📚 Solving Quadratic Equations | 解二次方程
A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in many exam questions, from factorisation to graph sketching and word problems. This article will guide you through the key methods of solving quadratic equations, the discriminant, and common pitfalls, step by step.
二次方程是 IGCSE 数学中最重要的主题之一。它出现在许多考试题中，从因式分解到画图，再到应用题。本文将一步步带你掌握解二次方程的常用方法、判别式以及常见易错点。
1. What Is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is an equation that can be written in the general form:
二次方程是可以写成一般形式的方程：
ax² + bx + c = 0, where a ≠ 0
Here x is the unknown, and a, b, c are numbers. The highest power of x is 2, so the graph is a curve called a parabola.
其中 x 是未知数，a、b、c 是常数。x 的最高次数是 2，因此图像是一条称为抛物线的曲线。
3x + 1 = 0 is a quadratic equation, while 3x - 1 = 0 is linear, and x³ + x² = 0 is cubic.
例如，2x² - 3x + 1 = 0 是二次方程，而 3x - 1 = 0 是一次方程，x³ + x² = 0 是三次方程。
2. Expanding and Factorising Quadratics | 展开与因式分解二次式
Before solving, you need to be confident with factorising. Factorising is the reverse of expanding brackets.
在解题之前，你需要熟练因式分解。因式分解是展开括号的逆过程。
For an expression like (x + 2)(x + 3), expanding gives:
对于像 (x + 2)(x + 3) 这样的表达式，展开得到：
(x + 2)(x + 3) = x² + 5x + 6
To factorise x² + 5x + 6, we look for two numbers that multiply to 6 and add to 5. Those numbers are 2 and 3, so:
要因式分解 x² + 5x + 6，我们要找两个数，它们相乘等于 6，相加等于 5。这两个数是 2 和 3，所以：
x² + 5x + 6 = (x + 2)(x + 3)
the process can involve more careful pairing. For example, 2x² + 7x + 3 = (2x + 1)(x + 3).
当 x² 的系数不是 1 时，配对过程需要更仔细。例如，2x² + 7x + 3 = (2x + 1)(x + 3)。
3. Solving by Factorising | 因式分解法解方程
n be factorised, you can use the zero product property: if A × B = 0, then A = 0 or B = 0.
如果一个二次方程可以因式分解，你可以利用零积性质：如果 A × B = 0，那么 A = 0 或 B = 0。
Solve x² - 5x + 6 = 0. First factorise:
解方程 x² - 5x + 6 = 0。首先因式分解：
(x - 2)(x - 3) = 0
Then set each bracket to zero:
然后令每个括号等于零：
x - 2 = 0 ⇒ x = 2
x - 3 = 0 ⇒ x = 3
lutions are x = 2 and x = 3. Always check by substituting back into the original equation.
所以解是 x = 2 和 x = 3。一定要代回原方程检验。
h a repeated root, such as x² - 4x + 4 = 0, factorising gives (x - 2)² = 0, so x = 2 only.
对于有重根的二次方程，如 x² - 4x + 4 = 0，因式分解得到 (x - 2)² = 0，所以只有一个解 x = 2。
4. Solving by Using the Quadratic Formula | 公式法解方程
or impossible to factorise. In this case, use the quadratic formula. For ax² + bx + c = 0:
有些二次方程很难因式分解甚至无法因式分解。此时可以使用求根公式。对于 ax² + bx + c = 0：
x = (-b ± √b² - 4ac) / (2a)
Solve 2x² - 3x - 2 = 0 using the formula. Here a = 2, b = -3, c = -2.
用公式法解 2x² - 3x - 2 = 0。这里 a = 2, b = -3, c = -2。
First calculate the discriminant inside the square root:
先计算根号内的判别式：
b² - 4ac = (-3)² - 4×2×(-2) = 9 + 16 = 25
Then substitute:
然后代入：
x = (3 ± √25) / 4 = (3 ± 5) / 4
So x = (3 + 5)/4 = 2, or x = (3 - 5)/4 = -1/2.
所以 x = (3 + 5)/4 = 2，或 x = (3 - 5)/4 = -1/2。
5. Solving by Completing the Square | 配方法解方程
Completing the square rewrites a quadratic in the form (x + p)² + q. This is especially useful for finding the turning point of a parabola.
配方法将二次式改写为 (x + p)² + q 的形式。这对于求抛物线的顶点尤其有用。
Solve x² + 6x + 8 = 0 by completing the square.
用配方法解 x² + 6x + 8 = 0。
Take half of 6, which is 3, and write (x + 3)². Expanding (x + 3)² gives x² + 6x + 9. To keep the original expression, adjust by subtracting 9 and adding 8:
取 6 的一半，即 3，写出 (x + 3)²。展开 (x + 3)² 得到 x² + 6x + 9。为了保持原表达式不变，减去 9 再加上 8：
x² + 6x + 8 = (x + 3)² - 1
Now solve:
现在解方程：
(x + 3)² - 1 = 0
(x + 3)² = 1
x + 3 = ±1
x = -3 ± 1
So x = -2 or x = -4.
