📚 Solving Quadratic Equations | 解二次方程
Quadratic equations are one of the most frequently tested topics in IGCSE Mathematics. This revision guide covers everything you need: standard form, factorisation, the quadratic formula, completing the square, the discriminant, graphs, and common pitfalls — all with clear worked examples.
二次方程是 IGCSE 数学中考频最高的知识点之一。本复习指南涵盖你所需的一切：标准形式、因式分解法、求根公式、配方法、判别式、图像以及常见易错点，并配有清晰的例题讲解。
1. What Is a Quadratic Equation | 什么是二次方程
A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2. Its general form is:
二次方程是次数为 2 的多项式方程，即变量的最高次幂为 2。它的一般形式为：
ax² + bx + c = 0, 其中 a ≠ 0
ants, and x is the unknown variable. If a = 0, the equation becomes linear, not quadratic.
其中 a、b、c 为常数，x 为未知数。如果 a = 0，方程退化为一次方程，不再是二次方程。
- Examples: x² − 5x + 6 = 0; 2x² + 3x − 1 = 0; x² = 9.
- Non-examples: x + 2 = 0 (linear); x³ − 1 = 0 (cubic).
- 示例：x² − 5x + 6 = 0；2x² + 3x − 1 = 0；x² = 9。
- 反例：x + 2 = 0（一次方程）；x³ − 1 = 0（三次方程）。
2. Standard Form and Rearranging | 标准形式与整理方程
the standard form ax² + bx + c = 0. All terms must be on one side, with zero on the other.
在求解之前，始终将方程整理成标准形式 ax² + bx + c = 0。所有项移到一边，另一边为零。
Worked Example: Solve x² = 3x + 4.
例题：解方程 x² = 3x + 4。
Step 1: Bring all terms to the left: x² − 3x − 4 = 0.
第一步：将所有项移到左边：x² − 3x − 4 = 0。
Step 2: Now the equation is in standard form with a = 1, b = −3, c = −4.
第二步：现在方程成为标准形式，其中 a = 1，b = −3，c = −4。
Always check that the signs are correct after rearranging. A common error is forgetting to change the sign when moving terms across the equals sign.
整理后务必检查符号是否正确。一个常见错误是在移项时忘记变号。
3. Solving by Factorisation | 因式分解法
Factorisation is the fastest method when the quadratic has simple integer factors. The principle is: if the product of two expressions is zero, then at least one of them must be zero.
当二次式具有简单的整数因式时，因式分解是最快捷的方法。其原理是：若两个表达式的乘积为零，则其中至少有一个必为零。
若 p × q = 0，则 p = 0 或 q = 0
Worked Example: Solve x² − 5x + 6 = 0.
例题：解方程 x² − 5x + 6 = 0。
Step 1: Find two numbers whose product is +6 and whose sum is −5. Those numbers are −2 and −3.
第一步：找两个数，其乘积为 +6，其和为 −5。这两个数是 −2 和 −3。
Step 2: Write in factored form: (x − 2)(x − 3) = 0.
第二步：写成因式形式：(x − 2)(x − 3) = 0。
Step 3: Set each factor to zero: x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.
第三步：令每个因式为零：x − 2 = 0 或 x − 3 = 0，解得 x = 2 或 x = 3。
| Type | Factored Form | Example |
| Difference of squares | (x − p)(x + p) | x² − 9 = (x − 3)(x + 3) |
| Perfect square | (x ± p)² | x² + 6x + 9 = (x + 3)² |
| General trinomial | (mx + p)(nx + q) | 2x² + 5x + 2 = (2x + 1)(x + 2) |
If the coefficient of x² is not 1, use trial and error or the "cross method" to find the factors.
如果 x² 的系数不为 1，可用试错法或十字相乘法来寻找因式。
4. The Quadratic Formula | 求根公式法
dratic equation, even when factorisation is difficult or impossible. For ax² + bx + c = 0:
求根公式适用于所有二次方程，即使无法因式分解也能求解。对于 ax² + bx + c = 0：
x = (−b ± √b² − 4ac) / 2a
Worked Example: Solve 2x² + 3x − 5 = 0.
