Solving Quadratic Equations and Graphing Quadratics | 解二次方程与二次函数图像

📚 Solving Quadratic Equations and Graphing Quadratics | 解二次方程与二次函数图像

Quadratic equations and functions are a core part of the Edexcel IGCSE Mathematics syllabus. Mastering them opens the door to higher-level algebra, coordinate geometry, and problem-solving skills. In this revision article, we will break down every essential technique, from factoring to completing the square, and show you how to sketch quadratic graphs with confidence.

二次方程与二次函数是 Edexcel IGCSE 数学课程的核心内容。掌握它们能为你打开高阶代数、坐标几何和问题解决能力的大门。在这篇复习文章中，我们将逐一拆解每一种关键方法——从因式分解到配方法——并帮助你自信地绘制二次函数图像。

1. 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 real numbers, and a ≠ 0. The graph of a quadratic function y = ax² + bx + c is a parabola. If a > 0, the parabola opens upwards (a “U” shape); if a < 0, it opens downwards (an "n" shape).

二次方程是可以写成以下形式的方程：

ax² + bx + c = 0

其中 a、b、c 是实数，且 a ≠ 0。二次函数 y = ax² + bx + c 的图像是抛物线。若 a > 0，抛物线开口向上（呈“U”形）；若 a < 0，抛物线开口向下（呈“n”形）。

2. Solving by Factorising | 利用因式分解解方程

When a quadratic can be factorised, it is often the quickest method. Look for two numbers that multiply to give ac and add to give b. For equations where a = 1, find two numbers that multiply to c and add to b. Then use the zero product property.

当二次方程可以因式分解时，这通常是最快的方法。找到两个数，它们的乘积等于 ac，和等于 b。对于 a = 1 的情形，找到两个数，乘积等于 c，和等于 b。然后利用零乘积性质。

Example: Solve x² − 5x + 6 = 0.

例：解 x² − 5x + 6 = 0。

Find two numbers that multiply to 6 and add to −5. They are −2 and −3.
找到乘积为 6、和为 −5 的两个数，它们是 −2 和 −3。
Write: (x − 2)(x − 3) = 0
写出：(x − 2)(x − 3) = 0
So x = 2 or x = 3.
因此 x = 2 或 x = 3。

If the coefficient of x² is not 1, you may need a coefficient pair method or the ac method. For example, 2x² + 5x − 3 = 0. Here a = 2, c = −3, so ac = −6. Find two numbers multiplying to −6 and adding to 5: they are 6 and −1. Split the middle term:

如果 x² 的系数不是 1，你可能需要系数对法或 ac 法。例如，2x² + 5x − 3 = 0。这里 a = 2，c = −3，所以 ac = −6。找到乘积为 −6、和为 5 的两个数：它们是 6 和 −1。拆分中间项：

2x² + 6x − x − 3 = 0

2x(x + 3) − 1(x + 3) = 0

(2x − 1)(x + 3) = 0

Hence x = ½ or x = −3.

因此 x = ½ 或 x = −3。

3. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x + p)² + q. This is essential for finding the vertex of a parabola and solving equations without factorising.

配方法将二次式改写为 a(x + p)² + q 的形式。这是找到抛物线顶点、以及在不因式分解的情况下解方程的关键。

For x² + bx, add and subtract (b/2)². For example, x² + 6x + 2 becomes:

对于 x² + bx，加上并减去 (b/2)²。例如，x² + 6x + 2 变为：

x² + 6x + 9 − 9 + 2 = (x + 3)² − 7

When the coefficient of x² is not 1, factor it out first. For 2x² − 8x + 5:

当 x² 的系数不是 1 时，先把它提出来。例如 2x² − 8x + 5：

2(x² − 4x) + 5 = 2[(x − 2)² − 4] + 5 = 2(x − 2)² − 8 + 5 = 2(x − 2)² − 3

To solve an equation by completing the square, rearrange to (x + p)² = q, then take the square root of both sides. Remember to include positive and negative square roots.

要用配方法解方程，先化为 (x + p)² = q，然后两边开平方根。记住要同时取正负平方根。

4. The Quadratic Formula | 二次公式

Any quadratic equation ax² + bx + c = 0 can be solved using the formula:

x = (−b ± √(b² − 4ac)) / (2a)

This formula works for all quadratics, including those that do not factorise. On the Edexcel IGCSE exam, this formula is given on the formula sheet, but you must know how to substitute and simplify correctly.

任意二次方程 ax² + bx + c = 0 都可以用公式求解：

x = (−b ± √(b² − 4ac)) / (2a)

这个公式适用于所有二次方程，包括那些不能因式分解的。在 Edexcel IGCSE 考试中，该公式会印在公式表上，但你必须知道如何正确代值和化简。

Example: Solve x² + 4x + 1 = 0.

例：解 x² + 4x + 1 = 0。

Here a = 1, b = 4, c = 1.

