Solving Quadratic Equations: Techniques and Applications | 解二次方程：技巧与应用

📚 Solving Quadratic Equations: Techniques and Applications | 解二次方程：技巧与应用

A quadratic equation is one of the most fundamental topics in IGCSE Mathematics. It appears in algebra, geometry, trigonometry, and even in real-life problems involving area, motion, and optimisation. Mastering the techniques to solve quadratic equations will not only boost your exam confidence but also deepen your understanding of how mathematical models work.

二次方程是IGCSE数学中最基础的主题之一。它出现在代数、几何、三角学乃至涉及面积、运动和最优化的实际问题中。掌握解二次方程的技巧，不仅能提升你的考试信心，还能加深你对数学模型运作方式的理解。

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

A quadratic equation is a polynomial equation of degree 2. Its general form is written as ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The term quadratic comes from the Latin word “quadratus”, meaning square, because the highest power of the unknown is squared.

二次方程是最高次数为2的多项式方程。其一般形式写作 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。术语“二次”源自拉丁语“quadratus”，意为“平方”，因为未知数的最高次幂是平方。

For example, 2x² – 3x + 1 = 0 is a quadratic equation, while x² + 2x + 1 = 0 is another. If a = 0, the equation reduces to a linear one, so the condition a ≠ 0 is essential.

例如，2x² – 3x + 1 = 0 是二次方程，x² + 2x + 1 = 0 也是。如果 a = 0，方程就退化为一次方程，因此条件 a ≠ 0 至关重要。

In IGCSE exams, you will often be asked to solve quadratic equations, to sketch their graphs, or to apply them in word problems. Understanding the standard form is the first step to recognising them quickly.

在IGCSE考试中，你常会被要求解二次方程、画出它们的图像，或在应用题中运用它们。理解标准形式是快速识别二次方程的第一步。

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

The standard form of a quadratic equation is ax² + bx + c = 0. Here, a is called the quadratic coefficient, b is the linear coefficient, and c is the constant term. These coefficients can be positive, negative, or zero (except a).

二次方程的标准形式是 ax² + bx + c = 0。其中 a 称为二次项系数，b 称为一次项系数，c 称为常数项。这些系数可以是正数、负数或零（a 除外）。

A solution of a quadratic equation is a value of x that makes the equation true. For example, x = 1 solves x² – 2x + 1 = 0 because 1 – 2 + 1 = 0. A quadratic equation can have at most two real solutions, and sometimes only one or even none.

二次方程的解是使方程成立的 x 的值。例如，x = 1 是 x² – 2x + 1 = 0 的一个解，因为 1 – 2 + 1 = 0。二次方程最多有两个实数解，有时只有一个解，甚至没有实数解。

When a quadratic equation is not given in standard form, you must rearrange it into the form ax² + bx + c = 0 before solving. This is a very common first step in exam questions.

当二次方程没有以标准形式给出时，你必须先将其化为 ax² + bx + c = 0 的形式再求解。这是考试题目中非常常见的第一步。

3. Solving by Factoring | 因式分解法

Factoring is one of the quickest ways to solve a quadratic equation when the equation can be written as a product of two binomials. The principle is based on the zero product property: if AB = 0, then either A = 0 or B = 0.

因式分解是解二次方程最快的方法之一，前提是方程可以写成两个二项式的乘积。其原理基于零乘积性质：若 AB = 0，则 A = 0 或 B = 0。

For example, to solve x² – 5x + 6 = 0, we factor it as (x – 2)(x – 3) = 0. Then x – 2 = 0 or x – 3 = 0, giving x = 2 or x = 3.

例如，解 x² – 5x + 6 = 0，我们将其因式分解为 (x – 2)(x – 3) = 0。于是 x – 2 = 0 或 x – 3 = 0，得 x = 2 或 x = 3。

When the coefficient a is not 1, we need to find two numbers that multiply to ac and add up to b. For instance, 2x² + 7x + 3 = 0 can be factored as (2x + 1)(x + 3) = 0.

当系数 a 不是1时，我们需要找到两个数，使它们的乘积等于 ac，且和等于 b。例如，2x² + 7x + 3 = 0 可以分解为 (2x + 1)(x + 3) = 0。

Factoring is efficient, but it only works when the roots are rational numbers. For equations that cannot be factored easily, we turn to other methods.

因式分解法高效，但仅当根为有理数时才适用。对于无法简便分解的方程，我们需要使用其他方法。

4. Solving by Completing the Square | 配方法

Completing the square is a powerful technique that rewrites a quadratic expression in the form a(x + p)² + q. This method works for any quadratic equation and also reveals the vertex of the parabola.