例题：解方程 2x² + 3x − 5 = 0。
Step 1: Identify a = 2, b = 3, c = −5.
第一步：确定 a = 2，b = 3，c = −5。
Step 2: Substitute into the formula:
第二步：代入公式：
x = (−3 ± √(3² − 4 × 2 × (−5))) / (2 × 2)
Step 3: Simplify: x = (−3 ± √9 + 40) / 4 = (−3 ± √49) / 4 = (−3 ± 7) / 4.
第三步：化简：x = (−3 ± √9 + 40) / 4 = (−3 ± √49) / 4 = (−3 ± 7) / 4。
Step 4: So x = 4/4 = 1, or x = −10/4 = −2.5.
第四步：因此 x = 4/4 = 1，或 x = −10/4 = −2.5。
Always write the formula down first, then substitute carefully. A common mistake is mis-stating the sign of c when substituting.
务必先写出公式，再仔细代入。常见错误是代入时弄错 c 的符号。
5. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form (x + p)² + q. This is especially useful for finding turning points and solving equations without using the formula.
配方法将二次式改写为 (x + p)² + q 的形式。它在求顶点坐标和不解方程的情况下尤为有用。
Worked Example: Solve x² + 6x + 2 = 0 by completing the square.
例题：用配方法解方程 x² + 6x + 2 = 0。
Step 1: Halve the coefficient of x: 6 ÷ 2 = 3. Write (x + 3)².
第一步：将 x 的系数除以 2：6 ÷ 2 = 3。写出 (x + 3)²。
Expand (x + 3)² = x² + 6x + 9. This is 7 more than the original quadratic, so subtract 7:
第二步：展开 (x + 3)² = x² + 6x + 9。这比原式多了 7，因此减去 7：
(x + 3)² − 7 = 0
Step 3: Solve: (x + 3)² = 7, so x + 3 = ±√7, giving x = −3 ± √7.
第三步：求解：(x + 3)² = 7，所以 x + 3 = ±√7，得 x = −3 ± √7。
For quadratics where a ≠ 1, first divide the whole equation by a before completing the square.
当 a ≠ 1 时，先将整个方程除以 a，再进行配方。
6. The Discriminant | 判别式
The discriminant is the expression b² − 4ac inside the square root of the quadratic formula. It tells us the nature of the roots without solving the equation.
判别式是求根公式根号内的表达式 b² − 4ac。它无需解方程即可判断根的性质。
| Value of b² − 4ac | Nature of Roots | Graph |
| b² − 4ac > 0 | Two distinct real roots | Crosses the x-axis twice |
| b² − 4ac = 0 | One repeated real root | Touches the x-axis once |
| b² − 4ac < 0 | No real roots | Does not intersect the x-axis |
| b² − 4ac 的值 | 根的性质 | 图像 |
| b² − 4ac > 0 | 两个不等实根 | 与 x 轴交于两点 |
| b² − 4ac = 0 | 一个重根 | 与 x 轴相切于一点 |
| b² − 4ac < 0 | 无实根 | 与 x 轴无交点 |
Worked Example: Determine the nature of the roots of x² + 4x + 5 = 0.
例题：判断方程 x² + 4x + 5 = 0 的根的性质。
scriminant is 4² − 4 × 1 × 5 = 16 − 20 = −4. Since −4 < 0, the equation has no real roots.
这里 a = 1，b = 4，c = 5。判别式为 4² − 4 × 1 × 5 = 16 − 20 = −4。因为 −4 < 0，所以方程无实根。
7. Sum and Product of Roots | 根的和与积
quadratic equation ax² + bx + c = 0 with roots α and β, the following relationships hold:
对于根为 α 和 β 的二次方程 ax² + bx + c = 0，有以下关系：
α + β = −b/a
α × β = c/a
Worked Example: Find the sum and product of the roots of 2x² + 8x + 3 = 0.
