Solving Quadratic Equations & Graphing Parabolas | 解二次方程与抛物线图像

📚 Solving Quadratic Equations & Graphing Parabolas | 解二次方程与抛物线图像

Quadratic equations and their graphs form one of the most important topics in IGCSE Mathematics. Whether you are solving for unknown values or sketching a smooth U-shaped curve, mastering quadratics unlocks a wide range of problems in algebra, geometry, and real-world applications.

二次方程及其图像是 IGCSE 数学中最重要的主题之一。无论是求未知数的值，还是绘制平滑的 U 形曲线，掌握二次函数都能帮助你解决代数、几何以及现实生活中的大量问题。

1. What Is a Quadratic Equation? | 什么是二次方程？

A quadratic equation is any equation that can be written in the standard form:

二次方程是指可以写成以下标准形式的方程：

ax² + bx + c = 0, where a ≠ 0

Here, a, b, and c are constants, and x is the unknown variable. The highest power of x is 2, which is why it is called “quadratic” (from the Latin word quadratus, meaning square).

其中，a、b、c 是常数，x 是未知数。未知数 x 的最高次数为 2，因此被称为“二次”（源自拉丁语 quadratus，意为“平方”）。

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

例如，2x² + 5x − 3 = 0 就是一个二次方程，其中 a = 2，b = 5，c = −3。如果 a = 0，方程就变成一次方程（线性方程），而不是二次方程。

2. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method when the quadratic has simple integer roots. The idea is to rewrite ax² + bx + c as a product of two binomials.

当二次方程具有简单的整数根时，因式分解通常是最快的方法。其核心思想是将 ax² + bx + c 改写为两个二项式的乘积。

For a monic quadratic (a = 1), we look for two numbers that multiply to give c and add to give b:

对于首项系数为 1 的二次方程（a = 1），我们需要找到两个数，使它们的乘积等于 c，和等于 b：

x² + bx + c = (x + p)(x + q), where p × q = c and p + q = b

Example: Solve x² − 5x + 6 = 0. We need two numbers that multiply to 6 and add to −5: these are −2 and −3. Thus (x − 2)(x − 3) = 0, giving x = 2 or x = 3.

示例：解方程 x² − 5x + 6 = 0。我们需要找到两个数，乘积为 6，和为 −5：这两个数是 −2 和 −3。因此 (x − 2)(x − 3) = 0，得到 x = 2 或 x = 3。

For a non-monic quadratic (a ≠ 1), use the method of grouping or the “cross method”. For example, 2x² + 7x + 3 = 0 can be factored as (2x + 1)(x + 3) = 0, so x = −½ or x = −3.

对于首项系数不为 1 的二次方程（a ≠ 1），可以使用分组分解法或“十字相乘法”。例如，2x² + 7x + 3 = 0 可以分解为 (2x + 1)(x + 3) = 0，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 + h)² + k. This is especially useful for finding the vertex of a parabola and for solving equations that do not factorise easily.

配方法将二次式改写为 a(x + h)² + k 的形式。这在求抛物线顶点以及解不易因式分解的方程时特别有用。

For x² + bx, add and subtract (b/2)²:

对于 x² + bx，加上并减去 (b/2)²：

x² + bx = (x + b/2)² − (b/2)²

Example: Complete the square for x² + 6x + 1. Half of 6 is 3, and 3² = 9. So x² + 6x + 1 = (x + 3)² − 9 + 1 = (x + 3)² − 8.

示例：对 x² + 6x + 1 配方。6 的一半是 3，3² = 9。所以 x² + 6x + 1 = (x + 3)² − 9 + 1 = (x + 3)² − 8。

Solving (x + 3)² − 8 = 0 gives (x + 3)² = 8, so x + 3 = ±√8, hence x = −3 ± 2√2.

解方程 (x + 3)² − 8 = 0，得到 (x + 3)² = 8，因此 x + 3 = ±√8，即 x = −3 ± 2√2。

4. The Quadratic Formula | 求根公式

The quadratic formula solves any quadratic equation directly. For ax² + bx + c = 0:

求根公式可以直接求解任何二次方程。对于 ax² + bx + c = 0：

x = (−b ± √(b² − 4ac)) / (2a)

This formula is derived from completing the square. You should memorise it and know how to substitute a, b, and c carefully, especially when they are negative.

这个公式由配方法推导而来。你应该牢记它，并知道如何仔细代入 a、b、c 的值，尤其是当它们为负数时。

Example: Solve 2x² − 4x − 3 = 0 using the formula. Here a = 2, b = −4, c = −3. Substituting:

示例：用求根公式解 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。代入得：

x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = (4 ± 2√10) / 4 = 1 ± √10/2

So the two solutions are x ≈ 2.58 and x ≈ −0.58.

