📚 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
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
ways 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)
ery 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。
ple: 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。
f 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
gative, 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
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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