这里 a = 1，b = 4，c = 1。

x = (−4 ± √(16 − 4)) / 2 = (−4 ± √12) / 2 = (−4 ± 2√3) / 2 = −2 ± √3

So the solutions are x = −2 + √3 and x = −2 − √3. Always check whether the question asks for exact values or decimals. The answer in surd form is often preferred.

所以解为 x = −2 + √3 和 x = −2 − √3。始终检查题目要求精确值还是小数。通常以根式形式给出答案更受青睐。

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

The expression b² − 4ac inside the quadratic formula is called the discriminant. It tells you how many real roots a quadratic equation has:

二次公式中的表达式 b² − 4ac 称为判别式。它告诉你二次方程有多少个实数根：

Value of b² − 4ac	Nature of roots
Positive (Δ > 0)	Two distinct real roots
Zero (Δ = 0)	One repeated real root (equal roots)
Negative (Δ < 0)	No real roots

判别式的值 b² − 4ac（对应英文表格中的内容）：

判别式 > 0	两个不同实数根
判别式 = 0	一个重复实数根（等根）
判别式 < 0	无实数根

If the discriminant is a perfect square, the quadratic factorises over the integers. If it is positive but not a perfect square, the roots are irrational. If the discriminant is negative, the graph does not intersect the x-axis.

如果判别式是一个完全平方数，则二次方程可以用整数因式分解；如果它是正数但不是完全平方数，则根是无理数；如果判别式为负，则图像与 x 轴没有交点。

6. Key Features of a Quadratic Graph | 二次函数图像的关键特征

To sketch y = ax² + bx + c accurately, you need to find:

要准确地画出 y = ax² + bx + c 的草图，你需要找出：

The y-intercept: set x = 0, so y = c.
y 轴截距：令 x = 0，则 y = c。
The x-intercepts: solve ax² + bx + c = 0. These are the roots of the equation.
x 轴截距：解 ax² + bx + c = 0。这些就是方程的根。
The turning point (vertex): its x-coordinate is −b/(2a). Substitute this into the equation to find the y-coordinate.
顶点（极值点）：它的 x 坐标是 −b/(2a)。将其代入方程即可求得 y 坐标。
The axis of symmetry: the vertical line x = −b/(2a).
对称轴：垂直线 x = −b/(2a)。

If the quadratic is written in completed square form y = a(x + p)² + q, the vertex is simply (−p, q). The line of symmetry is x = −p.

如果二次函数写成配方法形式 y = a(x + p)² + q，顶点就是 (−p, q)。对称轴是 x = −p。

For example, y = 2(x − 3)² + 4 has vertex (3, 4) and opens upwards because a = 2 > 0. It does not touch the x-axis since the minimum y-value is 4.

例如，y = 2(x − 3)² + 4 的顶点为 (3, 4)，且因 a = 2 > 0 开口向上。它不与 x 轴相交，因为最小 y 值为 4。

7. Solving Quadratic Equations by Graph Sketching | 通过图像法解二次方程

Sketching the graph of y = ax² + bx + c allows you to estimate the solutions to ax² + bx + c = 0. The solutions are the x-coordinates where the graph crosses the x-axis. This method is useful for checking your algebraic work or when the roots are not exact.

画出 y = ax² + bx + c 的图像可以估算 ax² + bx + c = 0 的解。解就是图像与 x 轴交点的 x 坐标。该方法可用于检查代数计算结果，或当根不精确时进行估计。

You may also be asked to solve a quadratic equation that arises from a curve and a line. For example, to find the intersection of y = x² and y = 2x + 3, set x² = 2x + 3, then x² − 2x − 3 = 0, which factorises to (x − 3)(x + 1) = 0. So x = 3 or x = −1.

有时题目要求解由抛物线与直线相交而产生的二次方程。例如，求 y = x² 与 y = 2x + 3 的交点，令 x² = 2x + 3，则 x² − 2x − 3 = 0，分解为 (x − 3)(x + 1) = 0。所以 x = 3 或 x = −1。

8. Applications of Quadratics in Problem Solving | 二次方程在实际问题中的应用

Quadratic equations appear in areas such as geometry, projectile motion, and economics. For example:

二次方程出现在几何、抛体运动和经济学等领域。例如：

Area problems: The length of a rectangle is 3 cm more than its width, and the area is 28 cm². Let width = w, then length = w + 3. So w(w + 3) = 28, giving w² + 3w − 28 = 0. Factoring: (w + 7)(w − 4) = 0, so w = 4 (since width cannot be negative).
面积问题：矩形的长比宽多 3 cm，面积为 28 cm²。设宽为 w，则长为 w + 3。于是 w(w + 3) = 28，得到 w² + 3w − 28 = 0。因式分解：(w + 7)(w − 4) = 0，所以 w = 4（宽不能为负）。
Projectile motion: h(t) = −5t² + 20t + 1 models height in metres. To find when it hits the ground, set h(t) = 0 and solve for t. Use the quadratic formula or complete the square.
抛体运动：h(t) = −5t² + 20t + 1 表示以米为单位的的高度。求落地时间，令 h(t) = 0 并解 t。使用配方法或二次公式。

When solving word problems, always check whether both solutions are physically possible. Discard negative values if they do not make sense in the context.

