📚 Solving Quadratic Equations: Factorising, Formula and Graphs | 解二次方程：因式分解、公式法与图像
A quadratic equation is one of the most important algebraic skills in IGCSE Mathematics. It appears in topics ranging from algebraic manipulation and coordinate geometry to real-life problem solving. Mastering quadratic equations means you can factorise expressions, use the quadratic formula accurately, understand the discriminant, and link algebra to the shape and position of a parabola.
二次方程是 IGCSE 数学中最重要的代数技能之一。它出现在代数变形、坐标几何以及实际问题解决等多个主题中。掌握二次方程意味着你能够对表达式进行因式分解、准确使用求根公式、理解判别式，并将代数与抛物线的形状和位置联系起来。
1. What Is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is an equation where the highest power of the variable is 2. The general form is usually written with all terms on one side and zero on the other. In IGCSE problems, the variable is often x, but quadratic equations can also be written using other letters such as y or t.
二次方程是指未知数的最高次数为 2 的方程。一般形式通常将所有项放在方程的一边，另一边为零。在 IGCSE 题目中，未知数常用 x 表示，但二次方程也可以使用其他字母，如 y 或 t。
ax² + bx + c = 0
her than quadratic. For example, x² + 3x + 2 = 0 is quadratic, while 3x + 2 = 0 is linear.
在这个标准形式中，a、b 和 c 是常数，并且 a 不能为零，因为如果 a = 0，方程就变成一次方程而不是二次方程。例如，x² + 3x + 2 = 0 是二次方程，而 3x + 2 = 0 是一次方程。
2. Standard Form and Coefficients | 标准形式与系数
Before solving a quadratic equation, it is essential to rearrange the equation into standard form. This means collecting all terms on one side so that the other side is exactly zero. The coefficient of x² is called a, the coefficient of x is called b, and the constant term is called c.
在解二次方程之前，必须先把方程整理成标准形式。这意味着将所有项移到一边，使另一边恰好为零。x² 的系数称为 a，x 的系数称为 b，常数项称为 c。
5x + 7, it must first be rearranged to 3x² - 5x - 7 = 0 so that a = 3, b = -5 and c = -7.
例如，方程 2x² - 4x + 1 = 0 中 a = 2，b = -4，c = 1。如果给出的方程是 3x² = 5x + 7，则必须首先整理为 3x² - 5x - 7 = 0，因此 a = 3，b = -5，c = -7。
Being careful with signs is extremely important when identifying a, b and c. Many errors in the quadratic formula come from copying the wrong sign for b or c.
在确定 a、b 和 c 时，特别注意符号非常重要。求根公式中许多错误都来自 b 或 c 的符号抄写错误。
3. Solving by Factorising | 因式分解法
two linear brackets. For example, x² + 5x + 6 = 0 can be factorised as (x + 2)(x + 3) = 0.
解二次方程最快的方法之一就是因式分解法。当二次表达式可以写成两个一次括号的乘积时，这种方法非常有效。例如，x² + 5x + 6 = 0 可以因式分解为 (x + 2)(x + 3) = 0。
To factorise a simple quadratic of the form x² + bx + c, you need to find two numbers that multiply to give c and add to give b. In the example above, 2 and 3 multiply to give 6 and add to give 5, so the factors are (x + 2) and (x + 3).
要对 x² + bx + c 形式的简单二次式进行因式分解，你需要找到两个数，它们相乘等于 c，相加等于 b。在上面的例子中，2 和 3 相乘等于 6，相加等于 5，因此因式为 (x + 2) 和 (x + 3)。
If the coefficient of x² is not 1, such as in 2x² + 7x + 3, the process is slightly longer. You can use the ac method: multiply a and c to get 6, find two numbers that multiply to 6 and add to 7, which are 6 and 1, then split the middle term and factorise by grouping.
如果 x² 的系数不是 1，例如 2x² + 7x + 3，过程会稍长一些。你可以使用 ac 法：先将 a 和 c 相乘得到 6，找到相乘等于 6 且相加等于 7 的两个数，即 6 和 1，然后拆分中间项并通过分组进行因式分解。
2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)
4. The Zero Product Property | 零乘积性质
n at least one of the factors must be zero. In symbols, if A × B = 0, then A = 0 or B = 0.
将二次方程因式分解后，下一步就是使用零乘积性质。该规则指出，如果两个或多个因式的乘积为零，那么至少有一个因式必须为零。用符号表示，如果 A × B = 0，则 A = 0 或 B = 0。
ves x = -2, and x + 3 = 0 gives x = -3. Therefore the two solutions are x = -2 and x = -3.