因此两个解约为 x ≈ 2.58 和 x ≈ −0.58。

5. The Discriminant | 判别式

The expression b² − 4ac under the square root is called the discriminant, often denoted by Δ (Greek letter Delta). It tells us how many real roots a quadratic equation has.

根号下的表达式 b² − 4ac 称为判别式，通常用 Δ（希腊字母 Delta）表示。它告诉我们二次方程有多少个实数根。

判别式 Δ	根的个数	图像与 x 轴交点
Δ > 0	两个不同的实数根	两个交点
Δ = 0	一个实数根（重根）	一个切点
Δ < 0	无实数根	无交点

Example: For x² + 2x + 5 = 0, Δ = 2² − 4(1)(5) = 4 − 20 = −16 < 0, so the equation has no real roots, and the parabola does not intersect the x-axis.

示例：对于 x² + 2x + 5 = 0，Δ = 2² − 4(1)(5) = 4 − 20 = −16 < 0，因此方程没有实数根，抛物线不与 x 轴相交。

6. Sum and Product of Roots | 根的和与积（韦达定理）

If a quadratic equation ax² + bx + c = 0 has roots α and β, then the sum and product of the roots follow simple formulas:

如果二次方程 ax² + bx + c = 0 的两个根为 α 和 β，则它们的和与积遵循以下简洁公式：

α + β = −b/a, αβ = c/a

These relationships, known as Vieta’s formulas, allow you to check your solutions quickly or to construct a quadratic equation when the roots are known.

这些关系被称为韦达定理，可以帮助你快速检验答案，或者在已知根的情况下构造二次方程。

Example: If the roots of a quadratic are 2 and −5, then the sum is −3 and the product is −10. The monic equation is x² − (−3)x + (−10) = 0, i.e. x² + 3x − 10 = 0.

示例：如果一个二次方程的两个根为 2 和 −5，则它们的和为 −3，积为 −10。首项系数为 1 的方程为 x² − (−3)x + (−10) = 0，即 x² + 3x − 10 = 0。

7. Sketching Parabolas: Key Features | 绘制抛物线：关键特征

The graph of y = ax² + bx + c is a parabola. When sketching, you need to identify four key features:

函数 y = ax² + bx + c 的图像是一条抛物线。在绘制草图时，你需要确定以下四个关键特征：

Opening direction: If a > 0, the parabola opens upward (U shape); if a < 0, it opens downward (∩ shape).
开口方向：若 a > 0，抛物线开口向上（U 形）；若 a < 0，开口向下（∩ 形）。
y-intercept: The point (0, c), where the curve crosses the y-axis.
y 轴截距：曲线与 y 轴的交点，即 (0, c)。
x-intercepts (roots): The solutions of ax² + bx + c = 0, if they exist.
x 轴截距（根）：方程 ax² + bx + c = 0 的解（如果存在的话）。
Vertex (turning point): The maximum or minimum point of the parabola.
顶点（转折点）：抛物线的最大值点或最小值点。

Plotting these points and drawing a smooth curve through them gives an accurate sketch.

标出这些点，并用平滑曲线连接，即可得到准确的草图。

8. Vertex and Axis of Symmetry | 顶点与对称轴

The parabola is symmetric about a vertical line called the axis of symmetry. The x-coordinate of the vertex is found by:

抛物线关于一条垂直直线对称，这条直线称为对称轴。顶点的 x 坐标可以通过以下公式求出：

x = −b/(2a)

Once the x-coordinate is known, substitute it back into the equation to find the y-coordinate. Alternatively, if the equation is written in the form y = a(x − h)² + k, the vertex is simply at (h, k).

求出 x 坐标后，将其代回原方程即可得到 y 坐标。或者，如果方程写成 y = a(x − h)² + k 的形式，顶点直接就是 (h, k)。

Example: For y = x² − 4x + 1, the axis of symmetry is x = 4/(2 × 1) = 2. Substituting x = 2 gives y = 4 − 8 + 1 = −3. So the vertex is at (2, −3). Since a = 1 > 0, this vertex is a minimum point.

示例：对于 y = x² − 4x + 1，对称轴为 x = 4/(2 × 1) = 2。将 x = 2 代入得 y = 4 − 8 + 1 = −3。因此顶点为 (2, −3)。因为 a = 1 > 0，所以该顶点是最小值点。

9. Solving Quadratic Inequalities | 解二次不等式

Quadratic inequalities, such as ax² + bx + c > 0 or ≤ 0, are solved by first finding the roots and then determining which intervals satisfy the inequality.

