Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are equations of the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. In Edexcel IGCSE Mathematics, solving quadratic equations is a core skill that appears in both Paper 1 and Paper 2. In this article, we will review the standard form, three algebraic methods, the discriminant, graphical interpretation and word problems.

二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b、c 为常数且 a ≠ 0。在 Edexcel IGCSE 数学中，求解二次方程是核心技能，在 Paper 1 和 Paper 2 中都会出现。本文将复习标准形式、三种代数方法、判别式、图像解释以及应用题。

1. The Standard Form | 标准形式

Every quadratic equation can be rearranged into the standard form: ax² + bx + c = 0. Here, a is the coefficient of x², b is the coefficient of x, and c is the constant term. For example, 3x² – 5x + 2 = 0 has a = 3, b = -5 and c = 2.

每个二次方程都可以整理成标准形式：ax² + bx + c = 0。其中 a 是 x² 的系数，b 是 x 的系数，c 是常数项。例如，3x² – 5x + 2 = 0 中 a = 3，b = -5，c = 2。

Some equations may need rearrangement before solving. For instance, 2x² = 8 – 4x must be rewritten as 2x² + 4x – 8 = 0. Always collect all terms on one side first.

有些方程在求解前需要先整理。例如，2x² = 8 – 4x 必须改写为 2x² + 4x – 8 = 0。务必先将所有项移到一边。

2. The Zero Product Property | 零乘积原理

The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. In symbols: if AB = 0, then A = 0 or B = 0.

零乘积原理指出：如果两个因式的乘积为零，那么至少有一个因式必须为零。用符号表示：若 AB = 0，则 A = 0 或 B = 0。

This property is the key to solving quadratic equations by factorisation. Once the quadratic expression is factorised into two linear factors, we can set each factor equal to zero and solve.

这个原理是因式分解法求解二次方程的关键。将二次表达式分解为两个一次因式后，我们可以令每个因式等于零再求解。

3. Solution by Factorisation | 因式分解法

Factorisation is often the fastest method when the quadratic has simple integer roots. To factorise x² + bx + c, find two numbers that multiply to give c and add to give b.

当二次方程有简单的整数根时，因式分解通常是最快的方法。要分解 x² + bx + c，需要找两个数，它们的乘积为 c，和为 b。

For example, solve x² – 5x + 6 = 0. Two numbers that multiply to 6 and add to -5 are -2 and -3. So (x – 2)(x – 3) = 0, giving x = 2 or x = 3.

例如，解 x² – 5x + 6 = 0。两个数乘积为 6，和为 -5，它们是 -2 和 -3。因此 (x – 2)(x – 3) = 0，得到 x = 2 或 x = 3。

For expressions with a leading coefficient a ≠ 1, use the method of splitting the middle term. Solve 2x² – 7x + 3 = 0. Multiply a and c: 2 × 3 = 6. Find factors of 6 that add to -7: -1 and -6. Rewrite: 2x² – x – 6x + 3 = 0. Group: x(2x – 1) – 3(2x – 1) = 0, so (2x – 1)(x – 3) = 0. Hence x = 1/2 or x = 3.

对于首项系数 a ≠ 1 的表达式，使用拆中项法。解 2x² – 7x + 3 = 0。将 a 和 c 相乘：2 × 3 = 6。找到 6 的两个因式，和为 -7：-1 和 -6。改写为：2x² – x – 6x + 3 = 0。分组：x(2x – 1) – 3(2x – 1) = 0，所以 (2x – 1)(x – 3) = 0。因此 x = 1/2 或 x = 3。

4. The Quadratic Formula | 求根公式

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

任何一个二次方程 ax² + bx + c = 0 都可以用求根公式求解：

x = (-b ± √(b² – 4ac)) / (2a)

The formula works for all quadratics, including those that cannot be factorised easily. For example, solve 2x² + 3x – 1 = 0. Here a = 2, b = 3, c = -1. Substitute into the formula: x = (-3 ± √(9 + 8)) / 4 = (-3 ± √17) / 4.

求根公式适用于所有二次方程，包括那些不容易因式分解的方程。例如，解 2x² + 3x – 1 = 0。这里 a = 2，b = 3，c = -1。代入公式：x = (-3 ± √(9 + 8)) / 4 = (-3 ± √17) / 4。

5. Completing the Square | 配方法

Completing the square transforms a quadratic into the form (x + p)² + q. This is useful for solving equations and for finding the turning point of a parabola.

配方法将二次式化为 (x + p)² + q 的形式。这可用于解方程，也可用于求抛物线的顶点。

To solve x² + 6x + 4 = 0, first move the constant term: x² + 6x = -4. Then add the square of half the coefficient of x, which is (6/2)² = 9, to both sides: x² + 6x + 9 = 5. This gives (x + 3)² = 5. Therefore x + 3 = ±√5, so x = -3 ± √5.

解 x² + 6x + 4 = 0，先将常数项移项：x² + 6x = -4。然后加上 x 系数一半的平方，即 (6/2)² = 9，两边同时加 9：x² + 6x + 9 = 5。得到 (x + 3)² = 5。因此 x + 3 = ±√5，所以 x = -3 ± √5。

For a quadratic with leading coefficient a, first factor out a before completing the square.

对于首项系数不为 1 的二次式，先提取 a 再配方。

6. The Discriminant | 判别式

The discriminant is the part of the quadratic formula under the square root: Δ = b² – 4ac. It tells us how many real roots a quadratic equation has.

判别式是求根公式中根号下的部分：Δ = b² – 4ac。它告诉我们二次方程有多少个实数根。

If Δ > 0, there are two distinct real roots. If Δ = 0, there is exactly one repeated real root. If Δ < 0, there are no real roots.

若 Δ > 0，方程有两个不同的实数根；若 Δ = 0，方程有一个重根；若 Δ < 0，方程没有实数根。

For example, the equation x² – 4x + 4 = 0 has Δ = 16 – 16 = 0, so it has one solution x = 2. The equation x² + x + 1 = 0 has Δ = 1 – 4 = -3, so it has no real solutions.

例如，方程 x² – 4x + 4 = 0 的 Δ = 16 – 16 = 0，所以它有一个解 x = 2。方程 x² + x + 1 = 0 的 Δ = 1 – 4 = -3，所以它没有实数解。

7. Graphical Interpretation | 图像解释

The graph of a quadratic function y = ax² + bx + c is a parabola. The x-intercepts of the graph are the roots of the equation ax² + bx + c = 0.

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

更多咨询请联系16621398022（同微信）

Solving Quadratic Equations | 解二次方程