearranged into this form is also a quadratic equation.

字母 a 称为二次项系数，b 称为一次项系数，c 称为常数项。任何可以整理成这种形式的方程都是二次方程。

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

Before solving a quadratic equation, you must always rewrite it in the standard form ax² + bx + c = 0. For example, the equation 2x² = 8 − 3x must be rearranged to 2x² + 3x − 8 = 0.

在求解二次方程之前，你必须先将它改写为标准形式 ax² + bx + c = 0。例如，方程 2x² = 8 − 3x 必须整理为 2x² + 3x − 8 = 0。

The solutions of a quadratic equation are called the roots, or zeros. A quadratic equation has at most two real roots, but sometimes it may have only one root or no real roots at all.

二次方程的解称为根或零点。一个二次方程最多有两个实数根，但有时也可能只有一个根，或者没有实数根。

x = 2 和 x = 3 都是 x² − 5x + 6 = 0 的根

You should be able to check a root by substituting it back into the original equation; if the left side equals 0, the value is correct.

你应该能够通过将根代入原方程进行检验；如果左边等于 0，则该值是正确的。

3. Solving by Factorisation | 因式分解法求解

Factoring is often the quickest method for solving a quadratic, provided the expression factorises into linear factors over the integers.

因式分解法通常是求解二次方程的最快捷方法，前提是表达式能够分解为整数系数的线性因式。

For the equation x² − 5x + 6 = 0, we look for two numbers that multiply to give 6 and add to give −5. These numbers are −2 and −3, so the equation becomes:

对于方程 x² − 5x + 6 = 0，我们寻找两个数，它们的乘积为 6，和为 −5。这两个数是 −2 和 −3，因此方程变为：

(x − 2)(x − 3) = 0

Using the zero product property, we set each factor equal to zero: x − 2 = 0 or x − 3 = 0. Hence x = 2 or x = 3.

利用零积性质，我们令每个因式等于零：x − 2 = 0 或 x − 3 = 0。因此 x = 2 或 x = 3。

Always expand your factors to check your answer before moving on.

在继续之前，务必展开你的因式来检查答案。

4. Solving by Completing the Square | 配方法求解

Completing the square is a powerful technique that works for any quadratic equation. It involves rewriting the equation in the form (x + p)² = q.

配方法是一种强大的技巧，适用于任何二次方程。它涉及将方程改写为 (x + p)² = q 的形式。

Take the equation x² + 6x + 5 = 0. First, move the constant term to the other side: x² + 6x = −5.

以方程 x² + 6x + 5 = 0 为例。首先，将常数项移到另一边：x² + 6x = −5。

Add (b/2)² to both sides. Here b = 6, so (6/2)² = 9. This gives x² + 6x + 9 = 4, which simplifies to (x + 3)² = 4.

在等式两边同时加上 (b/2)²。这里 b = 6，所以 (6/2)² = 9。得到 x² + 6x + 9 = 4，即 (x + 3)² = 4。

Now take the square root of both sides: x + 3 = ±2. Therefore x = −1 or x = −5.

现在对两边开平方根：x + 3 = ±2。因此 x = −1 或 x = −5。

Remember that when taking the square root, you must include both the positive and negative square roots.

请记住，开平方根时必须同时包括正平方根和负平方根。

5. The Quadratic Formula | 求根公式

For any quadratic equation ax² + bx + c = 0, the solutions can be found using the quadratic formula. This formula is derived from completing the square and is always reliable.

对于任何二次方程 ax² + bx + c = 0，你都可以使用求根公式来求解。这个公式由配方法推导而来，并且永远可靠。

x = (−b ± √(b² − 4ac)) / (2a)

To use the formula, identify the values of a, b and c from the standard form, then substitute them carefully. For example, in 2x² − 3x − 2 = 0, we have a = 2, b = −3, c = −2.

