Solving Quadratic Equations and Functions | 二次方程与函数的解法与应用

📚 Solving Quadratic Equations and Functions | 二次方程与函数的解法与应用

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in many forms and require a solid understanding of algebraic manipulation and graphical interpretation.

二次方程是IGCSE数学中最重要的主题之一。它们以多种形式出现，需要扎实的代数和图形理解能力。

1. Standard Form of a Quadratic Equation | 二次方程的标准形式

A quadratic equation in one variable is an equation that can be written in the form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. The highest power of the variable is 2.

一元二次方程是可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 是实数且 a ≠ 0。变量的最高次幂是2。

The coefficient a is called the leading coefficient. It determines whether the parabola opens upward (a > 0) or downward (a < 0).

系数 a 称为首项系数。它决定抛物线开口向上（a > 0）还是向下（a < 0）。
The constant term c gives the y-intercept of the graph, because when x = 0, y = c.

常数项 c 给出图像的 y 轴截距，因为当 x = 0 时，y = c。
If a = 0, the equation is not quadratic but linear. So a must not be zero.

如果 a = 0，方程就不是二次的，而是线性的。因此 a 不能为零。

2. Solving by Factorisation | 因式分解法

Factorisation is the quickest method of solving a quadratic equation when the expression can be written as the product of two linear factors.

因式分解是当表达式可以写成两个一次因式的乘积时，求解二次方程最快的方法。

Set the equation to zero and factorise the quadratic expression.

将方程化为零，并对二次表达式进行因式分解。
Then use the zero product property: if AB = 0, then A = 0 or B = 0.

然后利用零乘积性质：如果 AB = 0，那么 A = 0 或 B = 0。

x² + 5x + 6 = (x + 2)(x + 3) = 0 → x = −2 or x = −3

For example, x² + 5x + 6 = 0. Factorising gives (x + 2)(x + 3) = 0, so the solutions are x = −2 and x = −3.

例如，x² + 5x + 6 = 0。因式分解得到 (x + 2)(x + 3) = 0，所以解为 x = −2 和 x = −3。

3. Completing the Square | 配方法

Completing the square is a powerful technique that works for any quadratic equation, even when factorisation is not easy.

配方法是一种强大的技巧，适用于任何二次方程，即使因式分解很困难时也有效。

The key idea is to rewrite ax² + bx + c in the form a(x + h)² + k, which reveals the turning point of the parabola.

关键思想是将 ax² + bx + c 重写为 a(x + h)² + k 的形式，这样可以揭示抛物线的顶点。

Divide both sides by a if a ≠ 1, or factor out a from the first two terms.

如果 a ≠ 1，就将方程两边除以 a，或从前两项中提出 a。
Add and subtract the square of half the coefficient of x inside the bracket.

在括号内加上并减去 x 项系数一半的平方。

x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7

Thus the equation becomes (x + 3)² = 7, and then you solve for x by taking square roots.

因此方程变为 (x + 3)² = 7，然后通过开平方来求解 x。

4. The Quadratic Formula | 二次公式

For any quadratic equation ax² + bx + c = 0, the solutions are given by the quadratic formula.

对于任意二次方程 ax² + bx + c = 0，解可由二次公式给出。

x = (−b ± √(b² − 4ac)) / (2a)

This formula works for all quadratic equations, including those that cannot be factorised. You just substitute the values of a, b, and c.

这个公式适用于所有二次方程，包括那些不能因式分解的。只需代入 a、b、c 的值即可。

Be careful with negative signs when substituting. Brackets help avoid mistakes.

代入时注意负号，使用括号有助于避免错误。
The symbol ± means you calculate two values: one with plus and one with minus.

符号 ± 表示你要计算两个值：一个用加号，一个用减号。

5. The Discriminant | 判别式

The expression inside the square root in the quadratic formula is called the discriminant and is denoted by Δ.

二次公式中根号内的表达式称为判别式，用 Δ 表示。

Δ = b² − 4ac

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

如果 Δ > 0，方程有两个不相等的实数根。
If Δ = 0, there is exactly one real root (a repeated root).

如果 Δ = 0，方程有一个实数根（重根）。
If Δ < 0, there are no real roots; the solutions are complex numbers.

如果 Δ < 0，方程没有实数根；解是复数。

In IGCSE, you are often asked to determine the number of roots without solving the equation fully.

在IGCSE考试中，你经常需要在不完全求解方程的情况下判断根的个数。

6. Graphs of Quadratic Functions | 二次函数的图像

A quadratic function y = ax² + bx + c produces a parabola. Its shape depends on a and its position depends on b and c.

二次函数 y = ax² + bx + c 的图像是抛物线。它的形状取决于 a，位置取决于 b 和 c。

When a > 0, the parabola opens upward and has a minimum point.

当 a > 0 时，抛物线开口向上，有最低点。
When a < 0, the parabola opens downward and has a maximum point.

