📚 Solving Quadratic Equations | 解二次方程
. Solving such equations is a core skill in IGCSE Mathematics, as it appears in algebra, graphs, and word problems.
二次方程是次数为2的多项式方程，通常写作 ax² + bx + c = 0，其中 a ≠ 0。解这类方程是 IGCSE 数学的核心技能，在代数、图像和应用题中都会出现。
1. Understanding Quadratic Equations | 理解二次方程
A quadratic equation has the standard form ax² + bx + c = 0. The values of a, b, and c are constants, and x is the unknown. The highest power of x is 2, which gives the equation its name.
二次方程的标准形式为 ax² + bx + c = 0。其中 a、b、c 是常数，x 是未知数。x 的最高次数为2，因此得名“二次”。
For example, 2x² – 5x + 3 = 0 is a quadratic equation. Here, a = 2, b = –5, and c = 3. Not every equation with x² is quadratic; the coefficient a must not be zero.
例如，2x² – 5x + 3 = 0 是一个二次方程。这里 a = 2，b = –5，c = 3。并非所有含 x² 的方程都是二次方程，系数 a 不能为零。
A quadratic equation may have two real roots, one repeated root, or no real roots.
二次方程可能有两个实根、一个重根，或没有实数根。
The roots are the values of x that satisfy the equation.
根是满足方程的 x 值。
2. Solving by Factorisation | 因式分解法
Factorisation is the simplest method when the quadratic expression can be written as a product of two linear factors. The principle is: if the product of two expressions is zero, then at least one of them must be zero.
当二次表达式可以写成两个一次因式的乘积时，因式分解是最简单的方法。其原理是：若两个表达式的乘积为零，则至少其中一个必须为零。
For example, solve x² – 5x + 6 = 0. Factorise to get (x – 2)(x – 3) = 0. Then x – 2 = 0 or x – 3 = 0, so x = 2 or x = 3.
例如，解 x² – 5x + 6 = 0。因式分解得 (x – 2)(x – 3) = 0。于是 x – 2 = 0 或 x – 3 = 0，所以 x = 2 或 x = 3。
Always rearrange the equation to the form ax² + bx + c = 0 before factorising.
因式分解前，务必先将方程整理成 ax² + bx + c = 0 的形式。
Check your answers by substituting them back into the original equation.
将答案代回原方程进行检验。
3. Solving by the Quadratic Formula | 公式法
For any quadratic equation ax² + bx + c = 0, the roots are given by the quadratic formula:
x = (–b ± √b² – 4ac) / (2a)
This method works for all quadratics, even those that cannot be factorised easily. You must substitute the values of a, b, and c carefully.
这一方法适用于所有二次方程，包括那些不易因式分解的方程。代入 a、b、c 的值时必须仔细。
For example, solve 2x² + 3x – 2 = 0. Here a = 2, b = 3, c = –2. Substitute into the formula:
x = (–3 ± √(3² – 4×2×(–2))) / (2×2) = (–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。
4. Completing the Square | 配方法
Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This is useful for solving equations and for finding the vertex of a parabola.
配方法将 ax² + bx + c 改写为 a(x + p)² + q 的形式。这既可用于解方程，也可用于求抛物线的顶点。
or example, solve x² + 6x + 1 = 0 by completing the square. First write x² + 6x = (x + 3)² – 9. Then (x + 3)² – 9 + 1 = 0, so (x + 3)² = 8. Hence x + 3 = ±√8, so x = –3 ± 2√2.
例如，用配方法解 x² + 6x + 1 = 0。先将 x² + 6x 写成 (x + 3)² – 9。于是 (x + 3)² – 9 + 1 = 0，即 (x + 3)² = 8。因此 x + 3 = ±√8，所以 x = –3 ± 2√2。
The completed square form directly gives the turning point (–p, q).
配方后的形式直接给出转向点 (–p, q)。
This method is especially helpful when solving inequalities or sketching graphs.
这一方法在解不等式或画图时尤为有用。
5. The Discriminant | 判别式
The discriminant is the part of the formula under the square root: Δ = b² – 4ac. It determines the nature of the roots without solving the full equation.
判别式是公式中根号内的部分：Δ = b² – 4ac。它无需解完整方程即可判断根的性质。
If Δ > 0, there are two distinct real roots.
