Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations appear frequently in IGCSE Mathematics, from simple factorisation to the quadratic formula and word problems. Mastering this topic is essential for higher-level algebra and coordinate geometry.

二次方程在IGCSE数学中频繁出现，从简单的因式分解到求根公式和实际问题。掌握这一主题对更高层次的代数和坐标几何至关重要。

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 form ax² + bx + c = 0, where a, b and c are constants, and a ≠ 0. The highest power of the variable is 2.

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

For example, 2x² − 5x + 3 = 0 is quadratic, while x³ − 2x = 0 is not.

例如，2x² − 5x + 3 = 0 是二次方程，而 x³ − 2x = 0 不是。

2. Solving by Factorisation | 因式分解法

The simplest method is to factorise the quadratic into two linear factors. If ax² + bx + c = (px + q)(rx + s) = 0, then either px + q = 0 or rx + s = 0.

最简单的方法是将二次式分解为两个一次因式。如果 ax² + bx + c = (px + q)(rx + s) = 0，那么要么 px + q = 0，要么 rx + s = 0。

Example: Solve x² − 7x + 10 = 0.

例：解 x² − 7x + 10 = 0。

(x − 2)(x − 5) = 0 → x = 2 or x = 5

Check: 2² − 7(2) + 10 = 4 − 14 + 10 = 0; 5² − 7(5) + 10 = 25 − 35 + 10 = 0.

检验：2² − 7(2) + 10 = 4 − 14 + 10 = 0；5² − 7(5) + 10 = 25 − 35 + 10 = 0。

3. Special Case: Difference of Two Squares | 特例：平方差公式

When a quadratic has the form x² − k² = 0, it factorises as (x − k)(x + k) = 0. This is called the difference of two squares.

当二次方程具有 x² − k² = 0 的形式时，可以分解为 (x − k)(x + k) = 0。这称为平方差公式。

Example: Solve 4x² − 9 = 0.

例：解 4x² − 9 = 0。

(2x − 3)(2x + 3) = 0 → x = 3/2 or x = −3/2

Always look for a common factor first. For instance, 3x² − 12 = 3(x² − 4) = 3(x − 2)(x + 2).

始终先寻找公因式。例如，3x² − 12 = 3(x² − 4) = 3(x − 2)(x + 2)。

4. The Quadratic Formula | 求根公式

If factorisation is difficult or impossible, use the quadratic formula:

如果因式分解困难或无法进行，使用求根公式：

x = (−b ± √(b² − 4ac)) / (2a)

This formula works for every quadratic equation ax² + bx + c = 0. The symbol √ represents the positive square root.

这个公式适用于所有二次方程 ax² + bx + c = 0。符号 √ 表示正平方根。

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

例：利用公式解 2x² + 3x − 2 = 0。这里 a = 2，b = 3，c = −2。

x = (−3 ± √(9 + 16)) / 4 = (−3 ± 5) / 4

So x = (−3 + 5)/4 = 1/2, or x = (−3 − 5)/4 = −2.

因此 x = (−3 + 5)/4 = 1/2，或 x = (−3 − 5)/4 = −2。

5. Completing the Square | 配方法

Completing the square rewrites x² + bx + c as (x + p)² + q. For x² + bx, add and subtract (b/2)².

配方法将 x² + bx + c 改写为 (x + p)² + q。对于 x² + bx，加上并减去 (b/2)²。

Example: Solve x² + 6x + 1 = 0 by completing the square.

例：用配方法解 x² + 6x + 1 = 0。

(x + 3)² − 9 + 1 = 0 → (x + 3)² = 8

x + 3 = ±√8 → x = −3 ± 2√2

This method is also useful for finding the vertex of a parabola or solving equations when the coefficient of x² is 1.

这种方法也适用于求抛物线的顶点，或在 x² 的系数为1时解方程。

6. The Discriminant | 判别式

The expression b² − 4ac is called the discriminant. It tells us the number of real roots without solving the equation.

