Quadratic Equations: Solving and Graphing | 二次方程：求解与图像

📚 Quadratic Equations: Solving and Graphing | 二次方程：求解与图像

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, geometry, and even in real-world problems involving area, motion, and profit. Mastering the skills of solving and sketching quadratics is essential for success in the examinations.

二次方程是 IGCSE 数学中最重要的主题之一。它们出现在代数、几何中，也出现在涉及面积、运动和利润的实际问题中。掌握求解与绘制二次函数图形的技能，对考试成功至关重要。

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 x is the unknown, and a, b, and c are constants, with a ≠ 0. If a = 0, the equation becomes linear and is no longer quadratic.

二次方程是指可以写成 ax² + bx + c = 0 形式的方程，其中 x 是未知数，a、b、c 是常数，且 a ≠ 0。如果 a = 0，方程就变成了一次方程，不再是二次方程。

The highest power of x is 2, which gives the equation its name “quadratic”. Some textbooks call the variable “x” but it may also be represented by other letters such as t or y, depending on the context.

x 的最高次数是 2，这也就是“二次”这个名称的来源。有些教科书用“x”，但在不同问题中也可能用 t 或 y 等其他字母表示未知数。

For example, 2x² + 5x – 3 = 0 is quadratic, while x + 4 = 0 is not. A quadratic equation can have zero, one, or two real solutions, which are also called roots.

例如，2x² + 5x – 3 = 0 是二次方程，而 x + 4 = 0 不是。二次方程可能有零个、一个或两个实数解，这些解也叫做根。

2. Solving by Factorisation | 因式分解法

Factorisation is often the fastest method when the quadratic expression can be factored easily. The idea is to rewrite ax² + bx + c = 0 as a product of two linear factors, then use the zero product property: if A × B = 0, then A = 0 or B = 0.

因式分解法通常是在二次表达式可以轻松分解时最快的方法。其核心思想是将 ax² + bx + c = 0 改写成两个一次因式的乘积，然后利用零积性质：若 A × B = 0，则 A = 0 或 B = 0。

Consider the equation x² – 5x + 6 = 0. We look for two numbers that multiply to 6 and add to -5. Those numbers are -2 and -3. Therefore x² – 5x + 6 = (x – 2)(x – 3) = 0.

考虑方程 x² – 5x + 6 = 0。我们要寻找两个数，它们相乘等于 6，相加等于 -5。这两个数是 -2 和 -3。因此 x² – 5x + 6 = (x – 2)(x – 3) = 0。

所以 x = 2 或 x = 3。

x = 2 或 x = 3

For quadratics with a leading coefficient other than 1, such as 2x² + 7x + 3 = 0, you may need to use the “ac method” or trial and error. Check that (2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3, so the solutions are x = -1/2 or x = -3.

对于首项系数不是 1 的二次方程，如 2x² + 7x + 3 = 0，你可能需要使用“ac 法”或试错法。验证 (2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3，因此解为 x = -1/2 或 x = -3。

3. Solving by Completing the Square | 配方法

Completing the square is a technique that rewrites a quadratic expression as a perfect square plus a constant. It is especially useful for solving equations that do not factor nicely and for finding the turning point of a quadratic graph.

配方法是一种将二次表达式改写为“完全平方加常数”的技巧。它特别适用于求解不易因式分解的方程，也常用于求二次函数图像的顶点。

Start with x² + 6x + 8 = 0. Take the coefficient of x, which is 6, halve it to get 3, and square it to get 9. Then rewrite the first two terms as (x + 3)² – 9. The equation becomes (x + 3)² – 9 + 8 = 0, so (x + 3)² – 1 = 0.

从 x² + 6x + 8 = 0 开始。取 x 的系数 6，除以 2 得 3，再平方得 9。于是前两项改写为 (x + 3)² – 9。方程变为 (x + 3)² – 9 + 8 = 0，即 (x + 3)² – 1 = 0。

(x + 3)² = 1

Taking square roots gives x + 3 = ±1, so x = -3 + 1 = -2 or x = -3 – 1 = -4.

