Mastering Quadratic Equations and Functions | 掌握二次方程与函数

📚 Mastering Quadratic Equations and Functions | 掌握二次方程与函数

Quadratic equations and functions are a cornerstone of the IGCSE Mathematics syllabus. Understanding their structure, solutions, and graphs will not only help you score highly but also build a strong foundation for advanced topics.

二次方程与函数是 IGCSE 数学教学大纲的基石。理解它们的结构、解法和图像，不仅能帮助你在考试中获得高分，还能为更高级的数学主题打下坚实基础。

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1. The General Form of a Quadratic Function | 二次函数的一般形式

A quadratic function is any function that can be written in the form f(x) = ax² + bx + c, where a, b and c are constants with a ≠ 0. The highest power of x is always 2.

二次函数是指可以写成 f(x) = ax² + bx + c 形式的函数，其中 a、b、c 为常数，且 a ≠ 0。x 的最高次数始终为 2。

If a > 0, the graph opens upwards like a smile. If a < 0, the graph opens downwards like a frown. The value of a also controls how “steep” the curve is.

当 a > 0 时，图像向上开口，如同微笑；当 a < 0 时，图像向下开口，如同皱眉。a 的值还控制着曲线的“陡峭”程度。

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2. Standard Form and the Shape of the Parabola | 标准形式与抛物线形状

The standard form y = ax² + bx + c is easy to evaluate but not easy to sketch. To understand the shape, we find three important features: the y-intercept, the roots (x-intercepts), and the vertex (turning point).

标准形式 y = ax² + bx + c 便于求值，但不便于画图。要理解其形状，我们需要找到三个重要特征：y 截距、根（x 截距）以及顶点（转向点）。

• The y-intercept is always at (0, c).
• y 截距总是在 (0, c) 处。
• The x-intercepts are the solutions of ax² + bx + c = 0.
• x 截距是方程 ax² + bx + c = 0 的解。
• The vertex lies on the axis of symmetry x = −b / (2a).
• 顶点位于对称轴 x = −b / (2a) 上。

Axis of symmetry: x = −b ÷ (2a)

对称轴：x = −b ÷ (2a)

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3. Solving by Factorisation | 因式分解法求解

Factorisation is the fastest method when the quadratic has simple integer roots. We rewrite the quadratic as a product of two linear brackets.

当二次方程具有简单的整数根时，因式分解是最快的方法。我们将二次式重写为两个一次括号的乘积。

For example, solve x² − 5x + 6 = 0. We look for two numbers that multiply to 6 and add to −5. These are −2 and −3, so (x − 2)(x − 3) = 0. Therefore x = 2 or x = 3.

例如，解 x² − 5x + 6 = 0。我们需要找到两个数，它们乘积为 6，和为 −5。这两个数是 −2 和 −3，所以 (x − 2)(x − 3) = 0。因此 x = 2 或 x = 3。

Always rearrange the equation into the form ax² + bx + c = 0 before factorising. If the coefficient of x² is not 1, use the method of splitting the middle term or trial and error.

在因式分解之前，务必先将方程整理成 ax² + bx + c = 0 的形式。如果 x² 的系数不为 1，可以使用拆中项法或试错法。

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

Completing the square transforms ax² + bx + c into a(x + h)² + k. This form reveals the vertex directly: the vertex is (−h, k).

配方法将 ax² + bx + c 变换为 a(x + h)² + k。这种形式直接揭示顶点：顶点为 (−h, k)。

For example, complete the square for x² + 6x + 1. Take half of 6, which is 3, and square it to get 9. Then x² + 6x = (x + 3)² − 9. Hence x² + 6x + 1 = (x + 3)² − 8.

例如，对 x² + 6x + 1 配方。取 6 的一半等于 3，平方得到 9。于是 x² + 6x = (x + 3)² − 9。因此 x² + 6x + 1 = (x + 3)² − 8。

To solve an equation, set the completed-square form equal to zero and take the square root of both sides.