例题：求方程 2x² + 8x + 3 = 0 的根的和与积。
Sum = −8/2 = −4. Product = 3/2.
和 = −8/2 = −4。积 = 3/2。
This relationship is useful when checking answers or when constructing a quadratic equation from given roots.
这个关系可用于检验答案，也可用于根据已知根构造二次方程。
8. Quadratic Graphs | 二次函数图像
x + c is a parabola. Its shape and position depend on the signs of a and the discriminant.
y = ax² + bx + c 的图像是一条抛物线。其形状和位置取决于 a 的符号和判别式。
- If a > 0: the parabola opens upwards (U shape) and has a minimum point.
- If a < 0: the parabola opens downwards (∩ shape) and has a maximum point.
- 若 a > 0：抛物线开口向上（U 形），存在最小值点。
- 若 a < 0：抛物线开口向下（∩ 形），存在最大值点。
can be found by completing the square. For y = (x + p)² + q, the turning point is (−p, q).
顶点坐标可通过配方法求得。对于 y = (x + p)² + q，顶点坐标为 (−p, q)。
Worked Example: Find the turning point of y = x² − 6x + 5.
例题：求 y = x² − 6x + 5 的顶点坐标。
5 = (x − 3)² − 4. The turning point is (3, −4). Since a = 1 > 0, this is a minimum point.
配方：y = (x − 3)² − 9 + 5 = (x − 3)² − 4。顶点坐标为 (3, −4)。因为 a = 1 > 0，这是最小值点。
9. Word Problems | 应用题
Quadratic equations often appear in geometry and number problems. The key steps are: define the variable, form the equation, solve it, and check the validity of answers.
二次方程常出现在几何和数字问题中。关键步骤为：设未知数、列方程、解方程、检验答案的合理性。
Worked Example: A rectangle has length 4 cm longer than its width. Its area is 96 cm². Find the dimensions.
例题：一个长方形的长比宽长 4 cm，面积为 96 cm²。求其尺寸。
Step 1: Let the width be x cm. Then the length is (x + 4) cm.
第一步：设宽为 x cm，则长为 (x + 4) cm。
Step 2: Area = x(x + 4) = 96, so x² + 4x − 96 = 0.
第二步：面积 = x(x + 4) = 96，即 x² + 4x − 96 = 0。
Step 3: Factorise: (x + 12)(x − 8) = 0, giving x = −12 or x = 8.
第三步：因式分解：(x + 12)(x − 8) = 0，得 x = −12 或 x = 8。
p 4: Reject x = −12 because length cannot be negative. So width = 8 cm and length = 12 cm.
第四步：舍去 x = −12，因为长度不能为负。所以宽 = 8 cm，长 = 12 cm。
Always check whether each solution makes sense in the context of the problem.
务必检查每个解是否符合实际问题的情境。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Here are the most frequent errors students make in the exam, and how to avoid them.
以下是学生在考试中最常见的错误以及如何避免它们。
- Mistake 1: Forgetting to write the equation in standard form before factorising. Fix: Always rearrange to ax² + bx + c = 0 first.
- Mistake 2: Sign errors when substituting into the quadratic formula. Fix: Use brackets around negative numbers.
- Mistake 3: Dividing by a variable that may be zero. Fix: Never divide both sides by x unless x ≠ 0 is guaranteed.
- Mistake 4: Losing one solution. Fix: Every quadratic has up to two solutions; always write both.
- 错误一：因式分解前忘记整理成标准形式。对策：先移项整理为 ax² + bx + c = 0。
- 错误二：代入求根公式时符号出错。对策：负数代入时加括号。
- 错误三：除以可能为零的变量。对策：除非确定 x ≠ 0，否则不要两边同时除以 x。
- 错误四：漏掉一个解。对策：每个二次方程最多有两个解，务必都写出来。
In the exam, if factorisation does not work within a few seconds, switch to the quadratic formula. For multiple-choice questions, check answers by substituting back into the original equation.
考试中，如果因式分解几秒内无头绪，立即改用求根公式。对于选择题，可将答案代回原方程进行验证。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图