当 a < 0 时，抛物线开口向下，有最高点。
The graph crosses the y-axis at the point (0, c).

图像在点 (0, c) 处与 y 轴相交。
The x-intercepts (if they exist) are the real roots of the equation ax² + bx + c = 0.

x 轴截距（如果存在）就是方程 ax² + bx + c = 0 的实数根。

7. Vertex and Line of Symmetry | 顶点与对称轴

Every parabola has a line of symmetry that passes through its vertex. The x-coordinate of the vertex can be found using a simple formula.

每条抛物线都有一条通过其顶点的对称轴。顶点的 x 坐标可以通过一个简单公式求出。

Vertex x-coordinate = −b / (2a)

Once you have this value, substitute it back into the function to find the y-coordinate of the vertex.

得到这个值后，将其代回函数中即可求出顶点的 y 坐标。

The vertical line x = −b/(2a) is the axis of symmetry.

垂直线 x = −b/(2a) 就是对称轴。
If you use completing the square, the vertex can be read directly from the form y = a(x − h)² + k. The vertex is (h, k).

如果使用配方法，可以直接从 y = a(x − h)² + k 的形式读出顶点，顶点就是 (h, k)。

8. Solving Quadratic Equations by Graphical Method | 图像法解二次方程

You can solve a quadratic equation by sketching or plotting the graph of the corresponding function and reading the x-intercepts.

你可以通过绘制对应函数的草图或图形并读出 x 轴截距来解二次方程。

Draw the graph of y = ax² + bx + c on a coordinate grid.

在坐标网格上绘制 y = ax² + bx + c 的图像。
The points where the graph crosses the x-axis are the solutions to ax² + bx + c = 0.

图像与 x 轴的交点就是 ax² + bx + c = 0 的解。
If the graph never touches the x-axis, the equation has no real roots.

如果图像不与 x 轴接触，那么方程没有实数根。

This method is useful for approximating roots or when the equation is difficult to factorise.

这种方法适用于近似求根或方程难以因式分解时。

9. Quadratic Equation from Given Roots | 由已知根构造二次方程

If you know the roots r₁ and r₂ of a quadratic equation, you can write the equation as (x − r₁)(x − r₂) = 0.

如果你知道二次方程的两个根 r₁ 和 r₂，则可以将方程写成 (x − r₁)(x − r₂) = 0 的形式。

Expanding gives x² − (r₁ + r₂)x + r₁r₂ = 0.

展开后得到 x² − (r₁ + r₂)x + r₁r₂ = 0。

The sum of the roots is −b/a and the product is c/a for the general form ax² + bx + c = 0.

对于一般形式 ax² + bx + c = 0，两根之和为 −b/a，两根之积为 c/a。
These relationships are often used to check solutions or to build equations without solving.

这些关系常用于检查解或在不解方程的情况下构造方程。

10. Word Problems with Quadratic Equations | 二次方程应用题

Many real-world problems can be modelled using quadratic equations. Problems involving area, projectile motion, and number relationships often lead to quadratics.

许多现实问题可以用二次方程来建模。涉及面积、抛体运动和数字关系的问题通常会产生二次方程。

Define a variable clearly and use the given conditions to write an equation.

清楚定义变量，并根据给定条件写出方程。
Solve the equation using any suitable method.

使用合适的方法解方程。
Check whether each solution makes sense in the context of the problem. Reject negative lengths or distances if they are not possible.

检查每个解在问题情境中是否有意义。如果负长度或负距离不可能，就舍弃它们。

Example: The area of a rectangle is 28 cm² and its length is 3 cm more than its width. Find the width.

Let the width be w cm. Then the length is w + 3. The area gives w(w + 3) = 28, which becomes w² + 3w − 28 = 0. Factorising gives (w + 7)(w − 4) = 0, so w = 4 or w = −7. The width must be positive, so w = 4 cm.

设宽度为 w 厘米。则长度为 w + 3。面积给出 w(w + 3) = 28，即 w² + 3w − 28 = 0。因式分解得 (w + 7)(w − 4) = 0，所以 w = 4 或 w = −7。宽度必须为正，因此 w = 4 厘米。

11. Common Mistakes and Tips | 常见错误与提示

Students often lose marks due to careless algebraic errors. Here are some common pitfalls and how to avoid them.

学生常常因为粗心的代数错误而失分。以下是一些常见陷阱及避免方法。

Do not forget to set the equation to zero before factorising.

因式分解前不要忘记将方程化为零。
When using the quadratic formula, write down the values of a, b, and c first.

使用二次公式时，先写下 a、b、c 的值。
When completing the square, remember that the sign inside the bracket is opposite to the vertex x-coordinate.

配方时，记得括号内的符号与顶点 x 坐标相反。
Check your solutions by substituting them back into the original equation.

将解代回原方程进行验算。
Always consider whether a solution is possible in word problems.

在应用题中始终考虑解是否可能。

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Solving Quadratic Equations and Functions | 二次方程与函数的解法与应用