要解方程，将配方后的形式设为零，然后两边开平方。

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5. The Quadratic Formula | 二次公式

The quadratic formula works for all quadratics, including those that cannot be factorised. For ax² + bx + c = 0:

二次公式适用于所有二次方程，包括无法因式分解的方程。对于 ax² + bx + c = 0：

x = [−b ± √(b² − 4ac)] ÷ (2a)

x = [−b ± √(b² − 4ac)] ÷ (2a)

Memorise this formula carefully. In the IGCSE Edexcel exam, you may be expected to use it without a calculator, so practise substituting values accurately.

请仔细记住这个公式。在 Edexcel IGCSE 考试中，你可能需要在不使用计算器的情况下使用它，因此要练习准确代入数值。

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6. The Discriminant | 判别式

The expression b² − 4ac inside the square root is called the discriminant. It tells us the number and type of roots without fully solving the equation.

根号内的表达式 b² − 4ac 称为判别式。它无需完整解方程就能告诉我们根的数量和类型。

• If b² − 4ac > 0, there are two distinct real roots.
• 若 b² − 4ac > 0，则有两个不同实根。
• If b² − 4ac = 0, there is one repeated real root.
• 若 b² − 4ac = 0，则有一个重根（两个相等实根）。
• If b² − 4ac < 0, there are no real roots.
• 若 b² − 4ac < 0，则没有实根。

The discriminant also helps in sketching: positive means the parabola crosses the x-axis twice, zero means it touches once, and negative means it never crosses the x-axis.

判别式也有助于画图：正值意味着抛物线与 x 轴相交两次，零意味着相切一次，负值意味着永不与 x 轴相交。

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7. Sum and Product of Roots | 根的和与积

For a quadratic equation ax² + bx + c = 0 with roots α and β, we can find relationships without solving:

对于具有根 α 和 β 的二次方程 ax² + bx + c = 0，我们无需解方程即可找到关系：

Sum of roots: α + β = −b ÷ a

根的和：α + β = −b ÷ a

Product of roots: αβ = c ÷ a

根的积：αβ = c ÷ a

These formulas are useful for checking answers and for forming new quadratics with given roots. If the roots are α and β, the quadratic can be written as x² − (α + β)x + αβ = 0.

这些公式可用于检查答案，以及根据给定根构造新的二次方程。若根为 α 和 β，则可以写成 x² − (α + β)x + αβ = 0。

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8. Sketching Quadratic Graphs | 二次函数图像的画法

A good sketch of a quadratic needs five key features: shape (upwards or downwards), y-intercept, x-intercepts, vertex, and axis of symmetry.

一个高质量的二次函数草图需要五个关键特征：开口方向（向上或向下）、y 截距、x 截距、顶点和对称轴。

1. Determine the sign of a to decide the shape.
2. 确定 a 的正负以决定开口方向。
3. Find the y-intercept by substituting x = 0.
4. 通过代入 x = 0 求 y 截距。
5. Solve f(x) = 0 to find the x-intercepts.
6. 解 f(x) = 0 求 x 截距。
7. Use x = −b/(2a) to find the axis of symmetry, then substitute back to find the vertex.
8. 使用 x = −b/(2a) 求对称轴，然后代回求顶点。

If the quadratic has no real roots, you only need the vertex and the y-intercept, and you must label the graph clearly.

如果二次方程没有实根，你只需要顶点和 y 截距，但必须清楚地标注图像。

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9. Solving Quadratic Inequalities | 二次不等式

Quadratic inequalities require you to first solve the corresponding equation, then decide which intervals satisfy the inequality.

二次不等式需要你先解对应的方程，然后判断哪些区间满足不等式。

For example, solve x² − x − 6 < 0. Factorise: (x − 3)(x + 2) < 0. The critical points are x = −2 and x = 3. Test values in the three intervals:

例如，解 x² − x − 6 < 0。因式分解：(x − 3)(x + 2) < 0。临界点为 x = −2 和 x = 3。在三个区间内测试数值：

• x < −2: both brackets negative, product positive → not a solution.
• x < −2：两个括号均为负，乘积为正 → 不是解。
• −2 < x < 3: one negative, one positive, product negative → solution.
• −2 < x < 3：一负一正，乘积为负 → 是解。
• x > 3: both positive, product positive → not a solution.
• x > 3：两个括号均为正，乘积为正 → 不是解。

Solution: −2 < x < 3

解：−2 < x < 3

Be careful with the direction of the inequality when multiplying by a negative number.