表达式 b² − 4ac 称为判别式。它无需解方程即可告诉我们实数根的个数。

If b² − 4ac > 0, there are two distinct real roots.

如果 b² − 4ac > 0，方程有两个不同的实数根。
If b² − 4ac = 0, there is exactly one repeated real root.

如果 b² − 4ac = 0，方程有一个重根。
If b² − 4ac < 0, there are no real roots.

如果 b² − 4ac < 0，方程没有实数根。

Example: For x² − 2x + 5 = 0, discriminant = (−2)² − 4(1)(5) = 4 − 20 = −16, so no real roots.

例：对于 x² − 2x + 5 = 0，判别式 = (−2)² − 4(1)(5) = 4 − 20 = −16，因此没有实数根。

7. Solving Quadratic Equations by Factorising with a > 1 | 当二次项系数大于1时的因式分解

When a > 1, factorisation requires more care. For example, solve 3x² + 5x − 2 = 0.

当 a > 1 时，因式分解需要更加小心。例如，解 3x² + 5x − 2 = 0。

Look for factors of 3×(−2) = −6 that add to the coefficient of x, which is 5. Two such numbers are 6 and −1.

寻找3×(−2) = −6的因子，使其和为 x 的系数5。这两个数是6和−1。

3x² + 6x − x − 2 = 3x(x + 2) − 1(x + 2) = (3x − 1)(x + 2)

Thus (3x − 1)(x + 2) = 0, giving x = 1/3 or x = −2.

因此 (3x − 1)(x + 2) = 0，得到 x = 1/3 或 x = −2。

8. Solving Word Problems with Quadratics | 二次方程应用题

Many real-world problems lead to quadratic equations. Common contexts include area, projectile motion, and consecutive integers.

许多现实问题会引出二次方程。常见情境包括面积、抛体运动以及连续整数。

Example: The area of a rectangle is 48 cm². Its length is 2 cm longer than its width. Find the width.

例：一个矩形的面积为48 cm²。它的长比宽长2 cm。求宽。

Let width = w, then length = w + 2. The equation is w(w + 2) = 48.

设宽为 w，则长为 w + 2。方程为 w(w + 2) = 48。

w² + 2w − 48 = 0 → (w + 8)(w − 6) = 0

Since width cannot be negative, w = 6 cm. Always reject negative solutions when they do not make sense in context.

由于宽不能为负数，所以 w = 6 cm。当负数解在情境中没有意义时，务必舍去。

9. Graphical Interpretation | 图像意义

The roots of ax² + bx + c = 0 correspond to the x-coordinates where the parabola y = ax² + bx + c crosses the x-axis.

ax² + bx + c = 0 的根对应于抛物线 y = ax² + bx + c 与 x 轴交点的 x 坐标。

If the parabola does not cross the x-axis, the equation has no real roots. If it just touches the x-axis, there is one repeated root.

如果抛物线不与 x 轴相交，则方程没有实数根。如果它恰好与 x 轴相切，则有一个重根。

Discriminant	Number of real roots	Graph shape
b² − 4ac > 0	Two	Cuts x-axis at two points
b² − 4ac = 0	One	Touches x-axis at one point
b² − 4ac < 0	None	Does not reach x-axis

10. Common Mistakes to Avoid | 常见错误提醒

Forgetting that a cannot be zero in a quadratic equation.

忘记二次方程中 a 不能为零。
When using the quadratic formula, sign errors with negative b or c.

使用求根公式时，对负的 b 或 c 出现符号错误。
Missing one solution when taking square roots: x² = 9 gives x = ±3.

开平方时漏掉一个解：x² = 9 给出 x = ±3。
Incorrectly factorising when the constant term is negative.

当常数项为负时，因式分解出错。

Practice each method carefully. In an exam, choose the fastest reliable method: factorise when possible, otherwise use the formula.

请仔细练习每种方法。在考试中，选择最快且可靠的方法：能因式分解时使用因式分解，否则使用求根公式。

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Solving Quadratic Equations | 解二次方程