Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, geometry, and real-world problem solving. In this article, you will learn the key methods for solving quadratic equations, understand the discriminant, and avoid common mistakes.

二次方程是IGCSE数学中最重要的内容之一。它们出现在代数、几何以及现实问题求解中。在本文中，你将学习求解二次方程的关键方法、理解判别式，并避免常见错误。

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

A quadratic equation is a polynomial equation of degree 2. In standard form, it is written as ax² + bx + c = 0, where a, b and c are constants and a must not be zero. If a were zero, the equation would become linear, not quadratic.

二次方程是最高次数为2的多项式方程。标准形式为 ax² + bx + c = 0，其中 a、b、c 是常数，且 a 不能为零。如果 a 为零，方程就变成一次方程，而不是二次方程。

The coefficient a determines the curvature of the graph, b affects the position of the axis of symmetry, and c gives the y-intercept of the parabola. For example, x² – 5x + 6 = 0 is a quadratic equation with a = 1, b = -5 and c = 6.

系数 a 决定图像的弯曲程度，b 影响对称轴的位置，c 给出抛物线与 y 轴的交点。例如，x² – 5x + 6 = 0 是一个二次方程，其中 a = 1，b = -5，c = 6。

Not every quadratic equation is easy to solve directly. We need systematic methods. In the following sections, we will study four common approaches: factorisation, the quadratic formula, completing the square, and graphical methods.

并非所有二次方程都能直接求解。我们需要系统的方法。在接下来的小节中，我们将学习四种常用方法：因式分解法、求根公式、配方法和图像法。

2. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method when the quadratic expression can be written as the product of two linear factors. For example, x² – 5x + 6 = 0 can be factored as (x – 2)(x – 3) = 0. Since the product is zero, one of the factors must be zero.

因式分解法通常是最快的方法，适用于二次表达式能写成两个一次因式乘积的情形。例如，x² – 5x + 6 = 0 可以分解为 (x – 2)(x – 3) = 0。因为乘积为零，所以其中一个因式必然为零。

(x – 2)(x – 3) = 0 → x = 2 or x = 3

The key steps are: first, rearrange the equation to the form ax² + bx + c = 0; second, factorise the left-hand side; third, set each factor equal to zero; finally, solve the two linear equations.

关键步骤如下：首先，将方程整理为 ax² + bx + c = 0 的形式；其次，将左边因式分解；然后，令每个因式分别等于零；最后，解这两个一次方程。

When the coefficient a is not 1, factorisation can be harder. For example, 2x² + 5x – 3 = 0 factors as (2x – 1)(x + 3) = 0, giving x = 1/2 or x = -3. Always check by expanding your factors.

当 a 不为 1 时，因式分解会更困难。例如，2x² + 5x – 3 = 0 分解为 (2x – 1)(x + 3) = 0，得到 x = 1/2 或 x = -3。务必通过展开来验证你的因式。

3. The Quadratic Formula | 求根公式

The quadratic formula can solve any quadratic equation, even when factorisation is difficult or impossible. For ax² + bx + c = 0, the solutions are given by:

求根公式可以解任何二次方程，即使因式分解很困难或无法进行。对于 ax² + bx + c = 0，解为：

x = (−b ± √(b² − 4ac)) / 2a

To use the formula, identify a, b and c from the equation, substitute them into the formula, and simplify. For example, solve 2x² + 4x − 6 = 0. Here a = 2, b = 4, c = −6.

使用公式时，先确定方程中的 a、b、c，然后代入公式并化简。例如，解 2x² + 4x − 6 = 0，其中 a = 2，b = 4，c = −6。

x = (−4 ± √(4² − 4×2×(−6))) / (2×2) = (−4 ± √64) / 4 = (−4 ± 8) / 4

Thus x = 1 or x = −3. Notice that the formula gives two solutions because of the ± sign. Always write both solutions clearly.

因此 x = 1 或 x = −3。注意，由于 ± 号，公式给出两个解。务必把两个解都写清楚。

4. Completing the Square | 配方法

Completing the square rewrites a quadratic expression in the form (x + p)² + q. This method is useful for solving equations and also for finding the turning point of a parabola. For example, x² + 6x + 5 = 0 can be rewritten as (x + 3)² − 4 = 0.

配方法将二次表达式改写为 (x + p)² + q 的形式。这个方法常用于解方程，也用于求抛物线的顶点。例如，x² + 6x + 5 = 0 可以改写为 (x + 3)² − 4 = 0。

(x + 3)² − 4 = 0 → (x + 3)² = 4

Then take the square root of both sides: x + 3 = ±2. Therefore x = −1 or x = −5. Notice that completing the square requires halving the coefficient of x, squaring it, and then adjusting the constant term.

然后对方程两边开平方：x + 3 = ±2。因此 x = −1 或 x = −5。注意，配方法需要对 x 的系数取一半、平方，然后调整常数项。

If the coefficient of x² is not 1, first factor it out. For instance, 2x² − 8x + 3 = 0 becomes 2(x² − 4x) + 3 = 0, then complete the square inside the bracket.

如果 x² 的系数不为 1，先将其提出。例如，2x² − 8x + 3 = 0 变为 2(x² − 4x) + 3 = 0，然后在括号内配方。

5. The Discriminant | 判别式

In the quadratic formula, the expression under the square root, b² − 4ac, is called the discriminant, often denoted by Δ. The discriminant tells us how many real roots a quadratic equation has without solving it fully.

在求根公式中，根号下的表达式 b² − 4ac 称为判别式，通常用 Δ 表示。判别式告诉我们一个二次方程有多少个实数根，而无需完全求解。

Δ = b² − 4ac

If Δ > 0, there are two different real roots. If Δ = 0, there is exactly one real root (a repeated root). If Δ < 0, there are no real roots; the solutions are complex numbers, which are not part of the IGCSE syllabus.

