📚 Quadratic Functions and Equations | 二次函数与方程
Quadratic functions and equations form one of the most important foundations in IGCSE Mathematics. Mastering this topic is essential not only for exams but also for higher-level studies in algebra, calculus, and the physical sciences. This guide covers everything you need to know, from basic definitions to advanced problem-solving techniques.
二次函数与方程是IGCSE数学中最重要基础之一。掌握这一主题不仅对考试至关重要，也为将来的代数、微积分及自然科学学习奠定坚实基础。本指南涵盖从基本定义到高级解题技巧的全部要点，帮助您系统掌握这一核心知识模块。
1. Definition and Standard Form | 定义与标准形式
ith the condition that a ≠ 0. If a = 0, the function becomes linear rather than quadratic.
二次函数是最高次数为2的多项式函数，即变量的最高次幂为2。其一般形式为 f(x) = ax² + bx + c，其中 a、b、c 为常数，且必须满足 a ≠ 0。若 a = 0，该函数退化为线性函数而非二次函数。
The coefficient a determines the direction of the parabola: if a > 0, the parabola opens upward, resembling a U shape; if a < 0, it opens downward, resembling an inverted U. The constant term c represents the y-intercept, the point where the graph crosses the y-axis.
系数 a 决定抛物线的开口方向：当 a > 0 时，抛物线向上开口，呈"U"形；当 a < 0 时，抛物线向下开口，呈倒"U"形。常数项 c 表示 y 轴截距，即图像与 y 轴交点的纵坐标。
2. Complete the Square | 配方法
Completing the square is a powerful algebraic technique that transforms a quadratic expression into the form a(x + p)² + q. This form reveals the turning point of the parabola directly and is particularly useful when solving equations or sketching graphs.
配方法是重要的代数技巧，它将二次表达式转换为 a(x + p)² + q 的形式。这种形式直接揭示抛物线的顶点坐标，特别适用于解方程和绘制函数图像。
For the quadratic expression ax² + bx + c, the completed square form is:
a(x + b/2a)² + c − b²/4a
ical line x = −b/2a. This method also proves essential for deriving the quadratic formula.
其中，顶点（转向点）坐标为 (−b/2a, c − b²/4a)。对称轴为垂直直线 x = −b/2a。配方法也是推导二次公式的核心工具。
giving f(x) = (x + 3)² − 4. The vertex is at (−3, −4), and the line of symmetry is x = −3.
例如，对 f(x) = x² + 6x + 5 配方，我们加上并减去 (6/2)² = 9，得到 f(x) = (x + 3)² − 4。顶点坐标为 (−3, −4)，对称轴为 x = −3。
3. Solving Quadratic Equations by Factorisation | 因式分解法解二次方程
Solving quadratic equations is a fundamental skill. The factorisation method involves expressing the quadratic as a product of two linear factors and then setting each factor equal to zero. This method is fastest when the equation has rational roots.
解二次方程是基础技能。因式分解法将二次表达式写成两个一次因式的乘积，然后令每个因式等于零。当方程具有有理数根时，此方法最为快捷。
5x + 6 = 0, we find two numbers that multiply to 6 and add to 5, which are 2 and 3. Thus:
因式分解 ax² + bx + c 时，我们需要找到两个数，其乘积等于 ac，和等于 b。例如，解 x² + 5x + 6 = 0，我们需要找到乘积为 6、和为 5 的两个数，即 2 和 3。因此：
(x + 2)(x + 3) = 0
by substituting them back into the original equation. If the coefficient of x² is not 1, first try to factor out any common factor before applying this method.
令每个因式等于零，得 x = −2 或 x = −3。务必通过代回原方程来验证答案。若 x² 的系数不为 1，应首先提取公因式，再应用此法。
4. The Quadratic Formula | 二次公式求根法
ing any quadratic equation. For the equation ax² + bx + c = 0, the solutions are given by:
二次公式为解任意二次方程提供了通用方法。对于方程 ax² + bx + c = 0，其解为：
x = (−b ± √b² − 4ac) / 2a
s exactly one real root (a repeated root); if Δ < 0, it has no real roots but two complex roots.