开平方得 x + 3 = ±1，所以 x = -3 + 1 = -2 或 x = -3 – 1 = -4。

If the coefficient of x² is not 1, first divide the whole equation by that coefficient. For example, 2x² + 8x + 6 = 0 becomes x² + 4x + 3 = 0 before completing the square.

如果 x² 的系数不是 1，先让整个方程除以该系数。例如，2x² + 8x + 6 = 0 在配方前先化为 x² + 4x + 3 = 0。

4. Solving using the Quadratic Formula | 求根公式

The quadratic formula is a universal method that works for all quadratic equations, even when factorisation is impossible or difficult. For ax² + bx + c = 0, the solutions are given by:

求根公式是适用于所有二次方程的通用方法，即使因式分解不可能或很困难时也能使用。对于 ax² + bx + c = 0，解为：

x = (-b ± √(b² – 4ac)) / (2a)

To use the formula, identify a, b, and c from the equation. Take the equation 2x² – 4x – 6 = 0. Here a = 2, b = -4, c = -6. Substitute into the formula:

使用公式时，先从方程中识别 a、b、c。以方程 2x² – 4x – 6 = 0 为例，这里 a = 2，b = -4，c = -6。代入公式得：

x = (4 ± √(16 – 4 × 2 × (-6))) / (2 × 2)

x = (4 ± √(16 + 48)) / 4 = (4 ± √64) / 4 = (4 ± 8) / 4

Thus x = (4 + 8) / 4 = 3, or x = (4 – 8) / 4 = -1. The same result can be checked by factorising the original equation.

因此 x = (4 + 8) / 4 = 3，或 x = (4 – 8) / 4 = -1。原方程也可以用因式分解验证得到相同结果。

Always simplify the square root if possible. In the IGCSE exam, you are usually expected to write the answer in exact surd form rather than a rounded decimal.

应尽量化简平方根。在 IGCSE 考试中，通常要求以精确的根式形式书写答案，而不是四舍五入的小数。

5. The Discriminant | 判别式

The expression b² – 4ac under the square root in the quadratic formula is called the discriminant, often denoted by the Greek letter Δ. It tells us how many real roots the equation has without fully solving it.

求根公式中被开方的表达式 b² – 4ac 称为判别式，通常用希腊字母 Δ 表示。它可以告诉我们方程有多少个实数根，而无需完整求解。

Discriminant	Discriminant (Chinese)	Number of Real Roots	实数根个数
b² – 4ac > 0	判别式大于零	Two distinct real roots	两个不同的实数根
b² – 4ac = 0	判别式等于零	One repeated real root	一个二重实数根
b² – 4ac < 0	判别式小于零	No real roots	没有实数根

For example, for x² + x + 1 = 0, the discriminant is 1² – 4 × 1 × 1 = 1 – 4 = -3, which is negative. Therefore the equation has no real solutions.

例如，对于 x² + x + 1 = 0，判别式为 1² – 4 × 1 × 1 = 1 – 4 = -3，是负数，因此方程没有实数解。

When the discriminant is a positive perfect square, the roots are rational and factorisation is possible. When it is positive but not a perfect square, the roots are irrational and the quadratic formula is the best choice.

当判别式是正完全平方数时，根是有理数，可以用因式分解；当判别式为正但非完全平方数时，根是无理数，使用求根公式最为合适。

6. The Graph of a Quadratic Function | 二次函数图像

A quadratic function has the general form y = ax² + bx + c, and its graph is a curve called a parabola. The sign of a determines whether the parabola opens upward or downward.

二次函数的一般形式为 y = ax² + bx + c，其图像是一条称为抛物线的曲线。a 的符号决定了抛物线的开口方向。

If a > 0, the parabola opens upwards and has a minimum turning point.

若 a > 0，抛物线开口向上，有最小值顶点。
If a < 0, the parabola opens downwards and has a maximum turning point.

若 a < 0，抛物线开口向下，有最大值顶点。
The curve is symmetric about a vertical line through its turning point, called the axis of symmetry.

