Mastering Quadratic Equations | 掌握二次方程

📚 Mastering Quadratic Equations | 掌握二次方程

Quadratic equations appear in almost every IGCSE Mathematics paper, and mastering them is essential for topics ranging from algebra to coordinate geometry. This revision guide breaks down the key methods, graphs, inequalities, and common pitfalls so you can approach any quadratic problem with confidence.

二次方程几乎出现在每一份 IGCSE 数学试卷中，掌握它是学好代数、坐标几何等后续知识的基础。本复习指南将系统梳理二次方程的解法、图像、不等式及常见易错点，帮助你在考场上从容应对各类题型。

1. What Is a Quadratic Equation | 什么是二次方程

A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the unknown variable x is 2. Its general form is ax² + bx + c = 0, where a, b and c are constants and a must not be zero.

二次方程是最高次数为 2 的多项式方程，即未知数 x 的最高次幂为 2。它的一般形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a 不能为 0。

If a = 0, the equation becomes linear, because the x² term disappears. A quadratic equation always has two roots (solutions), although they may be equal or not real, as we will see in the discriminant section.

若 a = 0，方程会退化为一次方程，因为 x² 项消失了。二次方程总是有两个根（解），但这两个根可能相等，也可能不是实数，这将在后面“判别式”一节中详细讨论。

The standard form is ax² + bx + c = 0 with a ≠ 0; the curve represented by y = ax² + bx + c is called a parabola.
标准形式为 ax² + bx + c = 0，其中 a ≠ 0；由 y = ax² + bx + c 表示的曲线称为抛物线。

2. Solving by Factorisation | 因式分解法

Factorisation is often the fastest method when a, b and c are integers with simple factors. The idea is to rewrite ax² + bx + c as the product of two linear expressions, then use the zero product property: if AB = 0, then A = 0 or B = 0.

当 a、b、c 为整数且因式较简单时，因式分解法往往是最快的方法。其核心是将 ax² + bx + c 改写为两个一次式的乘积，然后利用零乘积性质：若 AB = 0，则 A = 0 或 B = 0。

Solve x² – 5x + 6 = 0. We look for two numbers that multiply to 6 and add to -5: those numbers are -2 and -3. Hence (x – 2)(x – 3) = 0, so x = 2 or x = 3.

解方程 x² – 5x + 6 = 0。我们需要找两个数，它们的乘积为 6，和为 -5：这两个数就是 -2 和 -3。因此 (x – 2)(x – 3) = 0，所以 x = 2 或 x = 3。

x² – 5x + 6 = 0 → (x – 2)(x – 3) = 0 → x = 2 或 x = 3

Always expand your factorised answer mentally to check: (x – 2)(x – 3) expands to x² – 3x – 2x + 6 = x² – 5x + 6. If the coefficient of x² is not 1, you may need to factorise by grouping or use trial and error.

完成后务必心算展开检查：(x – 2)(x – 3) 展开得 x² – 3x – 2x + 6 = x² – 5x + 6。若 x² 的系数不为 1，则可能需要分组分解或尝试因式分解。

3. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² + q. This method is powerful because it always works without relying on factorisation, and it directly reveals the turning point of the parabola.

配方法将二次式改写为 (x + p)² + q 的形式。这种方法非常强大，因为它不依赖因式分解，且能直接揭示抛物线的顶点坐标。

For x² + bx + c, take half of b, write (x + b/2)², then subtract (b/2)² and add the original c. For example, x² + 6x + 4 = 0 becomes (x + 3)² – 9 + 4 = 0, so (x + 3)² – 5 = 0.

对于 x² + bx + c，取 b 的一半，写出 (x + b/2)²，然后减去 (b/2)² 再加上原来的 c。例如，x² + 6x + 4 = 0 可化为 (x + 3)² – 9 + 4 = 0，即 (x + 3)² – 5 = 0。

x² + 6x + 4 = 0 → (x + 3)² = 5 → x = -3 ± √5

The vertex form y = a(x – h)² + k shows that the turning point is at (h, k), and the equation x = h is the axis of symmetry. In the example above, the vertex is (-3, -5).

顶点式 y = a(x – h)² + k 表明抛物线顶点为 (h, k)，直线 x = h 为对称轴。在上面的例子中，顶点坐标为 (-3, -5)。

4. The Quadratic Formula | 求根公式

When factorisation is difficult or impossible, the quadratic formula gives the solutions to any quadratic equation. For ax² + bx + c = 0, the roots are given by the following formula.