表达式 b² − 4ac 称为判别式，通常用 Δ 表示。判别式决定根的性质：若 Δ > 0，方程有两个不等实根；若 Δ = 0，方程有一个实根（重根）；若 Δ < 0，方程无实根，但有两个复数根。
dratic formula is always reliable, even when factorisation proves difficult or impossible.
例如，用公式解 2x² − 4x − 6 = 0，取 a = 2，b = −4，c = −6，得 Δ = 16 + 48 = 64，故 x = (4 ± 8)/4，解得 x = 3 或 x = −1。二次公式始终可靠，即使因式分解困难或无法进行时也能奏效。
5. The Discriminant and Nature of Roots | 判别式与根的性质
The discriminant is not just a computational tool; it provides deep insight into the behaviour of the quadratic. In IGCSE exams, questions frequently ask you to determine the number and type of roots without solving the equation.
判别式不仅是计算工具，它深刻揭示二次方程的行为特征。IGCSE考试中，常见题型是无需解方程而直接判断根的个数与类型。
actly one point, meaning the parabola is tangent to the axis. When Δ < 0, the parabola does not intersect the x-axis at all.
当 Δ > 0 且为完全平方数时，根为有理数且不等；当 Δ > 0 但不是完全平方数时，根为无理数且不等。当 Δ = 0 时，图像与 x 轴恰好相切于一点，抛物线在该点与轴相切。当 Δ < 0 时，抛物线完全不与 x 轴相交。
oots is −b/a and the product is c/a. These relationships are useful for constructing equations with specified root properties.
此外，判别式还帮助我们处理根的和与积。对于 ax² + bx + c = 0，两根之和为 −b/a，两根之积为 c/a。这些关系用于构造具有特定根性质的方程非常有用。
6. Graphing Quadratic Functions | 二次函数的图像
The graph of a quadratic function is a parabola with characteristics that can be identified from its equation. Key features include the vertex, the axis of symmetry, the y-intercept, and the x-intercepts (roots). To sketch a quadratic graph accurately, follow these steps.
二次函数图像是一条抛物线，其特征可从方程中识别。"关键特征"包括顶点、对称轴、y 轴截距和 x 轴截距（根）。要准确绘制二次函数草图，请遵循以下步骤。
- Determine the direction of opening: a > 0 means U-shaped, a < 0 means inverted U.
- Find the y-intercept directly from c.
- Find the x-intercepts by solving ax² + bx + c = 0, if real roots exist.
- Find the vertex using x = −b/2a and substitute to find the y-coordinate.
- Plot at least five points for an accurate sketch, including the vertex and intercepts.
- 确定开口方向：a > 0 为"U"形，a < 0 为倒"U"形。
- 直接从 c 得到 y 轴截距。
- 解方程 ax² + bx + c = 0 求 x 轴截距（若实根存在）。
- 利用 x = −b/2a 求顶点横坐标，代回求纵坐标。
- 至少描五个点（含顶点和截距）以保证草图准确。
ves x = 1 and x = 3. The vertex is at x = −4/(−2) = 2, and f(2) = 1. The vertex is (2, 1).
例如，绘制 f(x) = −x² + 4x − 3 的草图：a < 0 所以开口向下；y 轴截距为 −3；解 −x² + 4x − 3 = 0 得 x = 1 和 x = 3；顶点横坐标为 x = −4/(−2) = 2，f(2) = 1，顶点为 (2, 1)。
7. Problem Solving with Quadratics | 二次方程的应用题
Quadratic equations frequently appear in real-world applications, ranging from physics to geometry. In IGCSE examinations, problem-solving questions typically require you to translate a word problem into a quadratic equation and then solve it.
二次方程广泛应用于现实问题，涵盖物理与几何等众多领域。在IGCSE考试中，应用题通常要求将文字问题转化为二次方程再求解。
Common application areas include: projectile motion (height over time), area problems (dimensions of rectangles with given area and perimeter), and number problems (finding two consecutive integers with a given product). When solving word problems, clearly define your variable, set up the equation, solve it, and then interpret the answers in the context of the problem.
常见应用领域包括：抛体运动（高度随时间变化）、面积问题（给定面积和周长的矩形尺寸）、数字问题（寻找乘积为给定值的连续整数）。解应用题时，明确设定变量、建立方程、求解，然后在题目情境中解释答案。
so w = 5 (rejecting w = −8, as length cannot be negative). The rectangle is 5 cm by 8 cm.