曲线关于经过顶点的竖直线对称，这条线称为对称轴。

For example, y = x² – 2x + 3 opens upwards because a = 1 > 0. The axis of symmetry is x = -b / (2a) = 2 / (2 × 1) = 1.

例如，y = x² – 2x + 3 开口向上，因为 a = 1 > 0。对称轴为 x = -b / (2a) = 2 / (2 × 1) = 1。

Writing the equation in the completed square form y = a(x – h)² + k makes the turning point immediately visible at (h, k). This form is also known as vertex form.

将方程写成配平方形式 y = a(x – h)² + k 后，顶点可以立即看出为 (h, k)。这种形式也称为顶点式。

7. Roots, Intercepts and Turning Points | 根、截距与顶点

For the graph of y = ax² + bx + c, the x-intercepts are the real roots of the equation ax² + bx + c = 0. If the discriminant is negative, the graph does not cross the x-axis.

对于 y = ax² + bx + c 的图像，x 截距就是方程 ax² + bx + c = 0 的实数根。如果判别式为负，图像不与 x 轴相交。

The y-intercept is found by setting x = 0. This gives y = c, so the graph always crosses the y-axis at the point (0, c).

令 x = 0 可得 y 截距为 y = c，因此图像总是在点 (0, c) 处穿过 y 轴。

The turning point can be found by completing the square. For y = x² – 4x + 1, rewrite as y = (x – 2)² – 3. The minimum turning point is therefore at (2, -3).

顶点可以通过配方法求得。对于 y = x² – 4x + 1，写成 y = (x – 2)² – 3。因此最小顶点在 (2, -3)。

To sketch a quadratic graph, plot the y-intercept, the x-intercepts (if they exist), and the turning point. Then draw a smooth curve through these points, respecting the direction of opening.

要画二次函数的草图，先标出 y 截距、x 截距（如果存在）以及顶点，然后画一条平滑曲线穿过这些点，并注意开口方向。

8. Real-World Applications | 实际应用

Quadratic equations appear frequently in practical problems. For example, the area of a rectangular garden is given by a product of length and width, which may lead to a quadratic equation when one dimension is expressed in terms of the other.

二次方程经常出现在实际问题中。例如，矩形花园的面积等于长乘宽，当一个边长用另一个边长表示时，就可能得到一个二次方程。

Suppose a rectangle has length (x + 3) metres and width (x – 2) metres, and its area is 20 square metres. Then (x + 3)(x – 2) = 20, which expands to x² + x – 6 = 20, or x² + x – 26 = 0.

假设一个矩形的长为 (x + 3) 米，宽为 (x – 2) 米，面积为 20 平方米。那么 (x + 3)(x – 2) = 20，展开得 x² + x – 6 = 20，即 x² + x – 26 = 0。

Projectile motion also uses quadratics. The height h of a ball thrown upward is often modelled by h = -4.9t² + v₀t + h₀, where v₀ is the initial velocity and h₀ is the starting height.

抛体运动也会用到二次方程。向上抛出的球的高度 h 通常用 h = -4.9t² + v₀t + h₀ 建模，其中 v₀ 是初速度，h₀ 是起始高度。

When solving such problems, reject any negative or impossible solutions in the context, even if they are mathematically valid roots of the equation.

在解决这类问题时，即使某些根在数学上有效，也要舍弃在题意中不合理的负数解或不可能的解。

9. Common Mistakes | 常见错误

Many students make avoidable mistakes when working with quadratics. Being aware of these can dramatically improve your accuracy in the exam.

许多学生在处理二次方程时会犯可以避免的错误。留意这些常见错误可以显著提高考试中的正确率。

Sign errors when expanding brackets, such as writing (x – 2)(x + 3) as x² + x – 6 incorrectly.

展开括号时符号错误，例如把 (x – 2)(x + 3) 错误展开。
Forgetting to set the equation to zero before factorising. Always write ax² + bx + c = 0 first.

在因式分解前忘记将方程化为零。一定要先把方程写成 ax² + bx + c = 0。
Dividing both sides by x, which loses a root. For example, x² = x has solutions x = 0 and x = 1; dividing by x gives only x = 1.