当乘以负数时，要小心不等式方向的变化。

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10. Applications: Area and Projectile Motion | 实际应用：面积与抛体运动

Quadratic equations often appear in geometry. For example, a rectangle has length (x + 3) cm and width (x − 1) cm. If the area is 12 cm², find x.

二次方程经常出现在几何问题中。例如，一个矩形的长为 (x + 3) cm，宽为 (x − 1) cm。如果面积为 12 cm²，求 x。

Area = length × width, so (x + 3)(x − 1) = 12. Expanding gives x² + 2x − 3 = 12, so x² + 2x − 15 = 0. Factorising: (x + 5)(x − 3) = 0, so x = 3 or x = −5. Since a length cannot be negative, x = 3.

面积 = 长 × 宽，所以 (x + 3)(x − 1) = 12。展开得 x² + 2x − 3 = 12，即 x² + 2x − 15 = 0。因式分解得 (x + 5)(x − 3) = 0，所以 x = 3 或 x = −5。由于长度不能为负，因此 x = 3。

In physics, the height of a projectile is often modelled by h(t) = −4.9t² + vt + h₀. Setting h = 0 gives the time when it lands. Always interpret the roots in context and discard impossible values.

在物理中，抛体的高度通常用 h(t) = −4.9t² + vt + h₀ 模型表示。令 h = 0 可求出落地时间。始终根据实际背景解释根，并舍弃不可能的值。

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11. Common Mistakes to Avoid | 避免常见错误

Many students lose marks on quadratic questions due to avoidable errors. Here are the most frequent pitfalls:

许多学生在二次方程题目中因为可避免的错误而丢分。以下是最常见的陷阱：

• Forgetting to rearrange the equation to equal zero before factorising.
• 忘记先将方程整理成等于零的形式再进行因式分解。
• Misapplying the quadratic formula by using −b incorrectly.
• 错误地使用二次公式中的 −b。
• When completing the square, forgetting to divide all terms by a if a ≠ 1.
• 配方时，若 a ≠ 1，忘记将所有项除以 a。
• Swapping the signs when using the sum and product of roots.
• 使用根的和与积时弄错符号。
• In inequalities, not testing all intervals.
• 在不等式中，没有测试所有区间。

Always check your solutions by substituting back into the original equation.

始终通过代入原方程来检验你的解。

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12. Practice Summary and Exam Tips | 练习总结与考试技巧

Quadratic questions appear in both Paper 1 (non-calculator) and Paper 2 (calculator). A quick revision checklist:

二次方程题目在 Paper 1（非计算器）和 Paper 2（计算器）中都会出现。快速复习清单如下：

Skill | 技能	Key Point | 要点
Factorisation | 因式分解	Find two numbers with correct sum and product | 找两个数，使其和与积正确
Completing the square | 配方	Vertex = (−h, k) in a(x + h)² + k | 顶点 = (−h, k)
Quadratic formula | 二次公式	x = [−b ± √(b² − 4ac)] / (2a), memorise it | 牢记公式
Discriminant | 判别式	b² − 4ac determines number of real roots | 决定实根个数

In the exam, read the question carefully: does it ask for exact values or rounded to a certain degree? Draw clear graphs using a ruler for straight lines, but quadratic curves should be smooth and symmetric.

考试中请仔细读题：要求精确值还是四舍五入到指定精度？画图时直线用直尺，二次曲线要平滑且对称。

Practice every method until it becomes automatic. Mastering quadratics will guarantee many easy marks across both papers.

将每种方法练习到自动熟练的程度。掌握二次方程将确保你在两份试卷中稳定获得大量分数。

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