例如，一个矩形的长比宽多 3 厘米，面积为 40 平方厘米。设宽为 w，则 w(w + 3) = 40，即 w² + 3w − 40 = 0。因式分解得 (w + 8)(w − 5) = 0，故 w = 5（舍弃 w = −8，因为长度不能为负）。矩形尺寸为 5 厘米 × 8 厘米。
8. Inequalities with Quadratics | 二次不等式
Quadratic inequalities extend the solving techniques of quadratic equations to determine range of values. To solve ax² + bx + c > 0 or < 0, first solve the corresponding equation, then use a sign diagram to determine which intervals satisfy the inequality.
二次不等式将二次方程的求解技术扩展到取值范围问题。解 ax² + bx + c > 0 或 < 0 时，先解对应方程，然后利用符号图（数轴表）确定哪些区间满足不等式。
3, it is negative; for x > 3, it is positive. Therefore, the solution is x ≤ −1 or x ≥ 3.
考虑不等式 x² − 2x − 3 ≥ 0。因式分解得 (x − 3)(x + 1) ≥ 0。临界值为 x = 3 和 x = −1。分区间测试：当 x < −1 时，乘积为正；当 −1 < x < 3 时，乘积为负；当 x > 3 时，乘积为正。因此解为 x ≤ −1 或 x ≥ 3。
Remember: when multiplying or dividing an inequality by a negative number, you must reverse the inequality sign. However, this rule does not apply to the factorisation approach used above, as we are analysing the sign of the product over different intervals.
请记住：当不等式两边乘以或除以负数时，必须反向改变不等号方向。但上述因式分解法不涉及这一规则，因为我们是在分析乘积在不同区间上的符号。
9. Solving Simultaneous Equations with Quadratics | 二次方程与联立方程组
IGCSE examinations often include simultaneous equations where one equation is linear and the other is quadratic. Solving such systems requires substituting the linear expression into the quadratic equation and then solving the resulting quadratic equation.
IGCSE考试常包含一个线性方程与一个二次方程的联立方程组。解此类方程组需将线性表达代入二次方程，然后解所得到的二次方程。
1 = x² − x − 5, which simplifies to x² − 3x − 6 = 0. Using the quadratic formula, x = (3 ± √33)/2. Substituting these values back into y = 2x + 1 gives the corresponding y-values.
例如，解方程组 y = 2x + 1 与 y = x² − x − 5。由于两式都等于 y，令它们相等：2x + 1 = x² − x − 5，化简为 x² − 3x − 6 = 0。利用二次公式，x = (3 ± √33)/2。将这些 x 值代回 y = 2x + 1，得到相应的 y 值。
When solving such systems graphically, the x-coordinates of the intersection points of the straight line and the parabola represent the solutions. If the resulting quadratic has Δ < 0, it means the line does not intersect the parabola.
作图求解此类方程组时，直线与抛物线交点的横坐标即为解。若所得二次方程的 Δ < 0，说明直线与抛物线不相交。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Students often make repeated mistakes when working with quadratic functions. Recognising these common pitfalls will significantly improve your accuracy in examinations. Be especially careful with signs, algebraic manipulation, and notation.
学生在处理二次函数时常犯重复性错误。识别常见陷阱将显著提高考试正确率。尤其要注意符号、代数运算和书写规范。
- Do not forget the ± sign when taking square roots: √x² = |x|, not x.
- ing the quadratic formula, ensure the equation is in standard form ax² + bx + c = 0 first.
- When factorising, check signs carefully: (x − a)(x + b) expands differently from (x + a)(x − b).
- When completing the square, remember to add and subtract (b/2)² for monic quadratics.
- Always interpret solutions in context: negative lengths or times are usually rejected.
- Check your solutions by substituting back into the original equation.
- 开平方时不要忘记 ± 号：√x² = |x|，而不是 x。
- 使用二次公式前，确保方程已经化为标准形式 ax² + bx + c = 0。
- 因式分解时仔细检查符号：(x − a)(x + b) 的展开式与 (x + a)(x − b) 不同。
- 对首项系数为 1 的二次函数配方时，记得加、减 (b/2)²。
- 始终在题目情境中解释答案：负数长度或时间通常需要舍去。
- 将解代回原方程验证结果。
In the examination, read each question carefully to determine which method is expected. If the question says "factorise", do not use the formula; if it says "solve", any correct method is acceptable. For sketching questions, label the vertex, intercepts, and axis of symmetry clearly.
