Quadratic Functions and Equations: A Complete Guide for IGCSE | 二次函数与方程：IGCSE 完全指南

📚 Quadratic Functions and Equations: A Complete Guide for IGCSE | 二次函数与方程：IGCSE 完全指南

Quadratic functions and equations are one of the most important topics in the IGCSE Mathematics syllabus. They appear in almost every exam paper, often in both non-calculator and calculator sections. Understanding the core concepts, solving methods, and graphical interpretations is essential for achieving a high grade.

二次函数与方程是 IGCSE 数学大纲中最重要的主题之一。几乎每份试卷都会考到，既出现在非计算器部分，也出现在计算器部分。理解核心概念、求解方法以及图形解释，对取得高分至关重要。

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

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The value of a cannot be zero, because then the equation would become linear, not quadratic.

二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 是常数，且 a ≠ 0。a 的值不能为零，否则方程就会变成一次方程而不是二次方程。

a is the coefficient of x² and must be non-zero.
b is the coefficient of x.
c is the constant term.

a 是 x² 的系数，必须非零。
b 是 x 的系数。
c 是常数项。

For example, 2x² – 5x + 3 = 0 is a quadratic equation with a = 2, b = -5, c = 3. The solutions to this equation are called its roots.

例如，2x² – 5x + 3 = 0 是一个二次方程，其中 a = 2，b = -5，c = 3。这个方程的解称为它的根。

2. The Zero Product Property | 零乘积性质

The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. In algebraic terms, if AB = 0, then A = 0 or B = 0. This property is the foundation of solving quadratic equations by factorisation.

零乘积性质指出：如果两个因式的乘积为零，那么至少其中一个因式必须为零。用代数语言说，如果 AB = 0，则 A = 0 或 B = 0。这一性质是通过因式分解求解二次方程的基础。

This rule only works when the product equals exactly zero. If the product equals any other number, the property does not apply directly.

这条规则仅当乘积恰好等于零时成立。如果乘积等于其他任何数，该性质不能直接使用。

3. Solving by Factorisation | 因式分解法求解

Factorisation is the preferred method when the quadratic can be written as a product of two linear factors. For a typical equation x² + bx + c = 0, you need to find two numbers that multiply to give c and add to give b.

当二次方程可以写成两个一次因式的乘积时，因式分解是首选方法。对于典型的方程 x² + bx + c = 0，你需要找到两个数，它们相乘等于 c，相加等于 b。

Consider x² – 7x + 10 = 0. We need two numbers that multiply to 10 and add to -7. These numbers are -2 and -5. Therefore the equation becomes (x – 2)(x – 5) = 0. By the zero product property, x = 2 or x = 5.

考虑 x² – 7x + 10 = 0。我们需要找到两个数，相乘为 10，相加为 -7。这两个数是 -2 和 -5。因此方程变为 (x – 2)(x – 5) = 0。根据零乘积性质，x = 2 或 x = 5。

When the coefficient a is not 1, the process is slightly more involved, but the same principle applies: write the quadratic as a product of two brackets, then set each bracket equal to zero.

当系数 a 不等于 1 时，过程稍复杂，但原理相同：将二次式写成两个括号的乘积，然后让每个括号等于零。

4. Completing the Square | 配方法

Completing the square is another algebraic technique that rewrites a quadratic in the form a(x – h)² + k. This form is especially useful for finding the turning point of a parabola, as well as for solving equations where factorisation is complicated or impossible.

配方是另一种代数技巧，它将二次式改写为 a(x – h)² + k 的形式。这种形式在找抛物线顶点时特别有用，同时在因式分解复杂或无法分解时也用于解方程。

For example, take x² + 6x + 5 = 0. First, halve the coefficient of x, which is 3, and write (x + 3)². This equals x² + 6x + 9, which is 4 more than the original expression. So we adjust: (x + 3)² – 4 = 0. Then (x + 3)² = 4, so x + 3 = ±2, giving x = -1 or x = -5.

例如，取 x² + 6x + 5 = 0。先将 x 的系数除以 2，得到 3，写成 (x + 3)²。这等于 x² + 6x + 9，比原表达式多 4。因此要调整：写成 (x + 3)² – 4 = 0。然后 (x + 3)² = 4，所以 x + 3 = ±2，得到 x = -1 或 x = -5。

The expression inside the bracket is always x plus or minus half of b, and the adjustment term is the square of that half, adjusted by the original constant.