若 Δ > 0，则有两个不相等的实数根。
If Δ = 0, there is one repeated real root (or two equal real roots).
若 Δ = 0，则有一个重根（或两个相等的实数根）。
If Δ < 0, there are no real roots; the roots are complex.
若 Δ < 0，则没有实数根；根为复数。
For example, for x² – 4x + 4 = 0, Δ = (–4)² – 4×1×4 = 0, so there is one repeated root x = 2.
例如，对 x² – 4x + 4 = 0，Δ = (–4)² – 4×1×4 = 0，因此有一个重根 x = 2。
6. Quadratic Graphs | 二次函数图像
y = ax² + bx + c is a parabola. If a > 0, it opens upwards; if a < 0, it opens downwards.
二次函数 y = ax² + bx + c 的图像是一条抛物线。若 a > 0，开口向上；若 a < 0，开口向下。
The x-intercepts of the graph are the roots of the equation ax² + bx + c = 0.
图像与 x 轴的交点就是方程 ax² + bx + c = 0 的根。
The y-intercept is at (0, c).
与 y 轴的交点为 (0, c)。
The axis of symmetry is the vertical line x = –b/(2a).
对称轴是竖直线 x = –b/(2a)。
7. Finding the Vertex | 求顶点
tex (or turning point) of the parabola can be found from the completed square form y = a(x – h)² + k, where the vertex is (h, k). Alternatively, use the formula h = –b/(2a), then k = f(h).
抛物线的顶点可以通过配方形式 y = a(x – h)² + k 求得，其中顶点为 (h, k)。或者使用公式 h = –b/(2a)，再计算 k = f(h)。
For example, for y = x² – 4x + 1, h = –(–4)/(2×1) = 2, and k = 4 – 8 + 1 = –3. So the vertex is (2, –3).
例如，对 y = x² – 4x + 1，h = –(–4)/(2×1) = 2，k = 4 – 8 + 1 = –3。因此顶点为 (2, –3)。
8. Applications in Real Life | 实际应用
Quadratic equations appear in physics, geometry, and economics. Examples include projectile motion, area problems, and profit maximisation.
二次方程出现在物理、几何和经济学中。例如抛体运动、面积问题以及利润最大化问题。
Projectile motion: the height of an object is often h = –gt² + vt + s.
抛体运动：物体的高度常为 h = –gt² + vt + s。
Area problems: a rectangular field with a fixed perimeter may have area A = x(L – 2x), which is quadratic.
面积问题：固定周长的矩形场地，其面积 A = x(L – 2x)，属于二次函数。
Profit: P(x) = –x² + bx + c models profit against selling price.
利润：P(x) = –x² + bx + c 可用来表示利润与售价的关系。
9. Common Mistakes | 常见错误
Students often make errors when solving quadratics. Here are some pitfalls to avoid.
学生在解二次方程时常犯错误。以下是一些需要避免的陷阱。
Forgetting to rearrange to standard form before factorising.
因式分解前忘记整理成标准形式。
Misapplying the quadratic formula – ensure 4ac is subtracted after b².
套用公式时出错——务必先算 b²，再减去 4ac。
When taking the square root, forget the ± sign.
取平方根时漏掉 ± 号。
Confusing the vertex x-coordinate with the roots.
将顶点的 x 坐标与方程的根混淆。
10. Practice Questions | 练习
Try these IGCSE-style questions to check your understanding.
请尝试以下 IGCSE 风格的问题，检验你的理解。
| Question | Method |
| Solve x² – 7x + 10 = 0 | Factorise → (x – 2)(x – 5) = 0 |
| Solve 2x² + 5x – 3 = 0 | Use quadratic formula |
| Find the vertex of y = x² + 2x – 3 | Complete the square → (x + 1)² – 4 |
| Determine nature of roots for x² + 2x + 5 = 0 | Δ = 4 – 20 = –16 → no real roots |
Answers: 1) x = 2 or x = 5; 2) x = 1/2 or x = –3; 3) Vertex (–1, –4); 4) No real roots.
答案：1) x = 2 或 x = 5；2) x = 1/2 或 x = –3；3) 顶点 (–1, –4)；4) 无实数根。
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