使用公式时，先从标准形式中确定 a、b、c 的值，然后小心代入。例如，在 2x² − 3x − 2 = 0 中，我们有 a = 2，b = −3，c = −2。

Substitution gives x = (3 ± √(9 + 16)) / 4 = (3 ± 5) / 4. The two solutions are x = 2 and x = −0.5.

代入得 x = (3 ± √(9 + 16)) / 4 = (3 ± 5) / 4。两个解为 x = 2 和 x = −0.5。

Always simplify the square root first, then divide, to avoid calculation errors.

总是先化简平方根，再相除，以避免计算错误。

6. The Discriminant | 判别式

The expression under the square root in the quadratic formula is called the discriminant, denoted by Δ (delta). Its value tells us the nature of the roots without solving the equation.

求根公式中平方根下的表达式称为判别式，用 Δ（德尔塔）表示。它的值可以在不求解方程的情况下告诉我们根的性质。

Δ = b² − 4ac

If Δ > 0, the equation has two distinct real roots.
如果 Δ > 0，方程有两个不同的实数根。
If Δ = 0, the equation has exactly one real root (a repeated root).
如果 Δ = 0，方程恰好有一个实数根（一个重根）。
If Δ < 0, the equation has no real roots (the roots are complex).
如果 Δ < 0，方程没有实数根（根为复数）。

Knowing the discriminant helps you decide which method to use and lets you check whether your answers are plausible.

了解判别式有助于你选择求解方法，并能让你检查答案是否合理。

7. Roots and Coefficients | 根与系数的关系

There is a useful connection between the roots of a quadratic equation and its coefficients. If the roots are α (alpha) and β (beta), then we have the following relationships.

二次方程的根与系数之间存在有用的关系。如果方程的根为 α（阿尔法）和 β（贝塔），则有以下关系。

α + β = −b/a 且 αβ = c/a

For example, for x² − 5x + 6 = 0, the sum of the roots is 5 and the product is 6, which matches the roots 2 and 3.

例如，对于 x² − 5x + 6 = 0，两根之和为 5，两根之积为 6，这与根 2 和 3 相符。

These relationships allow you to form a quadratic equation when the roots are given, without solving for the coefficients directly.

这些关系允许你在已知根的情况下构造二次方程，而无需直接求解系数。

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

The graph of y = ax² + bx + c is a parabola. When a > 0, the parabola opens upwards like a U shape; when a < 0, it opens downwards like an inverted U.

函数 y = ax² + bx + c 的图像是抛物线。当 a > 0 时，抛物线开口朝上，像字母 U；当 a < 0 时，开口朝下，像倒置的 U。

The x-intercepts of the graph correspond to the real roots of the equation ax² + bx + c = 0. If the discriminant is negative, the graph does not cross the x-axis at all.

图像与 x 轴的交点对应方程 ax² + bx + c = 0 的实数根。如果判别式为负，则图像完全不与 x 轴相交。

The turning point (vertex) can be found by completing the square. For y = (x − p)² + q, the vertex is (p, q).

可以通过配方法找到顶点。对于 y = (x − p)² + q，顶点为 (p, q)。

Sketching a quadratic graph requires the root locations, the y-intercept, and the vertex; these all contribute to a clear picture.

画二次函数草图需要知道根的位置、y 轴截距以及顶点；这些都是构成清晰图像所必需的。

9. Modelling with Quadratics | 二次方程建模应用

Quadratics appear in many real-world situations. The path of a projectile under gravity is approximately parabolic, so we can use quadratic equations to model height over time.

二次函数出现在许多现实场景中。重力作用下抛射体的路径近似为抛物线，因此我们可以用二次方程来模拟高度随时间的变化。

Another common application is area optimisation. Suppose you have 20 m of fencing to make a rectangular enclosure against a wall. If one side needs no fencing, the area as a function of the width is quadratic.

另一个常见应用是面积最优化。假设你有 20 米围栏，靠墙围成一个矩形区域。如果一边不需要围栏，那么面积作为宽度的函数是二次的。

A(w) = w(20 − 2w) = 20w − 2w²

The maximum area occurs at the vertex of the parabola, which can be found using w = −b/(2a) = 5. This gives a maximum area of 50 m².