配方法是强大的技巧，它能将二次表达式改写为 a(x + p)² + q 的形式。这一方法适用于任何二次方程，同时还能揭示抛物线的顶点。

To solve x² + 6x + 2 = 0 by completing the square, we first move the constant to the other side: x² + 6x = -2. Then we add (6/2)² = 9 to both sides: x² + 6x + 9 = 7, which gives (x + 3)² = 7. Taking square roots yields x = -3 ± √7.

用配方法解 x² + 6x + 2 = 0，我们先将常数项移到另一边：x² + 6x = -2。然后在两边加上 (6/2)² = 9：x² + 6x + 9 = 7，即 (x + 3)² = 7。开平方得 x = -3 ± √7。

For a general equation ax² + bx + c = 0, completing the square leads directly to the quadratic formula. It is also used to find the maximum or minimum value of a quadratic function.

对于一般的 ax² + bx + c = 0，配方法可以直接推导出求根公式。它还可用于求二次函数的最大值或最小值。

x² + bx + (b/2)² = (x + b/2)²

This formula is the heart of completing the square. Remember to balance the equation by adding the same value to both sides.

这个公式是配方法的核心。切记通过在两边加上相同的值来保持方程平衡。

5. The Quadratic Formula | 求根公式

The quadratic formula is the most universal method for solving quadratic equations. It works for every quadratic equation, regardless of whether the roots are rational, irrational, or even complex (though complex roots are not required in IGCSE).

求根公式是解二次方程最通用的方法。它对所有二次方程都适用，无论根是有理数、无理数甚至复数（尽管IGCSE不要求复数根）。

For ax² + bx + c = 0, the solutions are given by:

对于 ax² + bx + c = 0，解为：

x = (-b ± √(b² – 4ac)) / (2a)

To use this formula, simply substitute the values of a, b, and c into the expression. Always simplify the square root first, then divide by 2a.

使用此公式时，只需将 a、b、c 的值代入表达式。先化简根号内的式子，再除以 2a。

Example: Solve 2x² – 4x – 3 = 0. Here a = 2, b = -4, c = -3. Substitution gives x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = (4 ± 2√10) / 4 = 1 ± √10/2.

例：解 2x² – 4x – 3 = 0。这里 a = 2，b = -4，c = -3。代入得 x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = (4 ± 2√10) / 4 = 1 ± √10/2。

6. The Discriminant | 判别式

The discriminant is the expression under the square root in the quadratic formula: Δ = b² – 4ac. It tells us the nature of the roots without actually solving the equation.

判别式是求根公式中根号内的表达式：Δ = b² – 4ac。它可以帮助我们不解方程而判断根的性质。

If Δ > 0, the equation has two distinct real roots.
如果 Δ > 0，方程有两个不相等的实数根。
If Δ = 0, the equation has exactly one real root (a repeated root).
如果 Δ = 0，方程有一个实数根（重根）。
If Δ < 0, the equation has no real roots.
如果 Δ < 0，方程没有实数根。

For example, for x² – 2x + 1 = 0, Δ = (-2)² – 4(1)(1) = 0, so there is one repeated root x = 1. For x² + 2x + 5 = 0, Δ = 4 – 20 = -16, so no real roots exist.

例如，对于 x² – 2x + 1 = 0，Δ = (-2)² – 4(1)(1) = 0，所以有一个重根 x = 1。对于 x² + 2x + 5 = 0，Δ = 4 – 20 = -16，因此没有实数根。

The discriminant also tells us whether the graph of the quadratic function crosses the x-axis at two points, touches it at one point, or never touches it.

判别式还告诉我们二次函数的图像是与x轴有两个交点、相切于一个点，还是与x轴无交点。

7. Graphing Quadratic Functions | 二次函数的图像

The graph of a quadratic function y = ax² + bx + c is a parabola. When a > 0, the parabola opens upwards and has a minimum point; when a < 0, it opens downwards and has a maximum point.

二次函数 y = ax² + bx + c 的图像是抛物线。当 a > 0 时，抛物线开口向上，有最低点；当 a < 0 时，开口向下，有最高点。

The x-coordinate of the vertex is found by x = -b/(2a). Substituting this value back into the equation gives the y-coordinate. The axis of symmetry is the vertical line through the vertex.

顶点的x坐标由 x = -b/(2a) 求得。将这一值代回方程可得到y坐标。对称轴是过顶点的竖直直线。

To sketch the graph, you can find the y-intercept (which is c), the x-intercepts (the roots, if they exist), and the vertex. These three features give a good outline of the parabola.

画图时，可以求出y截距（即 c）、x截距（若存在，即方程的根）以及顶点。这三个特征就能很好地勾勒出抛物线的大致形状。

For example, y = x² – 4x + 3 has vertex at x = 2, y = -1; the roots are x = 1 and x = 3; the y-intercept is 3. Plotting these points gives a clear graph.

