IGCSE Maths: Exploring Quadratic Equations & Graphs | IGCSE数学：探究二次方程与图像

📚 IGCSE Maths: Exploring Quadratic Equations & Graphs | IGCSE数学：探究二次方程与图像

A quadratic equation is an equation of the form ax² + bx + c = 0, where a ≠ 0. It is one of the most important topics in IGCSE Mathematics, bridging the gap between algebra and geometry. Understanding quadratic equations and their graphs is essential for solving many real-world problems and for success in your exams.

二次方程是形如 ax² + bx + c = 0 的方程，其中 a ≠ 0。它是IGCSE数学中最重要的主题之一，连接着代数与几何。理解二次方程及其图像对于解决许多实际问题以及在考试中取得好成绩至关重要。

1. The Standard Form | 标准形式

Every quadratic expression can be written in the standard form ax² + bx + c = 0. The coefficient a is called the leading coefficient, b is the linear coefficient, and c is the constant term. For a true quadratic equation, a must not equal zero; otherwise the equation becomes linear.

每个二次表达式都可以写成标准形式 ax² + bx + c = 0。系数 a 称为首项系数，b 称为一次项系数，c 称为常数项。对于真正的二次方程，a 不能等于零，否则方程就变成了一次方程。

For example, 2x² – 3x + 1 = 0 has a = 2, b = -3 and c = 1. Many IGCSE questions require you to rearrange a given equation into this form before solving.

例如，2x² – 3x + 1 = 0 中 a = 2，b = -3，c = 1。许多IGCSE考题要求你先将给定方程整理成这种形式，然后再求解。

When working with quadratic functions, we often write y = ax² + bx + c. This form is useful for identifying the y-intercept, which is simply the value of c.

在研究二次函数时，我们常写成 y = ax² + bx + c。这种形式便于识别 y 轴截距，其值就是 c。

2. Solving by Factorisation | 因式分解法

Factorisation is the quickest method when the quadratic has simple integer roots. The idea is to express ax² + bx + c as a product of two linear factors, then set each factor equal to zero.

当二次方程具有简单的整数根时，因式分解是最快捷的方法。其思想是将 ax² + bx + c 表示为两个一次因式的乘积，然后令每个因式为零。

For example, to solve x² – 5x + 6 = 0, look for two numbers that multiply to 6 and add to -5. They are -2 and -3, so (x – 2)(x – 3) = 0. Hence x = 2 or x = 3.

例如，解 x² – 5x + 6 = 0 时，寻找两个数相乘为 6 且相加为 -5。这两个数是 -2 和 -3，因此 (x – 2)(x – 3) = 0。所以 x = 2 或 x = 3。

Follow these steps:

Rearrange the equation so one side is zero.
Factorise the quadratic expression fully.
Set each factor equal to zero and solve for x.

遵循以下步骤：

将方程整理为一边等于零。
将二次表达式完全因式分解。
令每个因式等于零，并解出 x。

Always check your answers by substituting back into the original equation. If the quadratic does not factorise with integers, use a different method.

务必通过代回原方程来检验答案。如果二次式不能进行整数因式分解，则应使用其他方法。

3. Completing the Square | 配方法

Completing the square rewrites a quadratic as a perfect square plus a constant. This method is essential for solving equations that do not factorise and for finding the vertex of a parabola.

配方法将二次式改写为一个完全平方加一个常数的形式。这种方法对于解不能因式分解的方程以及求抛物线的顶点至关重要。

Take the expression x² + 6x + 2. Start by halving the coefficient of x, which is 3, and write (x + 3)². Then subtract 3² = 9 to keep the expression unchanged: x² + 6x + 2 = (x + 3)² – 9 + 2 = (x + 3)² – 7.

以表达式 x² + 6x + 2 为例。首先取 x 系数的一半，即 3，写成 (x + 3)²。然后减去 3² = 9 以保持表达式不变：x² + 6x + 2 = (x + 3)² – 9 + 2 = (x + 3)² – 7。

If a ≠ 1, factor out a before completing the square. For instance, 2x² + 8x + 5 = 2(x² + 4x) + 5 = 2[(x + 2)² – 4] + 5 = 2(x + 2)² – 3.

若 a ≠ 1，则先提取 a 再进行配方。例如，2x² + 8x + 5 = 2(x² + 4x) + 5 = 2[(x + 2)² – 4] + 5 = 2(x + 2)² – 3。

The completed square form y = a(x – h)² + k directly shows the vertex at (h, k). For y = 2(x + 2)² – 3, the vertex is (-2, -3).