最大面积出现在抛物线的顶点处，可以通过 w = −b/(2a) = 5 求得。因此最大面积为 50 平方米。

Solving such problems requires translating words into equations, then applying the algebraic techniques from this guide.

解决这类问题需要将文字转化为方程，然后应用本指南中的代数技巧。

10. Common Mistakes and Tips | 常见错误与技巧

Students often forget to rearrange the equation to standard form before applying the quadratic formula. Always check that the equation equals zero.

学生常常忘记在应用求根公式之前将方程整理为标准形式。务必检查方程是否等于零。

Another common mistake is losing a root when solving by factoring. Remember that the product equals zero means at least one factor must be zero, so you must consider both factors.

另一个常见错误是在因式分解时丢失一个根。请记住，乘积为零意味着至少有一个因式为零，所以你必须同时考虑两个因式。

When using the discriminant, take extra care with negative signs: for b = −3, b² = 9, while −4ac becomes −4 × 2 × (−2) = 16, not −16.

在使用判别式时，要特别小心负号：对于 b = −3，b² = 9，而 −4ac 变为 −4 × 2 × (−2) = 16，而不是 −16。

Finally, always check your solutions by substitution. This quick step can show up arithmetic errors before the exam is over.

最后，始终通过代入来检查你的解。这个快速的步骤可以在考试结束前暴露出计算错误。

11. Practice Questions | 练习问题

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

请自己尝试以下问题，然后核对下方的答案。

1. Solve x² − 7x + 12 = 0 by factorisation.

1. 使用因式分解法求解 x² − 7x + 12 = 0。

2. Solve 2x² + 5x − 3 = 0 using the quadratic formula.

2. 使用求根公式求解 2x² + 5x − 3 = 0。

3. Show that the equation x² − 4x + 5 = 0 has no real roots.

3. 证明方程 x² − 4x + 5 = 0 没有实数根。

4. A rectangle has length x + 4 and width x − 1. If the area is 30, form a quadratic equation and solve it to find x.

4. 一个矩形的长为 x + 4，宽为 x − 1。如果面积为 30，列出二次方程并求解 x。

Answers: 1. x = 3, 4; 2. x = 0.5, −3; 3. Δ = −4; 4. x = 11/2 or − 7? Check: (x+4)(x−1)=30 → x²+3x−34=0 → Δ=145, x≈4.79 or −7.09 (not exact integer).

答案：1. x = 3, 4；2. x = 0.5, −3；3. Δ = −4；4. 由 (x+4)(x−1)=30 得 x²+3x−34=0，判别式 Δ=145，因此 x ≈ 4.79 或 −7.09（非整数）。

Note that not all quadratic equations have nice integer solutions; the quadratic formula handles all cases.

请注意，并非所有二次方程都有漂亮的整数解；求根公式可以处理所有情况。

12. Summary | 总结

In this guide we covered the standard form of a quadratic equation, four important solution methods, the discriminant, the relationship between roots and coefficients, and several applications.

在本指南中，我们介绍了二次方程的标准形式、四种重要的求解方法、判别式、根与系数的关系，以及一些实际应用。

Remember to choose the most efficient method for each question: factorisation works well for simple integer roots, completing the square is useful for finding the vertex, and the quadratic formula always works. By practising regularly, you will build confidence and speed.

请记住，针对每个问题选择最有效的方法：因式分解适用于简单的整数根，配方法有助于求顶点，而求根公式永远有效。通过定期练习，你将建立起信心和速度。

Quadratic equations are not just an exam topic; they are a doorway to deeper mathematical thinking. Master them, and you will be well prepared for more advanced studies.

二次方程不仅仅是考试主题；它们是你深入数学思考的大门。掌握它们，你将为进一步的学习做好充分准备。

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Solving Quadratic Equations | 解一元二次方程