📚 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”）。
ation 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。
onic 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: y 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: 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: −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）
x + 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: d 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)
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: 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
ed 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: . 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: ising: (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 = −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)。
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