二次不等式，如 ax² + bx + c > 0 或 ≤ 0，通常先求出根，再确定满足不等式的区间。

Example: Solve x² − x − 6 > 0. Factorising gives (x − 3)(x + 2) > 0. The roots are x = 3 and x = −2. Testing intervals:

示例：解不等式 x² − x − 6 > 0。因式分解得 (x − 3)(x + 2) > 0。根为 x = 3 和 x = −2。检验各个区间：

x < −2: both factors negative, product positive ✓
x < −2：两个因子均为负，乘积为正 ✓
−2 < x < 3: one negative, one positive, product negative ✗
−2 < x < 3：一正一负，乘积为负 ✗
x > 3: both factors positive, product positive ✓
x > 3：两个因子均为正，乘积为正 ✓

Therefore the solution is x < −2 or x > 3. A sketch of the parabola helps visualise this immediately.

因此解为 x < −2 或 x > 3。画出抛物线草图可以立刻直观地看出结果。

10. Applications in Real Life | 实际应用

Quadratic equations appear in many real-world contexts. In physics, the height of a projectile over time follows a quadratic model. In geometry, area problems often lead to quadratic equations. In business, profit and revenue functions are frequently quadratic.

二次方程在许多现实场景中都会出现。在物理中，抛射体随时间变化的高度遵循二次模型；在几何中，面积问题常常导出二次方程；在商业中，利润和收入函数也往往是二次函数。

Example: A rectangular garden has length 3 metres more than its width. If the area is 40 m², find its width. Let the width be x. Then length = x + 3, so x(x + 3) = 40, i.e. x² + 3x − 40 = 0. Factorising: (x + 8)(x − 5) = 0, so x = 5 (rejecting the negative value). The width is 5 metres.

示例：一块矩形花园的长比宽多 3 米。如果面积为 40 平方米，求它的宽。设宽为 x，则长为 x + 3，因此 x(x + 3) = 40，即 x² + 3x − 40 = 0。因式分解得 (x + 8)(x − 5) = 0，所以 x = 5（舍去负值）。宽为 5 米。

11. Common Mistakes to Avoid | 避免常见错误

Students often lose marks on quadratics due to a few repeated errors. Being aware of these can make a significant difference in your exam score.

学生在二次方程相关题目中经常因为几个反复出现的错误而失分。了解这些错误可以显著提高你的考试成绩。

Forgetting to set the equation to zero before factorising or applying the formula.
忘记将方程化为等于零的形式再进行因式分解或代入公式。
Sign errors when substituting negative values into the quadratic formula.
符号错误：将负值代入求根公式时出错。
Dropping one root when solving ±√ — always write down both solutions.
漏掉一个根：解 ±√ 时要写出两个解。
Incorrect factorisation — always expand your factors to verify.
因式分解错误——始终将因子展开以验证是否正确。

Always check your answers by substituting them back into the original equation.

务必通过将答案代回原方程来检验是否正确。

12. Practice Set | 练习巩固

Work through these problems to test your understanding:

完成以下练习，检验你的理解程度：

1. Solve by factorisation: x² − 7x + 10 = 0

1. 用因式分解法解：x² − 7x + 10 = 0

2. Solve by completing the square: x² + 8x + 3 = 0

2. 用配方法解：x² + 8x + 3 = 0

3. Use the quadratic formula to solve 3x² + 5x − 2 = 0.

3. 用求根公式解 3x² + 5x − 2 = 0。

4. Find the discriminant of 2x² − 4x + 3 = 0 and state the number of real roots.

4. 求 2x² − 4x + 3 = 0 的判别式，并说明实数根的个数。

5. Sketch y = x² − 2x − 3, labelling the vertex, roots, and y-intercept.

5. 画出 y = x² − 2x − 3 的草图，标出顶点、根和 y 轴截距。

Answers: (1) x = 2 or x = 5. (2) x = −4 ± √13. (3) x = ⅓ or x = −2. (4) Δ = −8 < 0, no real roots. (5) Vertex (1, −4), roots x = 3 and x = −1, y-intercept (0, −3).

答案：(1) x = 2 或 x = 5。(2) x = −4 ± √13。(3) x = ⅓ 或 x = −2。(4) Δ = −8 < 0，无实数根。(5) 顶点 (1, −4)，根 x = 3 和 x = −1，y 轴截距 (0, −3)。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Solving Quadratic Equations & Graphing Parabolas | 解二次方程与抛物线图像