括号内的表达式总是 x 加上或减去 b 的一半，而修正项就是该一半的平方，再根据原常数进行调整。

5. The Quadratic Formula | 二次求根公式

For any quadratic equation ax² + bx + c = 0, the solutions can be found using the quadratic formula:

对于任意二次方程 ax² + bx + c = 0，可以使用二次求根公式来求解：

x = (−b ± √(b² − 4ac)) / (2a)

This formula works for all quadratic equations, whether the solutions are real or complex (though complex roots are generally not required at IGCSE level). You must be careful to substitute the values correctly and be precise with signs.

这个公式适用于所有二次方程，无论解是实数还是复数（不过 IGCSE 水平一般不要求复数根）。注意必须正确代入数值并小心处理符号。

Example: solve 2x² – 4x – 6 = 0 using the formula. Here a = 2, b = -4, c = -6. Substitute into the formula:

示例：用公式求解 2x² – 4x – 6 = 0。这里 a = 2，b = -4，c = -6。代入公式：

x = (−(−4) ± √((−4)² − 4×2×(−6))) / (2×2) = (4 ± √(16 + 48)) / 4 = (4 ± 8) / 4

This gives x = 3 or x = -1. Always check your answers by substituting them back into the original equation.

解得 x = 3 或 x = -1。务必把答案代回原方程进行验算。

6. The Discriminant and the Nature of Roots | 判别式与根的性质

The expression b² – 4ac inside the quadratic formula is called the discriminant. It tells us how many real roots a quadratic equation has without actually solving it.

二次求根公式中的 b² – 4ac 部分称为判别式。它不需要实际解方程就能告诉我们二次方程有多少个实数根。

Discriminant Δ = b² − 4ac	Nature of roots	根的性质
Δ > 0	Two distinct real roots	两个不同的实数根
Δ = 0	One repeated real root	一个重根（两个相等实数根）
Δ < 0	No real roots	没有实数根

For example, the equation x² – 6x + 9 = 0 has Δ = 36 – 36 = 0, meaning it has exactly one repeated root, x = 3.

例如，方程 x² – 6x + 9 = 0 的判别式 Δ = 36 – 36 = 0，意味着它只有一个重根 x = 3。

7. Graphs of Quadratic Functions | 二次函数图像

The graph of a quadratic function y = ax² + bx + c is a curve called a parabola. The sign of a determines the direction in which the parabola opens.

二次函数 y = ax² + bx + c 的图像是一条称为抛物线的曲线。a 的正负决定抛物线开口的方向。

If a > 0, the parabola opens upwards and has a minimum point.
If a < 0, the parabola opens downwards and has a maximum point.

如果 a > 0，抛物线开口向上，有最小值点。
如果 a < 0，抛物线开口向下，有最大值点。

The turning point of the parabola is also called the vertex. Its x-coordinate can be found using the formula x = -b / (2a). Substituting this back into the quadratic gives the y-coordinate of the vertex.

抛物线的顶点称为转向点。其 x 坐标可以用公式 x = -b / (2a) 求得，代回二次函数即可得到顶点的 y 坐标。

8. Transformations of Graphs | 图像的变换

In IGCSE, you may be asked to interpret transformations of the quadratic graph. The general form y = (x – h)² + k represents a parabola with vertex at the point (h, k).

在 IGCSE 中，你可能会被要求理解二次函数图像的变换。一般形式 y = (x – h)² + k 表示顶点在点 (h, k) 的抛物线。

y = f(x) + k shifts the graph vertically by k units (up if k > 0, down if k < 0).
y = f(x + h) shifts the graph horizontally by h units (left if h > 0, right if h < 0).
y = −f(x) reflects the graph in the x-axis.

y = f(x) + k 将图像垂直平移 k 个单位（k > 0 向上，k < 0 向下）。
y = f(x + h) 将图像水平平移 h 个单位（h > 0 向左，h < 0 向右）。
y = −f(x) 将图像关于 x 轴反射。

It is worth practising sketching these transformations on graph paper or using a graph-plotting tool to gain confidence.