两边同时除以 x，导致丢失一个根。例如 x² = x 的解为 x = 0 和 x = 1；除以 x 后只得到 x = 1。
Using the quadratic formula with the wrong values of a, b, or c. Always write down the equation before substituting.

使用求根公式时取错 a、b、c 的值。代入前务必先写出原方程。

Another common error is forgetting to take both the positive and negative square roots when solving (x + 3)² = 4. The correct result is x + 3 = ±2.

另一个常见错误是解 (x + 3)² = 4 时忘记取正负两个平方根。正确结果为 x + 3 = ±2。

10. Exam Tips | 考试技巧

In the IGCSE examination, question setters often award method marks as well as accuracy marks. Showing clear steps is essential, even if your final answer is wrong.

在 IGCSE 考试中，阅卷者除了给结果分之外还会给方法分。因此，即使最终答案错误，写出清晰的步骤也非常重要。

Always check whether the quadratic can be factorised before using the formula. Factorisation is usually quicker.

使用求根公式前先检查是否能用因式分解。因式分解通常更快。
If a question asks for answers to two decimal places, use the quadratic formula and give the final values to the required degree of accuracy.

如果题目要求将答案保留到两位小数，请使用求根公式并将最终值写到要求的精度。
Make sure you know the difference between “solve the equation” and “sketch the graph”. The first requires algebraic roots, the second requires key features such as intercepts and turning point.

务必区分“解方程”和“画函数草图”。前者需要求代数根，后者需要标出截距、顶点等关键特征。
Round only at the end of your working, not during intermediate steps, to avoid unnecessary errors.

只在最后一步四舍五入，不要在中间步骤中提前取近似值，以免产生不必要的误差。

Use your calculator carefully when entering negative numbers and square roots. If possible, verify your answer by substituting it back into the original equation.

使用计算器输入负数和平方根时要谨慎。若有可能，将答案代回原方程进行验证。

11. Practice Problems | 练习题

Here are some practice questions that cover the main skills discussed in this article. Try them before reading the final answers.

下面是一些涵盖本文主要技能的练习题。请先尝试作答，再阅读最终答案。

Solve x² – 3x – 10 = 0 by factorisation.

用因式分解法解 x² – 3x – 10 = 0。
Solve x² + 6x + 2 = 0 by completing the square, giving your answers in surd form.

用配方法解 x² + 6x + 2 = 0，答案写成根式形式。
Given the equation kx² + 4x + 1 = 0, find the value of k for which the equation has exactly one real root.

已知方程 kx² + 4x + 1 = 0，求该方程恰好有一个实数根时 k 的值。

Answers: The first equation factors as (x – 5)(x + 2) = 0, so x = 5 or x = -2. For the second, (x + 3)² – 7 = 0, so x = -3 ± √7. For the third, the discriminant is 16 – 4k = 0, giving k = 4.

答案：第一个方程因式分解为 (x – 5)(x + 2) = 0，所以 x = 5 或 x = -2。第二个方程为 (x + 3)² – 7 = 0，所以 x = -3 ± √7。第三个方程判别式为 16 – 4k = 0，得 k = 4。

12. Conclusion | 总结

Quadratic equations are a key part of IGCSE Mathematics. Depending on the problem, you can solve them by factorisation, completing the square, or using the quadratic formula. The discriminant helps you understand the nature of the roots without solving fully.

二次方程是 IGCSE 数学的核心内容。根据不同的问题，你可以选择因式分解法、配方法或求根公式来求解。判别式能在不解方程的情况下帮助你理解根的性质。

Equally important is the ability to interpret and sketch quadratic graphs. Always look for the intercepts, the turning point, and the direction of opening. With consistent practice, quadratic equations become one of the most reliable skill areas in your mathematical toolkit.

同样重要的是理解和绘制二次函数图像。时刻关注截距、顶点和开口方向。通过持续练习，二次方程将成为你数学工具箱中最得心应手的技能之一。

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Quadratic Equations: Solving and Graphing | 二次方程：求解与图像