📚 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
+ 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。
ertex 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
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