Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations appear in almost every IGCSE Mathematics examination. Whether you sit for Core or Extended papers, the ability to solve ax² + bx + c = 0 fluently is non-negotiable. This revision guide covers the three standard methods, the discriminant, the graph of a quadratic function, and the most common exam traps.

二次方程几乎出现在每一场 IGCSE 数学考试中。无论你参加 Core 还是 Extended 试卷，熟练求解 ax² + bx + c = 0 都是必备技能。本复习指南涵盖三种标准解法、判别式、二次函数图象以及最常见的考试陷阱。

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

A quadratic equation is any equation that can be rearranged into the general form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

二次方程是任何可以整理为一般形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。

The highest power of the unknown is 2, so the equation is said to be of degree 2.

未知数的最高次数为 2，因此称该方程为二次方程。
If a = 0, the equation becomes linear because the x² term disappears.

若 a = 0，x² 项消失，方程退化为一次方程。
A quadratic equation can have two distinct roots, one repeated root, or no real roots.

二次方程可以有两个不同的根、一个重根，或无实数根。
Every quadratic equation has exactly two roots if complex numbers are allowed, but IGCSE only considers real roots.

若允许复数，每个二次方程恰有两个根；但 IGCSE 只考虑实数根。

Before solving, always rearrange all terms to one side so the right-hand side is zero.

求解前，一定要把所有项移到等号一侧，使右侧为 0。

2. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the expression factorises cleanly. The idea is to write ax² + bx + c as a product of two linear brackets, then apply the zero-product property: if pq = 0, then p = 0 or q = 0.

当表达式能干净地分解时，因式分解是最快的方法。核心思想是把 ax² + bx + c 写成两个一次括号的乘积，然后运用零积性质：若 pq = 0，则 p = 0 或 q = 0。

Example 1: Solve x² − 5x + 6 = 0.

例 1：解 x² − 5x + 6 = 0。

x² − 5x + 6 = (x − 2)(x − 3) = 0

Therefore x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.

因此 x − 2 = 0 或 x − 3 = 0，即 x = 2 或 x = 3。

When the coefficient of x² is not 1, you must find two numbers whose product is a × c and whose sum is b, then split the middle term.

当 x² 的系数不为 1 时，需要找到两个数，使其乘积等于 a × c、和等于 b，然后拆中间项。

Example 2: Solve 2x² + 7x + 3 = 0.

例 2：解 2x² + 7x + 3 = 0。

Here a × c = 2 × 3 = 6. The factor pair 1 and 6 sums to 7, so split 7x as x + 6x.

这里 a × c = 2 × 3 = 6。因子对 1 和 6 的和为 7，因此将 7x 拆为 x + 6x。

2x² + x + 6x + 3 = 0 → x(2x + 1) + 3(2x + 1) = 0 → (x + 3)(2x + 1) = 0

Hence x = −3 or x = −½.

因此 x = −3 或 x = −½。

Always expand your brackets mentally to check the result before writing the final answer.

在写最终答案前，请务必在心里展开括号以检验结果。

3. The Quadratic Formula | 求根公式

When factorisation is difficult or impossible, the quadratic formula always works. For ax² + bx + c = 0, the roots are given by:

当因式分解困难或无法进行时，求根公式始终有效。对于 ax² + bx + c = 0，其根为：

x = (−b ± √(b² − 4ac)) / 2a

Memorise this formula exactly. A common error is to forget the “2a” in the denominator or to miss the ± sign.

务必准确记住这个公式。常见错误包括忘记分母中的 “2a” 或遗漏 ± 号。

Example: Solve 2x² − 4x − 3 = 0.

例：解 2x² − 4x − 3 = 0。

Here a = 2, b = −4, c = −3. Substitute into the formula:

这里 a = 2，b = −4，c = −3。代入公式：

x = (−(−4) ± √((−4)² − 4 × 2 × (−3))) / (2 × 2) = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4

x = (4 ± 2√10) / 4 = 1 ± ½√10

Always simplify surds in your final answer. Writing √40 instead of 2√10 will usually lose a method mark.

最终答案中务必化简根式。写成 √40 而不是 2√10 通常会扣掉方法分。

4. Completing the Square | 配方法

Completing the square rewrites a quadratic as a(x + p)² + q. It is essential for finding the turning point of a parabola and for deriving the quadratic formula.