当因式分解困难或无法进行时，求根公式可以给出任何二次方程的解。对于 ax² + bx + c = 0，其根由以下公式给出。

x = (-b ± √(b² – 4ac)) / (2a)

To use the formula, substitute the values of a, b and c carefully and simplify. Remember that the ± sign gives two solutions: one using the plus sign and one using the minus sign.

使用公式时，请仔细代入 a、b、c 的值并化简。注意 ± 符号代表两个解：一个取加号，一个取减号。

Solve 2x² + 3x – 2 = 0. Here a = 2, b = 3, c = -2. Substitution gives x = (-3 ± √(9 + 16)) / 4 = (-3 ± 5) / 4, so x = 1/2 or x = -2.

解方程 2x² + 3x – 2 = 0。这里 a = 2，b = 3，c = -2。代入得 x = (-3 ± √(9 + 16)) / 4 = (-3 ± 5) / 4，因此 x = 1/2 或 x = -2。

5. The Discriminant | 判别式

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

求根公式中的 b² – 4ac 称为判别式，通常用希腊字母 Δ 表示。它可以帮助我们在不解方程的情况下判断二次方程实数根的个数。

Δ = b² – 4ac

If Δ > 0, there are two distinct real roots. If Δ = 0, there is exactly one repeated real root (also called a double root). If Δ < 0, there are no real roots, meaning the graph never crosses the x-axis.

若 Δ > 0，方程有两个不相等的实数根；若 Δ = 0，方程有一个重根（等根）；若 Δ < 0，方程没有实数根，即图像与 x 轴没有交点。

判别式 Δ	根的情况	图像与 x 轴交点
Δ > 0	两个不等实根	两个交点
Δ = 0	一个重根	一个切点
Δ < 0	无实数根	无交点

A common IGCSE question asks: find the value of k for which kx² + 4x + 2 = 0 has equal roots. Set Δ = 16 – 8k = 0, giving k = 2.

常见的 IGCSE 考题：求 kx² + 4x + 2 = 0 有等根时 k 的值。令 Δ = 16 – 8k = 0，解得 k = 2。

6. Roots and Coefficients | 根与系数的关系

For a quadratic equation ax² + bx + c = 0, the sum and product of the roots have simple relationships with the coefficients. If the roots are α and β, then α + β = -b/a and αβ = c/a.

对于二次方程 ax² + bx + c = 0，两根之和与两根之积与系数之间存在简单关系。设两根为 α 和 β，则 α + β = -b/a，αβ = c/a。

α + β = -b/a，αβ = c/a

For x² – 5x + 6 = 0, the roots are 2 and 3. Their sum is 5 = -(-5)/1 and their product is 6 = 6/1, confirming the relationships. These identities are useful for constructing a quadratic equation when the roots are known.

对于 x² – 5x + 6 = 0，两根为 2 和 3。其和为 5 = -(-5)/1，积为 6 = 6/1，验证了上述关系。当已知根时，可利用这些关系构造二次方程。

If you know the roots are p and q, the equation can be written as x² – (p + q)x + pq = 0. For example, roots of 4 and -1 give x² – 3x – 4 = 0.

若已知两根为 p 和 q，则方程可写为 x² – (p + q)x + pq = 0。例如，根为 4 和 -1 时，方程为 x² – 3x – 4 = 0。

7. Sketching Quadratic Graphs | 二次函数图像

The graph of y = ax² + bx + c is a parabola. If a > 0, the parabola opens upward and has a minimum point; if a < 0, it opens downward and has a maximum point. The sign of a is therefore the first thing to check when sketching.

函数 y = ax² + bx + c 的图像是抛物线。若 a > 0，抛物线开口向上，有最小值点；若 a < 0，抛物线开口向下，有最大值点。因此画图时首先要看 a 的符号。

The key features of a quadratic graph are the roots (where the curve crosses the x-axis), the y-intercept (where it crosses the y-axis at (0, c)), the axis of symmetry x = -b/(2a), and the turning point. To find the turning point, complete the square or substitute x = -b/(2a) back into the equation.