建议在坐标纸上或使用绘图工具练习画出这些变换，以增强信心。

9. Intersection with the x-axis and y-axis | 与 x 轴和 y 轴的交点

The points where the curve y = ax² + bx + c crosses the x-axis are the solutions to the equation ax² + bx + c = 0. Therefore, finding the roots is equivalent to finding the x-intercepts.

曲线 y = ax² + bx + c 与 x 轴的交点正是方程 ax² + bx + c = 0 的解。因此，求根等价于求 x 轴截距。

The y-intercept is simply the value of y when x = 0, which is always c. This point is (0, c).

y 轴截距就是 x = 0 时的 y 值，始终是 c。该点为 (0, c)。

When the discriminant is positive, the curve crosses the x-axis at two distinct points. When the discriminant is zero, the curve touches the x-axis at exactly one point. When the discriminant is negative, the curve does not touch the x-axis at all.

当判别式为正时，曲线与 x 轴相交于两个不同点。判别式为零时，曲线与 x 轴恰好相切于一点。判别式为负时，曲线完全不与 x 轴相交。

10. Solving Quadratic Inequalities | 解二次不等式

Quadratic inequalities, such as ax² + bx + c > 0 or ax² + bx + c < 0, require solving the corresponding equation first, then determining the sign of the quadratic in different intervals.

二次不等式，如 ax² + bx + c > 0 或 ax² + bx + c < 0，需要先解对应的方程，再判断二次式在各个区间内的正负号。

Consider x² – 5x + 6 < 0. Factorise to get (x - 2)(x - 3) < 0. The critical points are x = 2 and x = 3. Test values in the intervals below 2, between 2 and 3, and above 3. The product is negative only between 2 and 3, so the solution is 2 < x < 3.

考虑 x² – 5x + 6 < 0。因式分解得 (x - 2)(x - 3) < 0。关键点为 x = 2 和 x = 3。在小于 2、2 到 3 之间、大于 3 的区间内分别取测试值。乘积仅在 2 到 3 之间为负，所以解为 2 < x < 3。

Sketching a quick graph of the parabola is often the most reliable way to avoid sign errors in inequalities.

快速画出抛物线草图通常是避免不等式符号错误的最可靠方法。

11. Applications in Problem Solving | 应用题中的实际应用

Quadratic functions frequently appear in word problems, such as finding the maximum area of a rectangle with a fixed perimeter, or determining when a projectile hits the ground. This requires translating the problem into a quadratic equation and interpreting the solutions in context.

二次函数经常出现在应用题中，例如在固定周长下求矩形的最大面积，或确定抛射物何时落地。这需要将问题转化为二次方程，并结合上下文解释解的含义。

For a rectangle with perimeter 20 m, if one side is x, the other side is 10 – x. The area is A = x(10 – x) = 10x – x². Completing the square gives A = -(x – 5)² + 25. The maximum area is therefore 25 m² when x = 5, meaning a square.

对于周长为 20 m 的矩形，若一边为 x，则另一边为 10 – x。面积为 A = x(10 – x) = 10x – x²。配方得到 A = -(x – 5)² + 25。因此最大面积为 25 m²，此时 x = 5，即正方形。

Always discard solutions that do not make sense physically, such as negative lengths or times.

务必舍弃物理意义上不合理的解，例如负数长度或负数时间。

12. Common Mistakes and Exam Tips | 常见错误与考试技巧

One common mistake is forgetting to set the equation to zero before factorising. Another is losing the ± sign when taking square roots. Carefully rewriting the quadratic formula and substituting each coefficient step by step helps avoid sign errors.

一个常见错误是忘记在因式分解前将方程化为等于零的形式。另一个是开平方时丢失 ± 号。认真重写二次公式并逐步代入每个系数，有助于避免符号错误。

When completing the square, remember to divide every term by a before halving the x-coefficient, if a is not 1. Also, when using the discriminant, be careful with negative values of b and c.

配方法中，如果 a 不等于 1，需要先把每一项除以 a，再对 x 系数取半。另外，使用判别式时，要小心 b 和 c 为负数的情况。

Finally, always show your working clearly. In IGCSE exams, many marks are awarded for correct method even when the final answer is wrong.

最后，务必清晰展示解题步骤。在 IGCSE 考试中，即使最终答案错误，正确的方法也能获得大量分数。

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Quadratic Functions and Equations: A Complete Guide for IGCSE | 二次函数与方程：IGCSE 完全指南