配方法将二次式改写为 a(x + p)² + q 的形式。它是求抛物线顶点和推导求根公式的必要工具。

Step 1: Halve the coefficient of x. For x² + 6x, half of 6 is 3.

第一步：将 x 的系数减半。对于 x² + 6x，6 的一半是 3。

Step 2: Write (x + 3)² and subtract 3² = 9 to keep the expression unchanged.

第二步：写出 (x + 3)²，并减去 3² = 9 以保证表达式不变。

Example: Solve x² + 6x + 2 = 0.

例：解 x² + 6x + 2 = 0。

x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7 = 0

(x + 3)² = 7

Take the square root of both sides, remembering the ± sign:

两边开平方，注意 ± 号：

x + 3 = ±√7 → x = −3 ± √7

If the coefficient a ≠ 1, factor a out of the first two terms first. For example, 2x² + 8x becomes 2(x² + 4x), then complete the square inside the bracket.

若二次项系数 a ≠ 1，先从前两项中提出 a。例如，2x² + 8x 化为 2(x² + 4x)，再在括号内配方。

5. The Discriminant | 判别式

The discriminant is the expression under the square root in the quadratic formula:

判别式是求根公式中根号内的表达式：

Δ = b² − 4ac

The value of Δ determines the number of real roots without solving the full equation.

Δ 的值可以不经完整求解就判断实数根的个数。

Discriminant 判别式	Number of Real Roots 实数根个数	Graph Interpretation 图象解释
Δ > 0	Two distinct real roots 两个不同的实数根	Parabola crosses x-axis twice 抛物线与 x 轴有两个交点
Δ = 0	One repeated real root 一个重根	Parabola touches x-axis once 抛物线与 x 轴相切
Δ < 0	No real roots 无实数根	Parabola lies entirely above or below x-axis 抛物线完全位于 x 轴上方或下方

Example: For x² + x + 1 = 0, Δ = 1² − 4 × 1 × 1 = −3 < 0, so the equation has no real roots.

例：对于 x² + x + 1 = 0，Δ = 1² − 4 × 1 × 1 = −3 < 0，故方程无实数根。

Examiners often ask for the range of values of k for which the equation has real roots. Set Δ ≥ 0 and solve the resulting inequality.

考官经常要求求 k 的范围，使方程有实数根。此时令 Δ ≥ 0 并解所得不等式即可。

6. Graph of y = ax² + bx + c | 二次函数图象

The graph of a quadratic function is a smooth curve called a parabola. The sign of a controls the orientation.

二次函数的图象是一条平滑曲线，称为抛物线。a 的符号决定开口方向。

If a > 0, the parabola opens upward and has a minimum point.

若 a > 0，抛物线开口向上，有最低点。
If a < 0, the parabola opens downward and has a maximum point.

若 a < 0，抛物线开口向下，有最高点。
The roots of ax² + bx + c = 0 are the x-intercepts of the graph.

方程 ax² + bx + c = 0 的根是图象与 x 轴的交点。
The axis of symmetry is the vertical line x = −b/(2a).

对称轴是竖直线 x = −b/(2a)。
The turning point (vertex) lies exactly on the axis of symmetry.

顶点恰好位于对称轴上。

After completing the square, y = a(x − h)² + k, the vertex is (h, k) and the axis of symmetry is x = h.

配方后得到 y = a(x − h)² + k，顶点为 (h, k)，对称轴为 x = h。

Example: Complete the square for y = x² − 8x + 15, then state the vertex.

例：对 y = x² − 8x + 15 配方，并写出顶点坐标。

y = (x − 4)² − 16 + 15 = (x − 4)² − 1

The vertex is (4, −1) and since a = 1 > 0, this is a minimum point.

顶点为 (4, −1)，且 a = 1 > 0，因此该点为最低点。

7. Sum and Product of Roots | 根与系数的关系

Let the roots of ax² + bx + c = 0 be α and β. Then two useful relationships hold.

设 ax² + bx + c = 0 的两根为 α 和 β，则以下两个重要关系成立。

α + β = −b/a

α × β = c/a

These formulas allow you to construct a quadratic equation when the roots are known, without expanding brackets.

这些公式允许你在已知根的情况下构造二次方程，而无需展开括号。

Example: If the roots are 2 and 5, then sum = 7 and product = 10, so the equation is x² − 7x + 10 = 0.

例：若两根为 2 和 5，则和为 7、积为 10，故方程为 x² − 7x + 10 = 0

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Solving Quadratic Equations | 解一元二次方程