Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations appear regularly in IGCSE Mathematics papers. From factorisation to the quadratic formula, this topic connects algebra, graphs, and real-world problem solving. This guide explains every core skill you need, with worked examples and common pitfalls to avoid.

二次方程是 IGCSE 数学考试中的常客。从因式分解到求根公式，这一课题将代数、图像与实际应用紧密相连。本指南将系统讲解你所需的核心技能，并配有例题与常见易错点提醒。

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

A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2. The standard form is:

二次方程是次数为 2 的多项式方程，即变量的最高次数为 2。其标准形式为：

ax² + bx + c = 0, where a ≠ 0

Here a is the coefficient of x², b is the coefficient of x, and c is the constant term. If a = 0, the equation becomes linear, so a is never zero in a genuine quadratic equation.

其中 a 是 x² 的系数，b 是 x 的系数，c 是常数项。若 a = 0，方程将退化为一次方程，因此在真正的二次方程中 a 永远不为零。

Key terms to remember:

需要牢记的关键术语：

Root (or solution): a value of x that satisfies the equation ax² + bx + c = 0.

根（或解）：满足方程 ax² + bx + c = 0 的 x 的值。
Leading coefficient: the coefficient of the highest degree term, which is a.

首项系数：最高次项的系数，即 a。
Parabola: the U-shaped curve that represents the quadratic function y = ax² + bx + c.

抛物线：表示二次函数 y = ax² + bx + c 的 U 形曲线。

2. Solving by Factorisation | 因式分解法

Factorisation is usually the quickest method when the quadratic has simple integer roots. The method relies on the zero-product property: if A × B = 0, then A = 0 or B = 0.

当二次方程具有简单的整数根时，因式分解通常是最快的方法。该方法依赖零乘积性质：若 A × B = 0，则 A = 0 或 B = 0。

Step 1: Rearrange the equation into standard form ax² + bx + c = 0.

步骤一：将方程整理为标准形式 ax² + bx + c = 0。
Step 2: Factorise the left-hand side into two brackets, for example (px + q)(rx + s) = 0.

步骤二：将等号左边分解为两个括号，例如 (px + q)(rx + s) = 0。
Step 3: Set each bracket equal to zero and solve the resulting linear equations.

步骤三：令每个括号等于零，并求解得到的两个一次方程。

Example: Solve x² – 5x + 6 = 0. We look for two numbers that multiply to 6 and add to -5; these are -2 and -3. Hence:

例：解方程 x² – 5x + 6 = 0。我们需要找两个数，它们相乘为 6、相加为 -5，即 -2 和 -3。因此：

(x – 2)(x – 3) = 0

x = 2 or x = 3

3. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² + q. This form directly reveals the vertex of the parabola and solves equations where factorisation is not obvious.

配方法将二次式改写为 (x + p)² + q 的形式。这一形式能直接显示抛物线的顶点，并能解决因式分解不明显的方程。

For an expression x² + bx, we add and subtract (b/2)²:

对于形如 x² + bx 的式子，我们加上并减去 (b/2)²：

x² + bx = (x + b/2)² – (b/2)²

Example: Solve x² + 6x + 8 = 0 by completing the square.

例：用配方法解方程 x² + 6x + 8 = 0。

x² + 6x + 8 = (x + 3)² – 9 + 8 = (x + 3)² – 1

(x + 3)² = 1

x + 3 = ±1, so x = -2 or x = -4

4. The Quadratic Formula | 求根公式

The quadratic formula solves any quadratic equation, including those that cannot be factorised easily. For ax² + bx + c = 0:

求根公式可以解任意二次方程，包括那些难以因式分解的方程。对于 ax² + bx + c = 0：

x = (-b ± √(b² – 4ac)) / (2a)

Example: Solve 2x² – 4x – 3 = 0 using the quadratic formula. Here a = 2, b = -4, c = -3.