所以 x = -2 或 x = -4。
6. The Discriminant: How Many Roots? | 判别式：有几个根？
The value of b² - 4ac is called the discriminant, often written as Δ. It tells you how many real roots a quadratic equation has.
b² - 4ac 的值称为判别式，通常记为 Δ。它告诉我们二次方程有多少个实数根。
If b² - 4ac > 0, there are two distinct real roots.
若 b² - 4ac > 0，则有两个不同的实数根。
If b² - 4ac = 0, there is one repeated real root.
若 b² - 4ac = 0，则有一个二重实数根。
If b² - 4ac < 0, there are no real roots.
若 b² - 4ac < 0，则没有实数根。
(-2)² - 4×1×5 = 4 - 20 = -16, so it has no real roots. Its graph never crosses the x-axis.
例如，方程 x² - 2x + 5 = 0 的判别式为 (-2)² - 4×1×5 = 4 - 20 = -16，所以没有实数根。它的图像不与 x 轴相交。
7. Quadratic Graphs and Roots | 二次函数图像与根
h y = ax² + bx + c. Solving the equation means finding where the curve crosses the x-axis.
二次方程 ax² + bx + c = 0 的实数根就是抛物线 y = ax² + bx + c 与 x 轴交点的横坐标。解方程就是求曲线与 x 轴的交点。
The turning point has x-coordinate -b/(2a). Substituting this into the equation gives the y-coordinate of the vertex.
顶点的 x 坐标为 -b/(2a)。将其代入方程即可得到顶点的 y 坐标。
the turning point is -(-4)/(2×1) = 2, and y = 2² - 4×2 + 3 = -1. So the vertex is (2, -1).
对于 y = x² - 4x + 3，因为 x² - 4x + 3 = (x - 1)(x - 3)，所以根是 x = 1 和 x = 3。顶点的 x 坐标为 -(-4)/(2×1) = 2，y = 2² - 4×2 + 3 = -1。所以顶点是 (2, -1)。
8. Word Problems Involving Quadratics | 二次方程应用题
Many exam problems require you to form a quadratic equation from a real-life situation.
许多考试题需要你根据实际情境建立二次方程。
Example: The area of a rectangle is 30 cm². Its length is 1 cm longer than its width. Find the width.
例：一个长方形的面积是 30 cm²，长比宽长 1 cm。求宽。
Let the width be x cm. Then the length is (x + 1) cm. Since area = length × width:
设宽为 x cm，那么长为 (x + 1) cm。因为面积 = 长 × 宽：
x(x + 1) = 30
x² + x - 30 = 0
Factorise:
因式分解：
(x + 6)(x - 5) = 0
= -6 or x = 5. Since width cannot be negative, the width is 5 cm, and the length is 6 cm.
所以 x = -6 或 x = 5。因为宽不能为负数，所以宽为 5 cm，长为 6 cm。
9. Common Mistakes and Exam Tips | 常见错误与考试技巧
Here are some frequent errors students make and how to avoid them.
以下是一些学生常犯的错误以及如何避免它们。
when using the quadratic formula or taking a square root. Always write x = ... or x = ....
使用求根公式或开平方时不要忘记 ± 符号。一定要写出两个解：x = ... 和 x = ...。
c = 0 before solving. If there are terms on both sides, move everything to one side first.
解二次方程前要先化为 ax² + bx + c = 0 的形式。如果等号两边都有项，先全部移到一边。
Checking whether the question asks for exact values or decimals. If the question says "give your answer correct to 2 decimal places", use the quadratic formula and round at the end.
注意题目要求精确值还是小数。如果题目说“答案精确到两位小数”，请使用求根公式并在最后四舍五入。
check the signs carefully. For example, x² - 3x - 4 = (x - 4)(x + 1), not (x + 4)(x - 1).
因式分解时仔细检查符号。例如，x² - 3x - 4 = (x - 4)(x + 1)，而不是 (x + 4)(x - 1)。
10. Practice Questions | 练习题
Try these problems on your own, then check your answers.
请独立尝试以下题目，然后再核对答案。
Solve x² - 7x + 10 = 0.
解方程 x² - 7x + 10 = 0。
Solve 3x² + 5x - 2 = 0 using the quadratic formula.
用求根公式解 3x² + 5x - 2 = 0。
Find the value(s) of k such that the equation x² + kx + 9 = 0 has exactly one real root.
求 k 的值，使得方程 x² + kx + 9 = 0 恰好有一个实数根。
Answers: 1. x = 2 or x = 5. 2. x = 1/3 or x = -2. 3. k = 6 or k = -6.
答案：1. x = 2 或 x = 5。2. x = 1/3 或 x = -2。3. k = 6 或 k = -6。
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