Solving Quadratic Equations and Graphing Quadratics | 二次方程求解与二次函数图像

📚 Solving Quadratic Equations and Graphing Quadratics | 二次方程求解与二次函数图像

Quadratic equations and quadratic functions are a central part of IGCSE Mathematics. You will need to solve quadratic equations by factorising, completing the square, and using the quadratic formula, as well as sketch and interpret their graphs. In this article, we will break down each method step by step and highlight the key skills you need for your exams.

二次方程与二次函数是 IGCSE 数学的核心内容。你需要掌握用因式分解、配方和二次公式求解二次方程，并能够描绘和解读它们的图像。在本文中，我们将逐步分解每一种方法，并强调你在考试中需要的关键技能。

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

A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2. In its standard form, it is written as ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

二次方程是次数为 2 的多项式方程，即变量的最高次数为 2。其标准形式写作 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。

For example, x² + 5x + 6 = 0 is a quadratic equation, but x³ + x² = 0 is not because the highest power is 3. Meanwhile, 2x + 1 = 0 is linear, not quadratic, because its highest power is 1.

例如，x² + 5x + 6 = 0 是二次方程，而 x³ + x² = 0 不是，因为最高次数为 3。同样，2x + 1 = 0 是一次方程而不是二次方程，因为其最高次数为 1。

When a quadratic is written as y = ax² + bx + c, it defines a quadratic function. Setting y = 0 turns it into a quadratic equation whose solutions are the x-intercepts of the graph.

当二次式写成 y = ax² + bx + c 时，它定义了一个二次函数。令 y = 0，它就变成一个二次方程，其解就是图像与 x 轴的交点。

---

2. Standard Form and Key Features | 标准形式与关键特征

In the standard form ax² + bx + c = 0, the coefficient a controls the shape of the graph. If a > 0, the parabola opens upward; if a < 0, it opens downward. The sign of a therefore tells us whether the vertex is a minimum or a maximum.

在标准形式 ax² + bx + c = 0 中，系数 a 控制图像的形状。若 a > 0，抛物线开口向上；若 a < 0，开口向下。因此 a 的符号告诉我们顶点是最小值还是最大值。

The constant c gives the y-intercept, since when x = 0, y = c. For the equation ax² + bx + c = 0, the value of c also affects the position of the graph but does not change the shape.

常数 c 给出 y 轴截距，因为当 x = 0 时，y = c。对于方程 ax² + bx + c = 0，c 的值影响图像的位置，但不会改变形状。

The solutions of the equation are called the roots. They are the x-coordinates where the graph crosses the x-axis. A quadratic can have two roots, one repeated root, or no real roots depending on the value of the discriminant.

方程的解称为根，它们是图像与 x 轴交点的 x 坐标。根据判别式的值，二次方程可以有两个根、一个重根，或者没有实数根。

---

3. Factorising Method | 因式分解法

Factorising a quadratic expression means writing it as a product of two linear expressions. For example, x² + 5x + 6 factorises to (x + 2)(x + 3). This is the reverse of expanding brackets.

将一个二次表达式因式分解，就是把它写成两个一次表达式的乘积。例如，x² + 5x + 6 可分解为 (x + 2)(x + 3)。这是展开括号的逆运算。

To factorise x² + bx + c, find two numbers that multiply to give c and add to give b. For x² + 5x + 6, the numbers 2 and 3 satisfy 2 × 3 = 6 and 2 + 3 = 5.

要分解 x² + bx + c，需要找到两个数，它们相乘等于 c，相加等于 b。对于 x² + 5x + 6，数字 2 和 3 满足 2 × 3 = 6 且 2 + 3 = 5。

Sometimes the coefficient of x² is not 1. In such cases, factorising may involve trial and error or grouping. For example, 2x² + 7x + 3 = (2x + 1)(x + 3). Always check by expanding.

有时 x² 的系数不为 1。在这种情况下，因式分解可能需要尝试或分组。例如，2x² + 7x + 3 = (2x + 1)(x + 3)。一定要通过展开来验证。

---

4. Solving by Factorising | 用因式分解法解方程

Once the quadratic is factorised, use the zero product property: if AB = 0, then A = 0 or B = 0. Solve the two linear equations to find the roots.

当二次式被因式分解后，利用零乘积性质：若 AB = 0，则 A = 0 或 B = 0。解这两个一次方程即可得到根。

Example: Solve x² + 5x + 6 = 0. Factorising gives (x + 2)(x + 3) = 0. Hence x + 2 = 0 or x + 3 = 0, so x = −2 or x = −3.