例如，y = x² – 4x + 3 的顶点在 x = 2，y = -1；根为 x = 1 和 x = 3；y截距为3。画出这些点即可得到清晰的图像。

8. Applications in Real Life | 实际应用

Quadratic equations appear in many real-world contexts, including physics, economics, and geometry. For instance, the height of a projectile over time can be modelled by a quadratic function, and the maximum height is found at the vertex.

二次方程出现在许多现实情境中，包括物理、经济学和几何学。例如，抛射体随时间变化的高度可以用二次函数建模，最大高度可通过顶点求得。

In business, profit functions are often quadratic, and the break-even points (where profit equals zero) are the roots of the equation. In geometry, problems involving areas of enlarged or reduced figures frequently lead to quadratic equations.

在商业中，利润函数通常是二次的，盈亏平衡点（利润为零）就是方程的根。在几何中，涉及面积放大或缩小的问题也常常会产生二次方程。

Example: A rectangular garden has a length that is 3 meters more than its width. If the area is 54 m², find its dimensions. Let the width be x, then length = x + 3. So x(x + 3) = 54, giving x² + 3x – 54 = 0. Solving by factoring: (x + 9)(x – 6) = 0, so x = 6 (ignoring negative width). The dimensions are 6 m by 9 m.

例：一个矩形花园的长比宽多3米。若面积为54平方米，求其尺寸。设宽为 x，则长为 x + 3。于是 x(x + 3) = 54，即 x² + 3x – 54 = 0。因式分解得 (x + 9)(x – 6) = 0，所以 x = 6（舍去负宽）。尺寸为6米乘9米。

9. Common Mistakes and Tips | 常见错误与技巧

Many students make the same types of errors when solving quadratic equations. Recognising these pitfalls can save you valuable marks in the exam.

许多学生在解二次方程时会犯相同的错误。识别这些陷阱可以帮助你在考试中保住宝贵的分数。

Forgetting to put the equation in standard form before solving.
求解前忘记将方程化为标准形式。
Confusing the signs of b and c in the quadratic formula.
在求根公式中弄错 b 和 c 的符号。
Not simplifying the square root fully.
没有完全化简根号。
Dropping the ± sign when taking square roots.
开平方时遗漏 ± 号。
Assuming that factors always have to be integers.
认为因式分解的结果一定都是整数。

A good tip is to check your solutions by substituting them back into the original equation. Also, if the question asks for answers to a specified degree of accuracy, remember to round only at the final step.

一个好的技巧是，将解代回原方程检验。另外，如果题目要求保留特定精度，请务必仅在最后一步进行四舍五入。

10. Practice Problems | 练习题

Here are some practice problems to test your understanding. Try to solve them using different methods and check your answers.

这里有一些练习题来测试你的理解。尝试用不同的方法解题并检查答案。

Solve x² – 7x + 12 = 0 by factoring.

用因式分解法解 x² – 7x + 12 = 0。
Solve 2x² + 5x – 3 = 0 using the quadratic formula.

用求根公式解 2x² + 5x – 3 = 0。
Find the vertex of y = x² – 6x + 10 and sketch its graph.

求 y = x² – 6x + 10 的顶点并画出它的图像。
Determine the nature of the roots of x² – 4x + 4 = 0 using the discriminant.

利用判别式判断 x² – 4x + 4 = 0 的根的性质。
The product of two consecutive positive integers is 132. Find the integers.

两个连续正整数的乘积是132。求这两个整数。

Answers: 1) x = 3, 4; 2) x = 1/2, -3; 3) vertex (3, 1); 4) one repeated root, x = 2; 5) 11 and 12.

答案：1) x = 3, 4；2) x = 1/2, -3；3) 顶点 (3, 1)；4) 一个重根，x = 2；5) 11 和 12。

11. Summary | 总结

In this article, we have explored the key techniques for solving quadratic equations: factoring, completing the square, and using the quadratic formula. We also discussed the discriminant, graph sketching, and applications.

在本文中，我们探讨了解二次方程的主要技巧：因式分解法、配方法和求根公式法。我们还讨论了判别式、图像画法和实际应用。

Remember that the method you choose may depend on the form of the equation and the desired accuracy. Factoring is fastest when roots are rational; completing the square is useful for finding vertices; the quadratic formula is the most reliable general method.

请记住，你选择的方法可能取决于方程的形式和所需的精度。当根为有理数时，因式分解最快；配方法对于求顶点很有用；求根公式是最可靠的一般方法。

With regular practice, solving quadratic equations will become second nature. Keep practising, and you will see steady improvement in your IGCSE Mathematics performance.

通过定期练习，解二次方程将变得得心应手。坚持练习，你会发现自己的IGCSE数学成绩稳步提升。

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Solving Quadratic Equations: Techniques and Applications | 解二次方程：技巧与应用