IGCSE Mathematics: Mastering Quadratic Equations and Their Graphs | IGCSE数学：掌握二次方程及其图像

📚 IGCSE Mathematics: Mastering Quadratic Equations and Their Graphs | IGCSE数学：掌握二次方程及其图像

A quadratic equation is a polynomial equation of degree 2, generally written as ax² + bx + c = 0. Understanding how to solve and graph quadratics is a core skill for IGCSE Mathematics, as it appears in algebra, coordinate geometry, and problem-solving questions throughout the syllabus.

二次方程是最高次数为 2 的多项式方程，通常写作 ax² + bx + c = 0。掌握二次方程的求解与图像绘制是 IGCSE 数学的核心技能，它在代数、坐标几何及全章应用题中反复出现。

1. The Standard Form of a Quadratic Equation | 二次方程的标准形式

The standard form of a quadratic equation is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The value of a determines whether the parabola opens upward or downward.

二次方程的标准形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。a 的正负决定了抛物线的开口方向。

If a > 0, the graph is a U-shaped parabola with a minimum point.
If a < 0, the graph is an upside-down U-shaped parabola with a maximum point.
The constant c gives the y-intercept of the graph.
The roots of the equation correspond to the x-intercepts of the graph.

若 a > 0，图像为开口向上的抛物线，存在最小值点。
若 a < 0，图像为开口向下的抛物线，存在最大值点。
常数 c 表示图像与 y 轴的交点。
方程的根对应图像与 x 轴的交点。

2. Solving Quadratic Equations by Factorisation | 用因式分解法解二次方程

Factorisation is often the quickest method for solving quadratics when simple integer factors exist. We rewrite ax² + bx + c as a product of two linear expressions and apply the zero product property.

当二次式存在简单的整数因式时，因式分解通常是最快的解法。我们将 ax² + bx + c 改写为两个一次式的乘积，并运用零乘积性质。

If p × q = 0, then p = 0 or q = 0.

若 p × q = 0，则 p = 0 或 q = 0。

For example, solve x² − 7x + 12 = 0. We look for two numbers that multiply to 12 and add to −7: they are −3 and −4. Thus (x − 3)(x − 4) = 0, so x = 3 or x = 4.

例如，解 x² − 7x + 12 = 0。寻找两个数，它们相乘为 12，相加为 −7：即 −3 和 −4。因此 (x − 3)(x − 4) = 0，所以 x = 3 或 x = 4。

Always rearrange the equation so that one side equals zero before factorising.
Check your solutions by substituting them back into the original equation.
If the coefficient a ≠ 1, factorise carefully using pairs of factors of ac.

因式分解前务必先将方程整理为一边等于零。
将解代回原方程进行验证。
若 a ≠ 1，需仔细利用 ac 的因数对进行分解。

3. Solving by Completing the Square | 用配方法解二次方程

Completing the square rewrites a quadratic in the form a(x + p)² + q, which is especially useful for finding the vertex and solving equations that cannot be factorised.

配方法将二次式改写为 a(x + p)² + q 的形式，这对于求顶点和解决无法因式分解的方程特别有用。

For example, solve x² + 6x + 2 = 0. We first write x² + 6x = (x + 3)² − 9, so the equation becomes (x + 3)² − 7 = 0. Then (x + 3)² = 7, hence x = −3 ± √7.

例如，解 x² + 6x + 2 = 0。先将 x² + 6x 写成 (x + 3)² − 9，于是方程变为 (x + 3)² − 7 = 0。则 (x + 3)² = 7，因此 x = −3 ± √7。

For x² + bx, add and subtract (b/2)².
If a ≠ 1, factor out a from the x² and x terms first.
The vertex of y = a(x + p)² + q is at (−p, q).

对于 x² + bx，需加上并减去 (b/2)²。
若 a ≠ 1，先将 a 从 x² 与 x 项中提取出来。
y = a(x + p)² + q 的顶点为 (−p, q)。

4. The Quadratic Formula | 求根公式

The quadratic formula works for any quadratic equation, even when factorisation is difficult or impossible. For ax² + bx + c = 0, the solutions are:

求根公式适用于任何二次方程，即使在因式分解困难或不可能时也有效。对于 ax² + bx + c = 0，其解为：

x = (−b ± √(b² − 4ac)) / (2a)

The expression under the square root, b² − 4ac, is called the discriminant. It determines the nature of the roots without fully solving the equation.

根号下的表达式 b² − 4ac 称为判别式。它无需完全解方程即可判断根的性质。

Discriminant	Nature of Roots
b² − 4ac > 0	Two distinct real roots
b² − 4ac = 0	One repeated real root
b² − 4ac < 0	No real roots

判别式	根的性质
b² − 4ac > 0	两个不相等的实数根
b² − 4ac = 0	两个相等的实数根（重根）
b² − 4ac < 0	没有实数根

5. The Graph of a Quadratic Function | 二次函数的图像

The graph of y = ax² + bx + c is a smooth, symmetric curve called a parabola. Its shape is determined entirely by the coefficient a and the position of the vertex.

y = ax² + bx + c 的图像是一条平滑对称的曲线，称为抛物线。其形状完全由系数 a 和顶点的位置决定。

The y-intercept is the point (0, c).
The roots are the x-coordinates where the graph crosses the x-axis.
The axis of symmetry is the vertical line x = −b/(2a).
The vertex lies on the axis of symmetry and represents the maximum or minimum value of the function.

