Quadratic Equations and Graphs (G-4 Student Book 500) | 二次方程与图像（G-4 学生用书 500）

📚 Quadratic Equations and Graphs (G-4 Student Book 500) | 二次方程与图像（G-4 学生用书 500）

Welcome to this focused revision guide for IGCSE Mathematics. This article covers the essential skills of solving quadratic equations and interpreting their graphs, based on the G-4 Student Book, page 500. You will learn step-by-step methods, key formulas, and common exam strategies.

欢迎阅读本篇 IGCSE 数学复习指南。本文围绕 G-4 学生用书第 500 页的内容，系统讲解解二次方程和解读函数图像的核心技能。你将学到分步解法、关键公式以及常见考试策略。

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1. The Standard Form of a Quadratic | 二次方程的标准形式

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b and c are constants, and a ≠ 0. The highest power of x is 2, which gives the equation its name.

二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数，且 a ≠ 0。未知数 x 的最高次数为 2，这也是“二次”名称的来源。

ax² + bx + c = 0

For example, 3x² – 5x + 2 = 0 is quadratic because a = 3, b = -5 and c = 2. If a = 0, the equation becomes linear, not quadratic.

例如，3x² – 5x + 2 = 0 是二次方程，因为 a = 3，b = -5，c = 2。如果 a = 0，方程就变成一次方程，而不是二次方程。

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2. Solving by Factorisation | 用因式分解法解方程

Factorisation is often the fastest method when the quadratic has simple integer roots. To solve x² + 5x + 6 = 0, we look for two numbers that multiply to 6 and add to 5. Those numbers are 2 and 3.

因式分解法是处理简单整数根时最快的方法。要解 x² + 5x + 6 = 0，我们寻找两个数，它们相乘等于 6，相加等于 5。这两个数就是 2 和 3。

(x + 2)(x + 3) = 0

Then set each bracket equal to zero: x + 2 = 0 gives x = -2, and x + 3 = 0 gives x = -3. So the solutions are x = -2 and x = -3.

然后令每个括号等于零：x + 2 = 0 得 x = -2，x + 3 = 0 得 x = -3。因此解为 x = -2 和 x = -3。

• Check the product of the constant terms gives c.
• Check the sum of the constant terms gives b.
• If the coefficient of x² is not 1, use the method of splitting the middle term.

• 验证常数项之积等于 c。
• 验证常数项之和等于 b。
• 如果 x² 的系数不为 1，可使用拆中项法。

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3. Solving by Completing the Square | 用配方法解方程

Completing the square rewrites x² + bx + c in the form (x + p)² + q. This is useful when factorisation is not obvious and for deriving the quadratic formula.

配方法将 x² + bx + c 改写为 (x + p)² + q 的形式。当因式分解不明显时，配方法很有用，也用于推导二次方程求根公式。

Start with x² + 6x + 2 = 0. Take half of 6, which is 3, then square it to get 9. Write x² + 6x = (x + 3)² – 9.

例如，解 x² + 6x + 2 = 0。取 6 的一半得 3，再平方得 9。把 x² + 6x 写成 (x + 3)² – 9。

(x + 3)² – 9 + 2 = 0 ⇒ (x + 3)² = 7

Then take square roots: x + 3 = ±√7, so x = -3 ± √7.

然后开平方：x + 3 = ±√7，所以 x = -3 ± √7。

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4. Solving by the Quadratic Formula | 用求根公式解方程

The quadratic formula works for every quadratic equation. For ax² + bx + c = 0, the solutions are given by:

求根公式适用于所有二次方程。对于 ax² + bx + c = 0，解为：

x = (−b ± √(b² − 4ac)) / (2a)

For example, solve 2x² – 4x – 3 = 0. Here a = 2, b = -4, c = -3. Substitute into the formula:

例如，解 2x² – 4x – 3 = 0。这里 a = 2，b = -4，c = -3。代入公式：

x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4

So x = (4 ± 2√10) / 4 = (2 ± √10) / 2. Always simplify your answers.

所以 x = (4 ± 2√10) / 4 = (2 ± √10) / 2。记得化简答案。

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5. The Discriminant and the Nature of Roots | 判别式与根的性质

The expression b² − 4ac is called the discriminant, often denoted by Δ. It tells us how many real roots a quadratic equation has.

表达式 b² − 4ac 称为判别式，通常记为 Δ。它告诉我们二次方程有多少个实数根。

• If Δ > 0: two different real roots.
• If Δ = 0: one repeated real root.
• If Δ < 0: no real roots (two complex roots).

• 若 Δ > 0：有两个不同实数根。
• 若 Δ = 0：有一个重根（两个相等的实数根）。
• 若 Δ < 0：没有实数根（有两个复数根）。

For the equation x² + 4x + 5 = 0, Δ = 16 – 20 = -4 < 0, so there are no real solutions. The graph never touches the x-axis.