例：用求根公式解方程 2x² – 4x – 3 = 0。这里 a = 2，b = -4，c = -3。

x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = (4 ± 2√10) / 4 = (2 ± √10) / 2

Thus x ≈ 2.581 or x ≈ -0.581. Always write exact values first, then approximate if asked.

因此 x ≈ 2.581 或 x ≈ -0.581。务必先写出精确值，再按题目要求取近似值。

5. The Discriminant | 判别式

The discriminant is the expression Δ = b² – 4ac under the square root in the quadratic formula. It tells us the nature of the roots without solving the whole equation.

判别式是求根公式中根号下的表达式 Δ = b² – 4ac。它无需完整求解即可判断根的性质。

Value of Δ	Nature of Roots	Graph Meaning
Δ > 0	Two distinct real roots（两个不同的实根）	Parabola crosses the x-axis twice（抛物线穿过 x 轴两次）
Δ = 0	One repeated real root（一个重根）	Parabola touches the x-axis once（抛物线切于 x 轴一次）
Δ < 0	No real roots（无实根）	Parabola does not meet the x-axis（抛物线不与 x 轴相交）

For example, the equation x² + 2x + 5 = 0 has Δ = 4 – 20 = -16 < 0, so it has no real solutions.

例如，方程 x² + 2x + 5 = 0 的判别式 Δ = 4 – 20 = -16 < 0，因此它没有实数解。

6. Sum and Product of Roots | 根与系数的关系

If a quadratic equation ax² + bx + c = 0 has roots α and β, then without solving the equation we can state:

若二次方程 ax² + bx + c = 0 的两根为 α 和 β，则无需解方程即可得到：

α + β = -b/a and αβ = c/a

For the simpler monic form x² + bx + c = 0, the sum of the roots is -b and the product is c. This is useful for checking answers and for forming a quadratic equation from given roots.

对于更简单的首一形式 x² + bx + c = 0，两根之和为 -b，两根之积为 c。这在检验答案以及根据已知根构造二次方程时十分有用。

Example: Find a quadratic equation with roots 3 and -4. The sum is 3 + (-4) = -1, and the product is 3 × (-4) = -12. Therefore the equation is x² + x – 12 = 0.

例：求一个以 3 和 -4 为根的二次方程。两根之和为 3 + (-4) = -1，两根之积为 3 × (-4) = -12。因此该方程为 x² + x – 12 = 0。

7. Sketching Quadratic Graphs | 绘制二次函数图像

To sketch y = ax² + bx + c accurately, identify four features: shape, y-intercept, roots, and vertex.

要准确绘制 y = ax² + bx + c 的图像，需要确定四个要素：开口方向、y 轴截距、根以及顶点。

Shape: if a > 0 the parabola opens upward; if a < 0 it opens downward.

开口方向：若 a > 0，抛物线开口向上；若 a < 0，抛物线开口向下。
y-intercept: the graph crosses the y-axis at the point (0, c).

y 轴截距：图像在点 (0, c) 处与 y 轴相交。
Roots: the x-values where y = 0, found by solving the quadratic equation.

根：令 y = 0 时得到的 x 值，可通过解二次方程求得。
Vertex: the x-coordinate is -b/(2a); substitute it back to find the y-coordinate.

顶点：x 坐标为 -b/(2a)，将其代回原式即可求出 y 坐标。

The vertical line x = -b/(2a) is the axis of symmetry of the parabola.

竖直直线 x = -b/(2a) 是抛物线的对称轴。

8. Applications in Real-World Problems | 实际应用问题

Quadratic equations model many physical and geometrical situations, such as projectile motion, area problems, and profit functions in economics.

二次方程可以建模许多物理与几何情境，例如抛体运动、面积问题以及经济学中的利润函数。

Example: A rectangle has length (x + 3) cm and width x cm. Its area is 28 cm². Find its dimensions.

例：一个长方形的长为 (x + 3) 厘米，宽为 x 厘米，面积为 28 平方厘米，求其尺寸。

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Solving Quadratic Equations | 解二次方程