配方的结果形式 y = a(x – h)² + k 直接给出顶点坐标 (h, k)。对于 y = 2(x + 2)² – 3，顶点为 (-2, -3)。

To solve an equation using this form, rearrange to isolate the square and then take the square root of both sides, remembering the ± symbol.

用这种形式解方程时，先通过变形孤立出完全平方，然后对方程两边同时开平方，切记加上 ± 符号。

4. The Quadratic Formula | 二次公式

The quadratic formula is a universal tool. It works for any quadratic equation, even those with irrational or complex roots. The formula is:

二次公式是一种通用工具。它对任何二次方程都有效，即使是具有无理数根或复数根的方程。公式为：

x = [-b ± √(b² – 4ac)] / (2a)

To use it, identify a, b and c from the standard form and substitute them into the formula carefully. Always simplify the square root first, then the fraction.

使用时，先从标准形式中确定 a、b、c，然后小心地代入公式。务必先化简根式，再化简分数。

For example, solve 2x² – 4x – 3 = 0 using the formula. Here a = 2, b = -4 and c = -3. Substitute to get x = [4 ± √(16 + 24)] / 4 = [4 ± √40] / 4 = [4 ± 2√10] / 4 = 1 ± (√10)/2.

例如，用公式法解 2x² – 4x – 3 = 0。这里 a = 2，b = -4，c = -3。代入得 x = [4 ± √(16 + 24)] / 4 = [4 ± √40] / 4 = [4 ± 2√10] / 4 = 1 ± (√10)/2。

In IGCSE exams, you may be asked to leave answers in surd form. This is often easier and more precise than a decimal approximation.

在IGCSE考试中，你可能会被要求以根式形式保留答案。这通常比十进制近似值更简单、更准确。

5. The Discriminant | 判别式

The discriminant is the expression b² – 4ac inside the quadratic formula. It tells us the number and type of roots without fully solving the equation.

判别式是二次公式中根号内的表达式 b² – 4ac。它不需要完全解方程就能告诉我们根的个数和类型。

If b² – 4ac > 0, the equation has two distinct real roots. If b² – 4ac = 0, it has exactly one repeated real root. If b² – 4ac < 0, there are no real roots.

若 b² – 4ac > 0，则方程有两个不同的实数根。若 b² – 4ac = 0，则方程恰有一个重根。若 b² – 4ac < 0，则没有实数根。

Discriminant	Nature of Roots	Graph
b² – 4ac > 0	Two distinct real roots	Crosses the x-axis twice
b² – 4ac = 0	One repeated root	Touches the x-axis once
b² – 4ac < 0	No real roots	Never meets the x-axis

The discriminant is one of the most frequently tested ideas in IGCSE quadratic questions, so make sure you can apply all three cases confidently.

判别式是IGCSE二次方程考题中最常考的概念之一，请务必熟练掌握应用这三种情况。

6. The Shape of a Quadratic Graph | 二次图像的基本形状

The graph of a quadratic function y = ax² + bx + c is a smooth curve called a parabola. Its shape is symmetric and has exactly one turning point, known as the vertex.

二次函数 y = ax² + bx + c 的图像是一条平滑曲线，称为抛物线。它是对称的，并且只有一个转向点，称为顶点。

If a > 0, the parabola opens upwards and the vertex is the lowest point, giving a U-shape. If a < 0, it opens downwards and the vertex is the highest point, giving an n-shape.

若 a > 0，抛物线开口向上，顶点为最低点，呈 U 形。若 a < 0，开口向下，顶点为最高点，呈 n 形。

The axis of symmetry is the vertical line passing through the vertex. Its equation is x = -b/(2a). This value is also the x-coordinate of the vertex.

对称轴是经过顶点的竖直线，其方程为 x = -b/(2a)。该值也是顶点的 x 坐标。

For y = x² – 4x + 3, a = 1, so the vertex x-coordinate is 4/2 = 2. Substituting x = 2 gives y = -1, so the vertex is (2, -1).

对于 y = x² – 4x + 3，a = 1，因此顶点 x 坐标为 4/2 = 2。代入 x = 2 得 y = -1，所以顶点为 (2, -1)。

Sketching a parabola requires only a few key points: the vertex, the y-intercept, and any x-intercepts if they exist.

画抛物线草图只需要几个关键点：顶点、y 轴截距以及存在的 x 轴截距。

7. Roots, Intercepts and the Vertex | 根、截距与顶点

The x-intercepts of a quadratic graph are the roots of the equation ax² + bx + c = 0. These are the points where y = 0. The y-intercept is always c, because when x = 0, y = c.

二次图像的 x 轴截距就是方程 ax² + bx + c = 0 的根。这些点是 y = 0 的位置。y 轴截距始终是 c，因为当 x = 0 时，y = c。

To find the vertex from the standard form, use x = -b/(2a), then substitute to find y. This method is quick and does not require completing the square.

