Mastering Quadratic Equations and Parabolic Graphs for IGCSE Mathematics | IGCSE 数学：掌握二次方程与抛物线图像

📚 Mastering Quadratic Equations and Parabolic Graphs for IGCSE Mathematics | IGCSE 数学：掌握二次方程与抛物线图像

Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebraic manipulation to coordinate geometry and real-world modelling. A strong understanding of quadratics helps you solve equations, sketch parabolas, and handle applied problems with confidence.

二次方程贯穿 IGCSE 数学大纲，从代数运算到坐标几何以及实际建模均有涉及。扎实掌握二次方程可以帮助你自信地解方程、绘制抛物线图像并解决应用题。

1. What You Need to Know | 你需要掌握的内容

In the IGCSE Mathematics specification, you are expected to recognise a quadratic expression, solve a quadratic equation by factorisation, completing the square, and the quadratic formula, and understand the shape and key features of the graph y = ax² + bx + c. You should also be able to use the discriminant to decide how many real roots an equation has.

在 IGCSE 数学考试中，你需要识别二次表达式，会用因式分解法、配方法和二次公式解二次方程，并理解图像 y = ax² + bx + c 的形状和关键特征。你还需要会用判别式判断方程有多少个实数根。

The standard form of a quadratic equation is ax² + bx + c = 0, where a ≠ 0. If a = 0, the equation becomes linear, so this condition is always assumed in quadratic problems.

二次方程的标准形式是 ax² + bx + c = 0，其中 a ≠ 0。如果 a = 0，方程就变成了一元一次方程，因此在二次方程问题中总是默认 a 不为零。

2. Standard Form and Key Terms | 标准形式与关键术语

A quadratic expression contains a term in x² as its highest power. The coefficient a controls the direction and width of the parabola. When a > 0, the graph is a ‘smile’ shape; when a < 0, it is a ‘frown’ shape. The constant term c gives the y-intercept, because substituting x = 0 gives y = c.

二次表达式的最高次数是 x² 项。系数 a 控制抛物线的开口方向和宽窄。当 a > 0 时，图像呈 ‘微笑’ 形状；当 a < 0 时，图像呈 ‘皱眉’ 形状。常数项 c 表示 y 轴截距，因为代入 x = 0 时 y = c。

The roots, solutions, or x-intercepts are the values of x that make y = 0. A quadratic equation can have two real roots, one repeated root, or no real roots. These cases correspond to the graph cutting the x-axis twice, touching it once, or not meeting it at all.

根、解或 x 轴截距是使 y = 0 的 x 值。二次方程可以有两个实数根、一个重根或没有实数根。这三种情况分别对应图像与 x 轴相交两次、相切一次或完全不相交。

3. Solving by Factorisation | 因式分解法

Factorisation is usually the fastest method when the quadratic can be written as a product of two linear brackets. For example, x² + 5x + 6 = 0 can be factorised as (x + 2)(x + 3) = 0. Setting each bracket to zero gives x = -2 or x = -3.

当二次方程可以写成两个一次括号的乘积时，因式分解通常是最快的方法。例如，x² + 5x + 6 = 0 可以因式分解为 (x + 2)(x + 3) = 0。令每个括号分别等于零，得到 x = -2 或 x = -3。

Always check that the equation is equal to zero before factorising. If it is not, rearrange all terms to one side first. Also check for common factors: for 2x² + 8x = 0, factor out 2x to get 2x(x + 4) = 0, giving x = 0 or x = -4.

因式分解前务必确认方程等于零。如果不为零，应先把所有项移到一边。还要检查公因式：例如 2x² + 8x = 0，先提取公因式 2x，得到 2x(x + 4) = 0，解得 x = 0 或 x = -4。

4. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x + p)² + q. This is useful when factorisation is not obvious and when you need to find the turning point. For example, x² + 6x + 2 = 0 can be written as (x + 3)² – 7 = 0.

配方法将二次式改写为 a(x + p)² + q 的形式。当因式分解不明显或需要求顶点时，配方法非常有用。例如，x² + 6x + 2 = 0 可以写成 (x + 3)² – 7 = 0。

To solve (x + 3)² – 7 = 0, add 7 to both sides, take the square root, and remember the ± sign: x + 3 = ±√7, so x = -3 ± √7. The same method reveals the vertex of the parabola y = x² + 6x + 2 as (-3, -7).

要解 (x + 3)² – 7 = 0，先两边加 7，再开平方，并记住 ± 号：x + 3 = ±√7，所以 x = -3 ± √7。同样的方法还可得出抛物线 y = x² + 6x + 2 的顶点坐标为 (-3, -7)。

5. The Quadratic Formula | 二次公式

The quadratic formula solves any quadratic equation of the form ax² + bx + c = 0. It states that x = (-b ± √(b² – 4ac)) / 2a. This formula is given in the IGCSE examination formula sheet, but you must know how to substitute values correctly.

二次公式可以解任何形如 ax² + bx + c = 0 的二次方程。公式为 x = (-b ± √(b² – 4ac)) / 2a。IGCSE 考试公式表会给出该公式，但你必须知道如何正确代入数值。

x = (-b ± √(b² – 4ac)) / 2a

For example, to solve 2x² – 3x – 5 = 0, identify a = 2, b = -3, c = -5. Then x = (3 ± √((-3)² – 4 × 2 × (-5))) / (2 × 2) = (3 ± √49) / 4 = (3 ± 7) / 4, giving x = 2.5 or x = -1.

例如，解方程 2x² – 3x – 5 = 0，先确定 a = 2，b = -3，c = -5。代入公式得 x = (3 ± √((-3)² – 4 × 2 × (-5))) / (2 × 2) = (3 ± √49) / 4 = (3 ± 7) / 4，解得 x

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