例如，由 (x + 2)(x + 3) = 0，我们令每个括号分别等于零：x + 2 = 0 得到 x = -2，x + 3 = 0 得到 x = -3。因此两个解为 x = -2 和 x = -3。
Always present your final answers clearly. In IGCSE, a quadratic equation usually has two roots, although sometimes the two roots are the same value, which is called a repeated root.
最后一定要清晰地写出答案。在 IGCSE 中，二次方程通常有两个根，尽管有时两个根的值相同，这种情况称为重根。
5. Solving by Completing the Square | 配方法
Completing the square is a method that rewrites a quadratic expression in the form (x + p)² + q. This form is especially useful for finding the vertex of a parabola and for solving equations that cannot easily be factorised.
配方法是一种将二次表达式改写成 (x + p)² + q 形式的方法。这种形式对于求抛物线的顶点以及解决不易因式分解的方程特别有用。
quare it to get 9, and add 9 to both sides: x² + 6x + 9 = -2 + 9. This gives (x + 3)² = 7.
对于 x² + 6x + 2 = 0，首先移项得 x² + 6x = -2。然后取 x 的系数的一半，即 3，将其平方得到 9，并在等式两边同时加上 9：x² + 6x + 9 = -2 + 9。于是得到 (x + 3)² = 7。
Taking the square root of both sides gives x + 3 = ±√7, so the solutions are x = -3 + √7 and x = -3 - √7. This method works for every quadratic equation, although it is most convenient when the coefficient of x² is 1.
对两边同时开平方，得到 x + 3 = ±√7，因此解为 x = -3 + √7 和 x = -3 - √7。这种方法适用于所有二次方程，但当 x² 的系数为 1 时最方便。
6. The Quadratic Formula | 求根公式
The quadratic formula is the most general method for solving any quadratic equation. It is derived from completing the square and gives the roots directly from the coefficients a, b and c. You should memorise it and be able to use it quickly and accurately.
求根公式是解任何二次方程最通用的方法。它由配方法推导而来，可以直接由系数 a、b 和 c 得到根。你应当牢记这个公式，并能快速准确地使用它。
x = (-b ± √b² - 4ac) ÷ 2a
-5. Substitute these into the formula: x = (-(-3) ± √((-3)² - 4(2)(-5))) ÷ (2 × 2) = (3 ± √9 + 40) ÷ 4 = (3 ± √49) ÷ 4 = (3 ± 7) ÷ 4.
例如，解方程 2x² - 3x - 5 = 0，确定 a = 2，b = -3，c = -5。将它们代入公式：x = (-(-3) ± √((-3)² - 4(2)(-5))) ÷ (2 × 2) = (3 ± √9 + 40) ÷ 4 = (3 ± √49) ÷ 4 = (3 ± 7) ÷ 4。
÷ 4 = 10 ÷ 4 = 2.5 and x = (3 - 7) ÷ 4 = -4 ÷ 4 = -1. So the roots are x = 2.5 or x = -1.
由此得到 x = (3 + 7) ÷ 4 = 10 ÷ 4 = 2.5，以及 x = (3 - 7) ÷ 4 = -4 ÷ 4 = -1。因此根为 x = 2.5 或 x = -1。
7. Discriminant and Number of Roots | 判别式与根的个数
The discriminant is the part of the quadratic formula under the square root sign: b² - 4ac. It tells you how many real solutions a quadratic equation has without actually solving the equation.
判别式是求根公式中平方根号下的部分：b² - 4ac。它可以在不实际解方程的情况下告诉你一个二次方程有多少个实数解。
uation has exactly one real root, also called a repeated root. If b² - 4ac < 0, the equation has no real roots, but it does have two complex roots in more advanced mathematics.
如果 b² - 4ac > 0，方程有两个不相等的实数根。如果 b² - 4ac = 0，方程恰好有一个实数根，也称为重根。如果 b² - 4ac < 0，方程没有实数根，但在更高深的数学中它有两个复数根。
it has one repeated root. In fact, x² - 4x + 4 = (x - 2)², so the only solution is x = 2.
例如，x² - 4x + 4 = 0 的判别式为 (-4)² - 4(1)(4) = 16 - 16 = 0，因此它有一个重根。事实上，x² - 4x + 4 = (x - 2)²，因此唯一解为 x = 2。
8. Graphs of Quadratic Functions | 二次函数图像
ax² + bx + c, and its graph is a curve called a parabola. If a is positive, the parabola opens upwards and has a minimum point. If a is negative, the parabola opens downwards and has a maximum point.
二次函数写作 y = ax² + bx + c，其图像是一条称为抛物线的曲线。如果 a 为正，抛物线开口向上并有一个最低点。如果 a 为负，抛物线开口向下并有一个最高点。
e y-coordinate is then found by substituting this x-value back into the original function.