对于方程 x² + 4x + 5 = 0，Δ = 16 – 20 = -4 < 0，因此没有实数解。图像不与 x 轴相交。

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6. Sum and Product of Roots | 根的和与积

If α and β are the roots of ax² + bx + c = 0, then their sum and product can be found directly from the coefficients:

若 α 和 β 是 ax² + bx + c = 0 的两个根，则它们的和与积可以直接由系数得到：

α + β = −b / a, αβ = c / a

This is useful for forming a quadratic with given roots. If the roots are 3 and -2, then the sum is 1 and the product is -6, so the equation is x² – x – 6 = 0.

这个关系可用于构造具有指定根的二次方程。若根为 3 和 -2，则和为 1，积为 -6，所以方程为 x² – x – 6 = 0。

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7. Quadratic Graphs: Vertex and Axis of Symmetry | 二次函数图像：顶点与对称轴

The graph of y = ax² + bx + c is a parabola. The axis of symmetry is a vertical line through the vertex, given by x = −b / (2a).

函数 y = ax² + bx + c 的图像是抛物线。对称轴是经过顶点的一条竖直直线，其方程为 x = −b / (2a)。

The vertex lies on this axis. To find its y-coordinate, substitute x = −b / (2a) into the equation.

顶点位于对称轴上。要得到顶点的 y 坐标，将 x = −b / (2a) 代入原方程即可。

Vertex: ( −b/(2a), c − b²/(4a) )

If a > 0, the parabola opens upwards and the vertex is a minimum point. If a < 0, it opens downwards and the vertex is a maximum point.

若 a > 0，抛物线开口向上，顶点为最低点。若 a < 0，抛物线开口向下，顶点为最高点。

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8. Interpreting Roots from the Graph | 从图像解读方程的根

The solutions of ax² + bx + c = 0 are the x-coordinates of the points where the graph y = ax² + bx + c crosses the x-axis. These points are called x-intercepts.

方程 ax² + bx + c = 0 的解，就是抛物线 y = ax² + bx + c 与 x 轴交点的 x 坐标。这些点称为 x 截距。

For example, if the graph of y = x² – x – 6 crosses the x-axis at x = 3 and x = -2, then the equation x² – x – 6 = 0 has solutions 3 and -2.

例如，若 y = x² – x – 6 的图像与 x 轴交于 x = 3 和 x = -2，那么方程 x² – x – 6 = 0 的解就是 3 和 -2。

• If the vertex lies above the x-axis and the parabola opens upwards, there are no x-intercepts.
• If the vertex lies on the x-axis, the equation has exactly one repeated root.
• If the vertex lies below the x-axis and the parabola opens upwards, there are two x-intercepts.

• 若抛物线开口向上且顶点在 x 轴上方，则没有 x 截距。
• 若顶点在 x 轴上，则方程只有一个重根。
• 若抛物线开口向上且顶点在 x 轴下方，则有两个 x 截距。

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9. Transformations of Quadratic Functions | 二次函数的图像变换

By writing a quadratic in the form y = a(x + p)² + q, we can describe its graph as a translation of y = ax². The vertex is at (-p, q), and the line x = -p is the axis of symmetry.

将二次函数写成 y = a(x + p)² + q 的形式，就可以把它的图像看作 y = ax² 经过平移得到的。顶点在 (-p, q)，对称轴为 x = -p。

Changing q moves the graph vertically up or down. Changing p moves it horizontally left or right. Changing a stretches or shrinks it vertically and can flip it upside down if a is negative.

改变 q 会上下平移图像；改变 p 会左右平移图像；改变 a 会垂直拉伸或压缩图像，若 a 为负值，图像还会上下翻转。

For example, y = 2(x – 1)² + 3 has its vertex at (1, 3), opens upwards with a steeper shape, and its axis of symmetry is x = 1.

例如，y = 2(x – 1)² + 3 的顶点在 (1, 3)，开口向上且形状更陡，对称轴为 x = 1。

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10. Exam Strategies and Common Pitfalls | 考试策略与常见错误

Always rearrange the equation into the standard form before applying a method. Forgetting to set the equation equal to zero is a common mistake.

在应用任何方法之前，务必先将方程整理成标准形式。常见的错误是忘记让方程等于零。

When using the quadratic formula, check the signs carefully. A negative b means −b becomes positive in the numerator.

使用求根公式时，要仔细检查符号。若 b 为负数，则分子中的 −b 会变成正数。

• When factorising, expand your answer to check it matches the original.
• When completing the square, half the coefficient of x, not the full coefficient.
• When sketching a graph, label the vertex, the axis of symmetry, and any intercepts.
• When solving word problems, reject any negative root if it has no real-world meaning.

• 因式分解后，展开答案以验证是否与原式一致。
• 配方时，取 x 系数的一半，而不是整个系数。
• 画图时，标出顶点、对称轴以及所有截距。
• 解应用题时，若根没有实际意义，要舍去负数根。

Practice with past paper questions to become confident. Remember that a correct graph can help you check the number of solutions before you solve algebraically.

通过练习历年真题来增强信心。记住，在代数求解之前，画出正确的图像可以帮助你判断解的数量。

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