解决应用题时，务必检查两个解在物理情境中是否都合理。如果负数不符合实际，应将其舍去。

9. Common Mistakes and How to Avoid Them | 常见错误与规避方法

Here are typical errors that lose marks in the IGCSE exam:

以下是在 IGCSE 考试中常见的丢分错误：

Missing the ± sign when taking square roots. Always write x = ±√k, not just x = √k.
漏掉 ± 号：取平方根时，应写 x = ±√k，而不是只写 x = √k。
Forgetting to rearrange the equation to equal zero before factoring. For example, x² + 2x = 3 must be rewritten as x² + 2x − 3 = 0.
忘记将方程化为等于零 后再进行因式分解。例如 x² + 2x = 3 必须改写为 x² + 2x − 3 = 0。
Dividing both sides by a variable that could be zero. Do not divide by x unless you are certain x ≠ 0.
等式两边同时除以一个可能为零的变量：除非确定 x ≠ 0，否则不要除以 x。
Mis-calculating the vertex: use x = −b/(2a), not −b/a.
顶点坐标计算错误：应使用 x = −b/(2a)，而不是 −b/a。
Incorrectly expanding (x + p)²: remember (x + p)² = x² + 2px + p², not x² + p².
错误展开 (x + p)²：注意 (x + p)² = x² + 2px + p²，而不是 x² + p²。

Reading the question carefully is also crucial. Some questions ask for roots, some for the turning point, and some for the discriminant. Underline exactly what is required.

仔细读题同样至关重要。有些题目要求根，有些要求顶点，有些要求判别式。请下划线标出题目具体需求。

10. Practice Questions for Quick Revision | 快速复习练习题

Try these questions yourself, then check the answers below.

请自己尝试以下题目，然后核对下方答案。

Solve x² − 7x + 10 = 0.
解 x² − 7x + 10 = 0。
Solve 2x² + 3x − 5 = 0 by factorising.
用因式分解法解 2x² + 3x − 5 = 0。
Write x² − 4x + 1 in the form (x − p)² + q.
将 x² − 4x + 1 写成 (x − p)² + q 的形式。
Find the discriminant of 3x² − 2x + 4 and state the nature of its roots.
求 3x² − 2x + 4 的判别式，并说明根的性质。
Sketch the graph of y = −(x − 1)² + 4, labeling the vertex, axis of symmetry, and intercepts.
画出 y = −(x − 1)² + 4 的图像，标出顶点、对称轴和截距。

Answers: 1. x = 2 or x = 5. 2. (2x + 5)(x − 1) = 0, so x = −2.5 or x = 1. 3. (x − 2)² − 3. 4. Discriminant = 4 − 48 = −44 < 0, no real roots. 5. Vertex at (1, 4), axis x = 1, y-intercept at (0, 3), x-intercepts at x = −1 and x = 3.

答案：1. x = 2 或 x = 5。2. (2x + 5)(x − 1) = 0，所以 x = −2.5 或 x = 1。3. (x − 2)² − 3。4. 判别式 = 4 − 48 = −44 < 0，无实数根。5. 顶点在 (1, 4)，对称轴 x = 1，y 轴截距在 (0, 3)，x 轴截距在 x = −1 和 x = 3。

11. Final Exam Tips for Quadratics | 二次方程的考场最后建议

On the Edexcel IGCSE paper, quadratics often appear in both Paper 1 and Paper 2. They can be standalone questions or embedded in longer problems. Here are some final tips to maximise your score:

在 Edexcel IGCSE 试卷中，二次方程常常同时出现在 Paper 1 和 Paper 2 中。它们可以是独立题目，也可能嵌入较长的应用题中。以下是一些最后提分建议：

Choose the most efficient method. For simple factorisations, factorise. For non-factorisable equations, use the formula. Completing the square is useful for graphs and solving where a is 1.
选择最高效的方法。简单因式分解用分解法；不能分解的用公式法。配方法对于图像问题以及 a = 1 时的求解很有用。
Show all working. Even if the final answer is wrong, method marks can be awarded. Write down the quadratic formula before substituting.
写出全部过程。即使最终答案错误，也可获得方法分。代入前先写出二次公式。
Always check your solutions by substituting back into the original equation if time permits.
如果时间允许，务必通过代回原方程检查你的解。
Use a calculator accurately for the discriminant and square roots. Round only at the final step, not in intermediate steps.
准确使用计算器计算判别式和平方根。只在最终步骤四舍五入，中间步骤不要四舍五入。

Understanding quadratics is not just about memorising formulas — it is about recognising patterns and choosing the right tool for each problem. With regular practice and careful attention to detail, you can master this topic and gain confidence for the Edexcel IGCSE Mathematics exam.

理解二次方程不仅仅是记住公式——更在于识别模式并为每个问题选择合适的工具。通过定期练习和对细节的仔细关注，你一定能够掌握该主题，并在 Edexcel IGCSE 数学考试中充满信心。

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Solving Quadratic Equations and Graphing Quadratics | 解二次方程与二次函数图像