二次函数图像的关键特征包括：根（曲线与 x 轴的交点）、y 轴截距（与 y 轴交于 (0, c)）、对称轴 x = -b/(2a) 以及顶点。求顶点坐标可通过配方法，或将 x = -b/(2a) 代回原方程计算。

Find the roots by setting y = 0 and solving the quadratic equation.
求根：令 y = 0，解对应的二次方程。
Find the y-intercept by setting x = 0, giving (0, c).
求 y 轴截距：令 x = 0，得到 (0, c)。
Find the turning point and plot it as the highest or lowest point of the curve.
求顶点坐标，并在图中标出曲线的最高点或最低点。

8. Quadratic Inequalities | 二次不等式

To solve a quadratic inequality such as x² – 5x + 6 > 0, first solve the corresponding equation to find the critical values. Then sketch the parabola or use a number line to determine which intervals satisfy the inequality.

解二次不等式（如 x² – 5x + 6 > 0）时，先解对应的方程以求出临界值，然后画抛物线草图或使用数轴来判断哪些区间满足不等式。

Since x² – 5x + 6 = (x – 2)(x – 3), the critical values are x = 2 and x = 3. The parabola opens upward, so the expression is positive outside the roots: x < 2 or x > 3.

因为 x² – 5x + 6 = (x – 2)(x – 3)，临界值为 x = 2 和 x = 3。抛物线开口向上，因此表达式在两根之外为正：x < 2 或 x > 3。

x² – 5x + 6 > 0 → x < 2 或 x > 3

For the inequality x² – 5x + 6 < 0, the solution lies between the critical values: 2 < x < 3. Remember to check the inequality sign carefully: strict inequalities use < or >, while non-strict inequalities use ≤ or ≥ and include the critical values.

对于不等式 x² – 5x + 6 < 0，解位于两个临界值之间：2 < x < 3。注意仔细核对不等号：严格不等号用 < 或 >，非严格不等号用 ≤ 或 ≥ 并包含临界值。

9. Word Problems | 实际应用题

Many IGCSE exam questions require translating a real-life situation into a quadratic equation. Read the problem carefully, define a variable, write the equation from the given relationships, and solve it.

许多 IGCSE 考题要求将实际情境转化为二次方程。请仔细读题，设出未知数，根据题中关系列出方程，然后求解。

A rectangle has a length that is 3 cm longer than its width, and its area is 28 cm². Let the width be x cm; then the length is (x + 3) cm, so x(x + 3) = 28.

一个长方形的长比宽长 3 cm，面积为 28 cm²。设宽为 x cm，则长为 (x + 3) cm，于是 x(x + 3) = 28。

x² + 3x – 28 = 0 → (x + 7)(x – 4) = 0 → x = 4 或 x = -7

The solution x = -7 is discarded because a length cannot be negative in this context. Since the problem asks for a physical dimension, only the positive solution x = 4 is valid, so the width is 4 cm and the length is 7 cm.

解 x = -7 因为长度不可能为负数而被舍去。由于题目求的是实际长度，只有正数解 x = 4 有效，因此宽为 4 cm，长为 7 cm。

10. Exam Tips and Common Mistakes | 考试要点与常见错误

One of the most frequent errors is forgetting that a must not be zero. Another is making sign mistakes when substituting negative values into the quadratic formula, especially when b is negative. Always write out each substitution step clearly.

最常见的错误之一是忘记 a 不能为 0。另一个常见错误是在求根公式中代入负数时出现符号错误，尤其是 b 为负数时。请务必清晰地写出每一步代入过程。

When a question says “equal roots” or “one repeated root”, this means Δ = 0. When it says “no real roots”, this means Δ < 0.
当题目说“等根”或“一个重根”时，意味着 Δ = 0；当题目说“无实数根”时，意味着 Δ < 0。
Always check factorised answers by expanding back. If the expansion does not match the original expression, the factorisation is wrong.
因式分解后务必展开检验。若展开结果与原式不符，则分解有误。
In word problems, interpret both solutions in the context of the question and reject impossible ones, such as negative lengths or times.
在应用题中，结合题意解释两个解，舍去不合理的解，例如负数的长度或时间。

Use your calculator to verify roots when allowed, but always show full working for method marks. Practising with past papers will help you recognise the common patterns of quadratic questions, from factorisation to discriminant problems.

在允许使用计算器时，可以用它验证根，但务必写出完整过程以获得方法分。通过练习历年真题，你将熟悉二次方程常见题型模式，从因式分解到判别式问题。

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Mastering Quadratic Equations | 掌握二次方程