考试时仔细审题，判断期望使用的解法。若题目要求"因式分解"，不要使用公式；若要求"求解"，任何正确方法均可。作图题需清晰标注顶点、截距和对称轴。
11. Practice Problems | 练习题目
The following practice problems cover the breadth of quadratic topics discussed in this article. Attempt each problem independently before checking the solutions provided. These problems reflect typical IGCSE examination standards.
以下练习覆盖本文所讨论的二次函数各知识点。请先独立尝试每道题目，再核对解答。这些题目反映了IGCSE考试的标准水平。
- Solve x² − 7x + 12 = 0 by factorisation.
- Write f(x) = x² + 8x + 15 in completed square form and state the vertex.
- Determine the nature of the roots of 3x² + 2x − 5 = 0 without solving.
- Solve 2x² + 3x − 7 = 0 using the quadratic formula, giving answers to 2 decimal places.
- Solve the inequality x² − 9 > 0.
- A rectangular garden has a perimeter of 28 m and an area of 48 m². Find its dimensions.
- 用因式分解法解 x² − 7x + 12 = 0。
- 将 f(x) = x² + 8x + 15 写成配方法形式，并指出顶点坐标。
- 不解方程，判断 3x² + 2x − 5 = 0 的根的性质。
- 用二次公式解 2x² + 3x − 7 = 0，答案保留两位小数。
- 解不等式 x² − 9 > 0。
- 矩形花园周长为 28 米，面积为 48 平方米，求其尺寸。
Solutions | 解答
1. (x − 3)(x − 4) = 0, so x = 3 or x = 4.
1. (x − 3)(x − 4) = 0，故 x = 3 或 x = 4。
2. f(x) = (x + 4)² − 1; the vertex is (−4, −1).
2. f(x) = (x + 4)² − 1；顶点为 (−4, −1)。
Δ = 4 − 4 × 3 × (−5) = 64 > 0, which is a perfect square, so two distinct rational roots.
3. Δ = 4 − 4 × 3 × (−5) = 64 > 0，为完全平方数，故有两个不等有理根。
4. x = (−3 ± √65)/4 ≈ 1.27 or x ≈ −2.77.
4. x = (−3 ± √65)/4 ≈ 1.27 或 x ≈ −2.77。
5. (x − 3)(x + 3) > 0, so x < −3 or x > 3.
5. (x − 3)(x + 3) > 0，故 x < −3 或 x > 3。
o w² − 14w + 48 = 0, yielding (w − 6)(w − 8) = 0. Therefore the dimensions are 6 m by 8 m.
6. 设长为 l、宽为 w。则 2(l + w) = 28 且 lw = 48。由第一式得 l + w = 14，故 l = 14 − w。代入得 w(14 − w) = 48，即 w² − 14w + 48 = 0，分解得 (w − 6)(w − 8) = 0。因此尺寸为 6 米 × 8 米。
12. Summary and Key Takeaways | 总结与核心要点
Quadratic functions and equations constitute a major topic in IGCSE Mathematics, appearing in both Paper 1 (non-calculator) and Paper 2 (calculator). Consistent practice with factorisation, the quadratic formula, and completing the square will build confidence and speed.
二次函数与方程是IGCSE数学的重要主题，在卷一（不使用计算器）和卷二（使用计算器）中均有出现。坚持练习因式分解法、二次公式和配方法将增强信心并提高速度。
Remember the three essential methods for solving quadratic equations: factorisation, completing the square, and the quadratic formula. The discriminant tells you the nature of the roots, while the completed square form reveals the vertex and line of symmetry. Always verify your solutions and interpret them in context.
牢记解二次方程的三种基本方法：因式分解法、配方法和二次公式。判别式揭示根的性质，配方法形式揭示顶点与对称轴。始终验证解答并在题目情境中解释其含义。
With regular practice, solving quadratic problems becomes intuitive and accurate. Master this topic, and you will build a strong foundation for IGCSE examinations and beyond.
通过规律性练习，解二次问题将变得得心应手且准确无误。掌握这一主题，您将为IGCSE考试及后续更高层次的学习奠定坚实基础。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图