如果 Δ > 0，方程有两个不同的实数根。如果 Δ = 0，方程恰有一个实数根（重根）。如果 Δ < 0，方程没有实数根；解为复数，复数不在IGCSE考纲范围内。

For example, the equation x² − 4x + 4 = 0 has Δ = (−4)² − 4×1×4 = 0, so it has one repeated root x = 2. The equation x² + x + 1 = 0 has Δ = 1 − 4 = −3, so it has no real roots.

例如，方程 x² − 4x + 4 = 0 的 Δ = (−4)² − 4×1×4 = 0，所以它有一个重根 x = 2。方程 x² + x + 1 = 0 的 Δ = 1 − 4 = −3，所以没有实数根。

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 of the roots is α + β = −b/a, and the product is αβ = c/a. These relationships are useful for checking solutions or constructing equations from given roots.

对于二次方程 ax² + bx + c = 0，设其两根为 α 和 β，则两根之和 α + β = −b/a，两根之积 αβ = c/a。这些关系可用于检验解或根据已知根构造方程。

α + β = −b/a, αβ = c/a

For example, the equation 2x² − 8x + 6 = 0 has roots with sum 8/2 = 4 and product 6/2 = 3. Indeed, the roots are 1 and 3, which sum to 4 and multiply to 3.

例如，方程 2x² − 8x + 6 = 0 的两根之和为 8/2 = 4，两根之积为 6/2 = 3。事实上，两根为 1 和 3，和为 4，积为 3。

This property is especially helpful when the roots are very large or when you need to form a quadratic equation with given roots. If the roots are α and β, the equation can be written as x² − (α + β)x + αβ = 0.

这个性质在根很大或需要根据已知根构造二次方程时特别有用。如果两根为 α 和 β，则方程可写为 x² − (α + β)x + αβ = 0。

7. Solving Word Problems | 应用题求解

Quadratic equations often arise from real-life situations such as areas, projectile motion, and number problems. The first step is to define a variable and translate the conditions into an equation. Then solve the equation and interpret the answer.

二次方程常出现在面积、抛体运动和数字问题等实际情境中。第一步是设未知数，并将条件转化为方程。然后解方程并解释答案。

Example: The area of a rectangle is 24 cm² and its length is 5 cm greater than its width. Let the width be x. Then the length is x + 5, so x(x + 5) = 24.

例如：一个矩形的面积为 24 cm²，长比宽大 5 cm。设宽为 x，则长为 x + 5，所以 x(x + 5) = 24。

x² + 5x − 24 = 0 → (x + 8)(x − 3) = 0 → x = 3 or x = −8

Since width cannot be negative, x = 3. The width is 3 cm and the length is 8 cm. Always reject negative or impossible solutions in context.

由于宽度不能为负，x = 3。宽为 3 cm，长为 8 cm。在具体情境中，一定要舍去负值或不可能的解。

When solving word problems, check that your answer satisfies the original conditions. It is also useful to state units clearly.

在解决应用题时，检查答案是否满足原始条件。同时应当写清单位。

8. Graphical Interpretation | 图形意义

The graph of a quadratic function y = ax² + bx + c is a parabola. The solutions of the equation ax² + bx + c = 0 correspond to the x-coordinates of the points where the parabola crosses the x-axis.

二次函数 y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的解对应抛物线与 x 轴交点的横坐标。

If the parabola intersects the x-axis at two points, the equation has two real roots. If the parabola just touches the x-axis at one point, there is one repeated root. If the parabola does not touch the x-axis, there are no real roots.

如果抛物线与 x 轴有两个交点，方程有两个实数根。如果抛物线只与 x 轴相切于一点，则有一个重根。如果抛物线与 x 轴没有交点，则没有实数根。

For example, y = x² − 4 crosses the x-axis at x = −2 and x = 2, so the equation x² − 4 = 0 has roots ±2. The vertex of the parabola can be found by completing the square: y = (x + 1)² − 4 gives a minimum point at (−1, −4).

例如，y = x² − 4 与 x 轴交于 x = −2 和 x = 2，所以方程 x² − 4 = 0 的根为 ±2。抛物线的顶点可以通过配方求得：y = (x + 1)² − 4 得到最低点为 (−1, −4)。

9. Common Mistakes and Tips | 常见错误与技巧

A frequent mistake is forgetting to rearrange the equation to the standard form ax² + bx + c = 0 before applying a method. For example, x² = 5x must be written as x² − 5x = 0, then factorised as x(x − 5) = 0.

一个常见错误是在应用方法之前忘记将方程整理成标准形式 ax² + bx + c = 0。例如，x² = 5x 必须写成 x² − 5x = 0，然后分解为 x(x − 5) = 0。

Another common error is dividing both sides by a variable that could be zero. Never cancel x from both sides unless you know x is not zero. Instead, factor out the common factor.

另一个常见错误是两边同时除以一个可能为零的变量。除非你知道 x 不等于零，否则绝不能在等号两边同时消去 x。正确的做法是提出公因式。

When using the quadratic formula, check your signs carefully, especially when b or c is negative. Also simplify square roots where possible. For example, √12 can be written as 2√3.

使用求根公式时，要仔细检查符号，尤其是当 b 或 c 为负数时。同时要尽可能化简根式。例如，√12 可以写成 2√3。

Finally, always verify your solutions by substituting them back into the original equation. This simple check can catch most errors.

最后，始终将解代回原方程进行验证。这一简单的检查能发现大多数错误。

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Solving Quadratic Equations | 解二次方程