抛物线的顶点是转折点。它的 x 坐标可以用公式 x = -b ÷ 2a 求得。然后将这个 x 值代回原函数即可求得 y 坐标。
efore, by drawing or sketching a quadratic graph, you can estimate the solutions of a quadratic equation visually.
图像与 x 轴的交点就是方程 ax² + bx + c = 0 的根。因此，通过绘制或勾勒二次函数图像，你可以直观地估算二次方程的解。
9. Using Graphs to Solve Equations | 利用图像解方程
In many IGCSE questions, you may be given a graph and asked to find the approximate solutions of a quadratic equation. The solutions correspond to the x-coordinates of the points where the graph cuts the x-axis.
在许多 IGCSE 题目中，你可能会看到给定图像并被要求求出二次方程的近似解。解对应于图像与 x 轴相交点的 x 坐标。
x-intercepts because it is entirely above the x-axis, so x² + 4 = 0 has no real solutions.
如果图像没有明显地与 x 轴相交，则对应的方程没有实数根。例如，y = x² + 4 的图像完全在 x 轴上方，因此没有 x 轴截距，所以 x² + 4 = 0 没有实数解。
wing a horizontal line y = k and reading off the x-coordinates of the intersection points.
你也可能被要求利用给定的图像来解形如 x² + bx + c = k 的方程。方法是在图像上画一条水平线 y = k，然后读取交点的 x 坐标。
10. Word Problems and Applications | 应用题与实际应用
Many real-life problems can be modelled by quadratic equations. Typical IGCSE examples involve areas of rectangles, projectile motion, product of numbers, and profit or cost calculations. The key is to translate the written information into an equation in standard form.
许多实际问题都可以用二次方程来建模。典型的 IGCSE 例题涉及矩形面积、抛体运动、数字的乘积以及利润或成本计算。关键是将文字信息转化为标准形式的方程。
28 = 0, which factorises as (x + 7)(x - 4) = 0. Since length cannot be negative, x = 4 cm.
例如，一个矩形的长比宽多 3 厘米。如果面积为 28 平方厘米，设宽为 x。则长为 x + 3，因此面积方程为 x(x + 3) = 28。展开得到 x² + 3x - 28 = 0，因式分解为 (x + 7)(x - 4) = 0。因为长度不能为负，所以 x = 4 厘米。
Always check that your solutions make sense in the original context. Negative roots often give mathematically correct values but are not valid for lengths, time, or quantities that must be positive.
务必检查你的解在原始情境中是否有意义。负根在数学上往往成立，但对于长度、时间或必须为正的量通常是无效的。
11. Common Mistakes to Avoid | 常见错误
t property cannot be used. You must first rearrange to x² + 3x - 4 = 0 and then factorise.
一个常见错误是在因式分解前忘记将方程设为零。如果你试图通过写成 x(x + 3) = 4 来解 x² + 3x = 4，那么零乘积性质就无法使用。你必须先整理为 x² + 3x - 4 = 0，然后再进行因式分解。
quals 3, but many students write -3. Writing down a, b and c first can help reduce these mistakes.
另一个常见错误是在求根公式中处理负号时出错。例如，如果 b = -3，那么 -b 等于 3，但许多学生却写成 -3。先把 a、b 和 c 写下来有助于减少这类错误。
Finally, when completing the square, do not forget to add the same value to both sides of the equation. Changing only one side changes the equation and leads to incorrect roots.
最后，在配方时，不要忘记在等式两边同时加上相同的值。只改变一边会改变方程，并导致错误的根。
12. Exam Tips for Quadratic Equations | 二次方程的考试技巧
In IGCSE examinations, show all your steps even if you use a calculator. Marks are often awarded for correct substitution into the quadratic formula or for the correct factorised form. Writing down a, b and c before using the formula can help you earn method marks.
在 IGCSE 考试中，即使使用计算器，也要展示所有步骤。正确代入求根公式或写出正确的因式分解形式通常都能得分。在使用公式前先写出 a、b 和 c，有助于你拿到方法分。
If a question asks for answers to a certain number of decimal places or significant figures, always round only at the final step. Use the full value or store intermediate results in your calculator to avoid rounding errors.
如果题目要求答案保留一定的小数位数或有效数字，务必只在最后一步进行四舍五入。使用完整数值或在计算器中存储中间结果，以避免舍入误差。
For graph-based questions, draw clear lines and label key points such as intercepts and the vertex. If you are asked to solve an equation using a graph, read the x-values carefully and state them with appropriate accuracy.
对于与图像相关的题目，要画出清晰的线条，并标出关键点，如截距和顶点。如果要求利用图像解方程，请仔细读取 x 值，并以适当的精度写出答案。
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