Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. It is one of the most important topics in IGCSE Mathematics.

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。这是 IGCSE 数学中最重要的主题之一。

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

Before solving, you must rewrite the equation so that one side is zero and the other side is arranged in descending powers of x: ax² + bx + c = 0.

在求解之前，必须将方程改写为一侧为零、另一侧按 x 的降幂排列：ax² + bx + c = 0。

Example: x² – 3x = 4 must be rewritten as x² – 3x – 4 = 0.
例如：x² – 3x = 4 必须改写为 x² – 3x – 4 = 0。
If a = 0, the equation becomes linear, not quadratic.
如果 a = 0，方程变成一次方程，而不是二次方程。

2. Solving by Factorisation | 因式分解法

When the quadratic expression can be factorised into two linear factors, set each factor equal to zero and solve.

当二次表达式可以分解为两个一次因式时，令每个因式等于零并求解。

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

Always check your solutions by substituting back into the original equation.

务必通过代入原方程来检验你的解。

3. 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不可能，则使用求根公式：

x = (-b ± √(b² – 4ac)) / (2a)

This formula works for every quadratic equation with real or complex solutions.

该公式适用于所有有实数解或复数解的二次方程。

Substitute the values of a, b and c carefully, paying attention to negative signs.

代入 a、b、c 的值时要格外小心，注意负号。

4. Completing the Square | 配方法

Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This is especially useful for finding turning points.

配方法将 ax² + bx + c 改写为 a(x + p)² + q 的形式。这在求顶点时特别有用。

x² + 6x + 5 = (x + 3)² – 9 + 5 = (x + 3)² – 4

Then solve (x + 3)² – 4 = 0 to get x = -1 or x = -5.

然后解 (x + 3)² – 4 = 0，得到 x = -1 或 x = -5。

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

The discriminant is Δ = b² – 4ac. Its sign tells us how many real roots the equation has.

判别式为 Δ = b² – 4ac。它的符号告诉我们方程有多少个实数根。

Δ = b² – 4ac	Nature of roots	根的性质
Δ > 0	Two distinct real roots	两个不相等的实数根
Δ = 0	One repeated real root	一个重根（两个相等的实数根）
Δ < 0	No real roots	没有实数根

6. Solving Quadratic Equations by Factorising Special Cases | 特殊情况下的因式分解求解

Recognise difference of squares: a² – b² = (a – b)(a + b).

识别平方差公式：a² – b² = (a – b)(a + b)。

x² – 9 = 0 → (x – 3)(x + 3) = 0 → x = 3 or x = -3

Perfect squares: x² + 2ab + b² = (x + b)², for example x² + 6x + 9 = (x + 3)².

完全平方式：x² + 2ab + b² = (x + b)²，例如 x² + 6x + 9 = (x + 3)²。

7. Graphical Interpretation | 图像解释

The solutions of ax² + bx + c = 0 are the x-coordinates where the parabola y = ax² + bx + c crosses the x-axis.

方程 ax² + bx + c = 0 的解就是抛物线 y = ax² + bx + c 与 x 轴交点的横坐标。

If the graph touches the x-axis at one point, there is one repeated root. If it never touches the x-axis, there are no real roots.

如果图像与 x 轴只有一个交点，则有一个重根。如果图像不与 x 轴相交，则没有实数根。

The vertex of the parabola can be found using completing the square or the formula x = -b/(2a).

抛物线的顶点可以通过配方法或公式 x = -b/(2a) 求得。

8. Applications to Word Problems | 应用题中的二次方程

Many real-world problems, such as areas, projectile motion and number puzzles, lead to quadratic equations.

许多现实问题，如面积、抛体运动和数字谜题，都会归结为二次方程。

Example: The length of a rectangle is 3 cm longer than its width, and its area is 70 cm². Find the width.

例：一个长方形的长比宽长 3 cm，面积为 70 cm²。求宽。

Let width = x, then length = x + 3 → x(x + 3) = 70 → x² + 3x – 70 = 0

Factorise: (x + 10)(x – 7) = 0. Since width cannot be negative, x = 7 cm.

因式分解：(x + 10)(x – 7) = 0。由于宽不能为负值，所以 x = 7 cm。

9. Common Mistakes and How to Avoid Them | 常见错误及避免方法

Forgetting to rearrange the equation into standard form before factorising or using the formula.
忘记在因式分解或使用公式前将方程整理为标准形式。
Misapplying the quadratic formula by using b instead of -b, or forgetting the ± sign.
在使用求根公式时误用 b 而非 -b，或忘记 ± 号。
Dividing both sides by x when x = 0 is a possible solution, which loses roots.
当 x = 0 是可能解时，两边除以 x，导致丢根。
Ignoring negative solutions when the context requires a positive answer.
当实际问题要求正数答案时，忽略负数解。

Always check your answers by substitution, and state the units or context when required.

始终通过代入检验答案，并在需要时注明单位或实际含义。

10. Practice Questions | 练习题

Try these problems to test your understanding:

尝试以下问题来测试你的理解：

Solve x² – 7x + 12 = 0.
解方程 x² – 7x + 12 = 0。
Use the quadratic formula to solve 2x² + 3x – 5 = 0.
使用求根公式解 2x² + 3x – 5 = 0。
Find the value of k for which x² + kx + 9 = 0 has exactly one root.
求 k 的值，使 x² + kx + 9 = 0 只有一个根。
A square has its side increased by 4 cm to form a new square with area 81 cm². Find the original side length.
一个正方形的边长增加 4 cm 后，新正方形的面积为 81 cm²。求原边长。

Answers: 1. x = 3 or 4 ; 2. x = 1 or -2.5 ; 3. k = ±6 ; 4. 5 cm

答案：1. x = 3 或 4 ；2. x = 1 或 -2.5 ；3. k = ±6 ；4. 5 cm

Published by TutorHao | 数学 Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Solving Quadratic Equations | 解二次方程