Solving Systems of Quadratic Equations | 二次联立方程组的求解策略

📚 Solving Systems of Quadratic Equations | 二次联立方程组的求解策略

Systems of quadratic equations are a fundamental topic in A-Level mathematics, appearing in pure mathematics, coordinate geometry, and even mechanics. Mastering the strategies to solve these systems is essential for achieving top marks in your examinations.

二次联立方程组是 A-Level 数学中的一个基础考点，出现在纯数学、坐标几何乃至力学中。掌握解这类方程组的策略，是考试中获得高分的关键。

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1. What Are Systems of Quadratic Equations? | 什么是二次联立方程组

A system of equations is quadratic if at least one equation in the system has degree 2. The most common type you will encounter is a linear-quadratic system, e.g., y = 2x + 1 combined with y = x² – 3x + 5. Less frequently, you will meet systems where both equations are quadratic.

如果一个方程组中至少有一个方程是二次的（即最高次数为 2），那么这个方程组就是二次的。最常见的类型是线性-二次方程组，例如 y = 2x + 1 与 y = x² – 3x + 5 联立。较少见的是两个方程均为二次的情形。

When solving such systems, you are looking for all ordered pairs (x, y) that satisfy every equation simultaneously. These solutions correspond to the intersection points of the curves in the xy-plane.

解这类方程组时，你要找出所有同时满足每个方程的有序数对 (x, y)。这些解对应着 xy 平面中曲线的交点。

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2. The Substitution Method | 代入消元法

The substitution method is the most powerful and frequently used technique. When one equation is linear, solve it for one variable and substitute the result into the quadratic equation.

代入消元法是最常用且最有力的技巧。当一个方程是线性时，先解出一个变量，再将结果代入二次方程。

Step 1: Rearrange the linear equation to make either x or y the subject.

第一步：将线性方程变形，用 x 或 y 表示另一个变量。

Step 2: Substitute this expression into the quadratic equation. This produces a single quadratic equation in one variable.

第二步：将该表达式代入二次方程，得到只含一个变量的一元二次方程。

Step 3: Solve this quadratic equation, typically by factorising, completing the square, or using the quadratic formula.

第三步：解这个一元二次方程，通常使用因式分解、配方法或求根公式。

Step 4: Substitute each value back into the linear equation to find the corresponding y-value (or x-value). Write each solution as an ordered pair.

第四步：将每个解代回线性方程，求出对应的 y 值（或 x 值）。将每一组解写成有序数对。

Example: Solve y = 2x – 1 and y = x² – 2x + 3

例：解联立方程 y = 2x – 1 与 y = x² – 2x + 3

Substitute: 2x – 1 = x² – 2x + 3 → x² – 4x + 4 = 0 → (x – 2)² = 0 → x = 2. Then y = 2(2) – 1 = 3. The system has exactly one solution: (2, 3).

代入得：2x – 1 = x² – 2x + 3 → x² – 4x + 4 = 0 → (x – 2)² = 0 → x = 2。再由 y = 2(2) – 1 = 3。该方程组恰有一个解：(2, 3)。

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3. The Elimination Method | 加减消元法

When both equations are quadratic, substitution may still work, but elimination can be more efficient-especially when the two quadratics have identical quadratic terms.

当两个方程都是二次时，代入法依然可行，但加减消元法往往更高效——尤其当两个二次方程含有相同的二次项时。

For example, consider x² + y² = 25 and x² + 2y² = 34. Subtracting the first from the second gives y² = 9, so y = ±3. Substituting back gives x = ±4 in each case, producing four solutions.

例如，考虑 x² + y² = 25 和 x² + 2y² = 34。将第二个方程减去第一个方程，得 y² = 9，即 y = ±3。代回得 x = ±4，共得四组解。

Key insight: If the quadratic terms do not match, you may still eliminate them by subtracting suitable multiples of the equations.

关键思路：如果二次项系数不一致，可以通过将方程乘以适当的倍数后再相减来消去二次项。

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4. Graphical Interpretation | 图像解释

Every solution to a system of equations is an intersection point between the graphs of the equations. This geometric view is invaluable for checking your answers and understanding the number of solutions.

联立方程组的每一组解，都是各方程图像的交点。这种几何视角对于检验答案和理解解的个数非常有价值。

A straight line and a parabola can intersect in 0, 1, or 2 points. A line and a circle can likewise meet in 0, 1 (tangent), or 2 points.

一条直线与一条抛物线可以有 0 个、1 个或 2 个交点。直线与圆也同样可以有 0 个、1 个（相切）或 2 个交点。

• Two distinct real solutions: the line cuts through the curve at two points.

  两个不同的实数解：直线与曲线相交于两点。

• One repeated solution: the line is tangent to the curve.

  一个重根：直线与曲线相切。

• No real solutions: the line misses the curve entirely.

  无实数解：直线与曲线完全不相交。

Sketching a rough graph before calculating can help you anticipate the result and avoid algebraic errors.

在计算前先画一个粗略的示意图，可以帮助你预测结果并避免代数错误。

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5. Using the Discriminant to Predict Solutions | 用判别式预判解的个数

After substitution, you will always reduce the system to a quadratic equation in one variable: ax² + bx + c = 0. The discriminant Δ = b² – 4ac determines how many real solutions exist.

经过代入消元后，系统总是化为一元二次方程 ax² + bx + c = 0。其判别式 Δ = b² – 4ac 决定实数解的个数。

Δ > 0 → two distinct real solutions (two intersection points)

Δ > 0 → 两个不同的实数解（两个交点）

Δ = 0 → one repeated real solution (tangency)

Δ = 0 → 一个重根（相切）

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