例：解 x² + 5x + 6 = 0。因式分解得 (x + 2)(x + 3) = 0。因此 x + 2 = 0 或 x + 3 = 0，所以 x = −2 或 x = −3。

For the equation x² − 4 = 0, you can use the difference of two squares: (x − 2)(x + 2) = 0, giving x = 2 or x = −2. This method is quick and works well for simple quadratics.

对于方程 x² − 4 = 0，可以利用平方差公式：(x − 2)(x + 2) = 0，得到 x = 2 或 x = −2。这种方法快捷，适合简单的二次方程。

---

5. Completing the Square | 配方法

Completing the square rewrites ax² + bx + c in the form a(x − h)² + k. This is useful for finding the vertex of a parabola and solving equations that do not factorise easily.

配方法将 ax² + bx + c 改写成 a(x − h)² + k 的形式。这有助于找到抛物线的顶点，并求解不易因式分解的方程。

For a simple quadratic x² + bx, add and subtract (b/2)². For example, x² + 6x can be written as (x + 3)² − 9 because (x + 3)² = x² + 6x + 9.

对于简单的二次式 x² + bx，加上并减去 (b/2)²。例如，x² + 6x 可写成 (x + 3)² − 9，因为 (x + 3)² = x² + 6x + 9。

Example: Solve x² + 6x + 2 = 0 by completing the square. First rewrite as (x + 3)² − 9 + 2 = 0, so (x + 3)² = 7. Then x + 3 = ±√7, so x = −3 ± √7.

例：用配方法解 x² + 6x + 2 = 0。先改写为 (x + 3)² − 9 + 2 = 0，即 (x + 3)² = 7。于是 x + 3 = ±√7，所以 x = −3 ± √7。

---

6. The Quadratic Formula | 二次公式

The quadratic formula solves any quadratic equation. For ax² + bx + c = 0, the solutions are given by the formula below. You should memorise this formula because it always works, even when factorising is difficult or impossible.

二次公式可解任何二次方程。对于 ax² + bx + c = 0，其解由下面的公式给出。你应该记住这个公式，因为它总是有效，即使因式分解困难或无法进行。

x = (−b ± √(b² − 4ac)) / (2a)

Example: Solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, c = −3. Substitute into the formula: x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = (4 ± 2√10) / 4 = 1 ± √10/2.

例：解 2x² − 4x − 3 = 0。此时 a = 2，b = −4，c = −3。代入公式：x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = (4 ± 2√10) / 4 = 1 ± √10/2。

Always simplify the square root if possible, and write your final answer in exact form. Do not round unless the question asks you to give a decimal approximation.

如果可能，始终化简根号，并以精确形式书写最终答案。除非题目要求给出小数近似值，否则不要四舍五入。

---

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

The expression under the square root in the quadratic formula, Δ = b² − 4ac, is called the discriminant. It tells us how many roots the equation has without fully solving it.

二次公式中根号下的表达式 Δ = b² − 4ac 称为判别式。它无需完全求解就能告诉我们方程有多少个根。

• If Δ > 0, there are two distinct real roots.

  若 Δ > 0，则有两个不同的实数根。

• If Δ = 0, there is one repeated real root.

  若 Δ = 0，则有一个重复的实数根。

• If Δ < 0, there are no real roots.

  若 Δ < 0，则没有实数根。

For example, the equation x² + 2x + 5 = 0 has Δ = 2² − 4 × 1 × 5 = 4 − 20 = −16 < 0, so it has no real roots. This means its graph never touches the x-axis.

例如，方程 x² + 2x + 5 = 0 的判别式为 Δ = 2² − 4 × 1 × 5 = 4 − 20 = −16 < 0，因此它没有实数根。这意味着它的图像从不接触 x 轴。

---

8. Solving Word Problems with Quadratics | 用二次方程解应用题

Many real-world problems can be modelled by quadratic equations. For example, area problems, projectile motion, and number puzzles often lead to a quadratic equation.

许多实际问题都可以用二次方程来建模，例如面积问题、抛体运动和数字谜题通常都会导出二次方程。

When solving word problems, define a variable, form the quadratic equation, solve it, and then check whether the answer makes sense in the context. Sometimes you must discard a negative or impossible solution.

解应用题时，先定义一个变量，建立二次方程，然后求解，最后检查答案是否符合实际背景。有时需要舍弃负数解或不可能的答案。

Example: The length of a rectangle is 3 cm longer than its width, and its area is 40 cm². Let the width

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）