与 y 轴的交点为 (0, c)。
根是图像与 x 轴交点的横坐标。
对称轴为垂直线 x = −b/(2a)。
顶点位于对称轴上，代表函数的最大值或最小值。

6. Sketching Parabolas | 绘制抛物线草图

A clear sketch of a quadratic graph should show the y-intercept, the roots, and the vertex. Follow these steps to produce an accurate graph.

绘制二次函数图像的清晰草图应显示 y 截距、根和顶点。请按以下步骤画出准确图形。

Find the y-intercept by substituting x = 0.
Find the roots by solving ax² + bx + c = 0.
Find the vertex using x = −b/(2a), then compute y.
Plot the points and draw a smooth curve through them.

代入 x = 0 求 y 截距。
解方程 ax² + bx + c = 0 求根。
利用 x = −b/(2a) 求顶点横坐标，再计算 y 值。
描点并通过这些点绘制平滑曲线。

If the discriminant is negative, the parabola has no x-intercepts. In such cases, use the vertex and y-intercept to sketch the curve.

若判别式为负，抛物线没有 x 截距。此时可利用顶点和 y 截距来绘制草图。

7. Finding the Equation of a Parabola | 求抛物线的方程

Given the roots r₁ and r₂, a quadratic function can be written in factorised form: y = a(x − r₁)(x − r₂). To determine a, substitute the coordinates of another known point on the graph.

已知根 r₁ 和 r₂ 时，二次函数可写成因式形式：y = a(x − r₁)(x − r₂)。要确定 a 的值，可代入图像上另一个已知点的坐标。

For example, a parabola crosses the x-axis at x = −1 and x = 3, and passes through (0, −6). Then y = a(x + 1)(x − 3). Substituting (0, −6): −6 = a(1)(−3) = −3a, so a = 2. Therefore y = 2(x + 1)(x − 3).

例如，一条抛物线与 x 轴交于 x = −1 和 x = 3，并经过点 (0, −6)。则 y = a(x + 1)(x − 3)。代入 (0, −6)：−6 = a(1)(−3) = −3a，所以 a = 2。因此 y = 2(x + 1)(x − 3)。

8. Solving Quadratic Inequalities | 解二次不等式

Quadratic inequalities are solved by finding the roots first, then determining the intervals where the quadratic expression is positive or negative.

解二次不等式需先求根，再确定二次表达式为正或负的区间。

For example, solve x² − x − 6 > 0. Factorising gives (x − 3)(x + 2) > 0. The critical points are x = −2 and x = 3. Testing intervals yields x < −2 or x > 3.

例如，解 x² − x − 6 > 0。因式分解得 (x − 3)(x + 2) > 0。关键点为 x = −2 和 x = 3。测试区间后得到 x < −2 或 x > 3。

For ax² + bx + c > 0 with a > 0, the solution lies outside the roots.
For ax² + bx + c < 0 with a > 0, the solution lies between the roots.
Sketching the parabola can help visualise the solution set.

当 a > 0 时，ax² + bx + c > 0 的解在两根之外。
当 a > 0 时，ax² + bx + c < 0 的解在两根之间。
绘制抛物线草图有助于直观理解解集。

9. Solving Word Problems with Quadratics | 用二次方程解应用题

Quadratic equations are often used to model real-world situations such as areas, motion, and number problems. The key is to translate the information into an equation.

二次方程常用于建模现实情境，如面积、运动与数字问题。关键在于将信息转化为方程。

Example: The length of a rectangle is 4 cm longer than its width, and its area is 96 cm². Find the dimensions.

例：一个长方形的长比宽长 4 cm，面积为 96 cm²。求长和宽。

Let the width be w cm. Then the length is w + 4, so w(w + 4) = 96, which gives w² + 4w − 96 = 0. Factorising: (w + 12)(w − 8) = 0. Thus w = 8 (reject w = −12). The rectangle is 8 cm by 12 cm.

设宽为 w cm。则长为 w + 4，所以 w(w + 4) = 96，即 w² + 4w − 96 = 0。因式分解得 (w + 12)(w − 8) = 0。因此 w = 8（舍去 w = −12）。该长方形为 8 cm × 12 cm。

Define a variable for the unknown quantity.
Write an equation based on the given conditions.
Solve the quadratic and reject any answer that does not make sense in context.

为未知量设定变量。
根据已知条件写出方程。
解二次方程并舍去在情境中不合理的答案。

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IGCSE Mathematics: Mastering Quadratic Equations and Their Graphs | IGCSE数学：掌握二次方程及其图像