要从标准形式求顶点，可使用 x = -b/(2a)，然后代入求 y。此方法快速，无需配方。

If the roots are known, the x-coordinate of the vertex is simply the midpoint of the two roots. For example, if the roots are 1 and 5, the vertex lies on the line x = 3.

如果根已知，顶点的 x 坐标就是两个根的中点。例如，若根为 1 和 5，则顶点位于直线 x = 3 上。

This relationship between roots and the vertex is very useful in exam questions that ask you to sketch a graph from given intercepts.

根与顶点之间的这种关系在“根据给定截距画图”的考题中非常有用。

8. Graphical Transformations | 图像变换

Quadratic graphs can be transformed by simple changes to the equation. Adding a constant outside the square moves the graph vertically, while adding a constant inside moves it horizontally.

二次图像可以通过对方程进行简单修改来变换。在完全平方外加上一个常数会使图像上下移动，而在括号内加上常数则会使图像左右移动。

For y = x² + k, the graph shifts up by k units if k > 0 and down by k units if k < 0. For y = (x - h)², the graph shifts right by h units if h > 0 and left by h units if h < 0.

对于 y = x² + k，若 k > 0 则图像上移 k 个单位，若 k < 0 则下移 k 个单位。对于 y = (x - h)²，若 h > 0 则图像右移 h 个单位，若 h < 0 则左移 h 个单位。

Multiplying by a negative value reflects the graph in the x-axis. For example, y = -x² is the reflection of y = x² with vertex still at the origin but opening downwards.

乘以负数会使图像关于 x 轴反射。例如，y = -x² 是 y = x² 的反射图像，顶点仍在原点，但开口向下。

Scaling by a factor a changes the steepness of the parabola. If |a| > 1, the graph is narrower; if 0 < |a| < 1, it is wider.

乘以比例因子 a 会改变抛物线的陡峭程度。若 |a| > 1，图像更窄；若 0 < |a| < 1，图像更宽。

9. Finding the Equation of a Quadratic Graph | 求二次图像的方程

Given a parabola on a graph, you can often work backwards to find its equation. The form you choose depends on which points are clearly visible.

给定一条抛物线，你通常可以反向求出其方程。选择哪种形式取决于图上哪些点清晰可见。

If you know the vertex (h, k), use the completed square form y = a(x – h)² + k. Then substitute another known point to find the value of a.

若你知道顶点 (h, k)，则使用顶点式 y = a(x – h)² + k。然后代入另一个已知点来求 a 的值。

If you know the two roots, say 2 and 5, write y = a(x – 2)(x – 5). Substitute a third point to find a. For example, if the graph passes through (0, 10), then 10 = a(-2)(-5) = 10a, so a = 1.

若你知道两个根，比如 2 和 5，可写成 y = a(x – 2)(x – 5)。代入第三个点求 a。例如，若图像经过 (0, 10)，则 10 = a(-2)(-5) = 10a，所以 a = 1。

Always use the y-intercept to verify your final equation. A quick check can prevent algebraic mistakes in your answer.

始终用 y 轴截距来验证最终方程。快速检查可以避免代数错误。

10. Real-World Modelling | 实际应用建模

Quadratic equations appear throughout science and business. Projectile motion, profit maximisation and area problems all give rise to quadratic models.

二次方程贯穿科学和商业领域。抛体运动、利润最大化和面积问题都会产生二次模型。

For a ball thrown upwards, the height h after t seconds is often h = -gt² + vt + h₀, where g is gravity, v is initial speed and h₀ is the starting height. The maximum height occurs at the vertex.

对于向上抛出的球，t 秒后的高度 h 通常为 h = -gt² + vt + h₀，其中 g 是重力，v 是初速度，h₀ 是初始高度。最大高度出现在顶点处。

To solve such problems, identify the meaningful variables from the question, write a quadratic equation, then use the method that is most efficient: factorising, completing the square or the quadratic formula.

要解决这类问题，先从题目中找出有意义的变量，写出二次方程，然后选择最有效的方法：因式分解、配方或二次公式。

Always consider the real-world context. A negative solution for time or length may be mathematically valid but physically meaningless, so reject it if necessary.

始终考虑实际情境。时间或长度为负数可能在数学上成立，但在物理上没有意义，因此必要时需舍去。

Practice with past paper questions is the best way to master quadratics. Try to solve each equation by two different methods to deepen your understanding.

练习历年真题是掌握二次方程的最佳途径。尝试用两种不同方法解同一方程，以加深理解。

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IGCSE Maths: Exploring Quadratic Equations & Graphs | IGCSE数学：探究二次方程与图像