Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations are a core topic in the IGCSE mathematics syllabus. They appear in algebra, coordinate geometry, and real-world problems. This article explains the key methods for solving quadratic equations, the use of the discriminant, and how to interpret quadratic graphs. By the end, you should be able to solve any quadratic equation confidently and avoid common pitfalls.

一元二次方程是 IGCSE 数学课程的核心内容，它们出现在代数、坐标几何和实际应用题中。本文将讲解求解一元二次方程的主要方法、判别式的使用以及如何理解二次函数图像。读完本文后，你应该能够自信地求解任何一元二次方程，并避免常见错误。

1. What is a Quadratic Equation? | 什么是一元二次方程？

A quadratic equation is any equation that can be arranged in the standard form ax² + bx + c = 0, where a, b and c are constants, and a ≠ 0. The highest power of the variable is 2, which gives the equation its name. If a = 0, the equation becomes linear, not quadratic.

一元二次方程是任何可以整理成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 是常数，并且 a ≠ 0。变量的最高次数是 2，因此得名。如果 a = 0，方程就变成一次方程，而不是二次方程。

Examples of quadratic equations:

一元二次方程的例子：

2x² – 3x + 1 = 0

2x² – 3x + 1 = 0
x² – 7x = 0

x² – 7x = 0
4x² – 9 = 0

4x² – 9 = 0

Notice that a quadratic equation may have an x term, a constant term, or both, but the x² term must always be present. Expressions like 2x³ – x = 0 are not quadratic because the highest power is 3.

注意，一元二次方程可以没有一次项，也可以没有常数项，但 x² 项必须存在。像 2x³ – x = 0 这样的表达式不是二次的，因为最高次数是 3。

2. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method to solve a quadratic equation when the expression can be written as a product of two linear factors. The method relies on the zero product property: if A × B = 0, then A = 0 or B = 0.

因式分解通常是解决二次方程最快的方法，前提是表达式可以写成两个一次因式的乘积。该方法依赖于零乘积性质：如果 A × B = 0，那么 A = 0 或 B = 0。

The steps are:

步骤如下：

Rearrange the equation into the form ax² + bx + c = 0.

将方程整理为 ax² + bx + c = 0 的形式。
Factorise the quadratic expression into two linear factors.

将二次表达式分解为两个一次因式。
Set each factor equal to zero and solve for x.

令每个因式等于零，并解出 x。

For example, solve x² – 7x + 12 = 0.

例如，解 x² – 7x + 12 = 0。

We need two numbers whose product is 12 and whose sum is -7. The numbers are -3 and -4, so the equation becomes (x – 3)(x – 4) = 0. Therefore x = 3 or x = 4.

我们需要两个数，其乘积为 12，和为 -7。这两个数是 -3 和 -4，因此方程化为 (x – 3)(x – 4) = 0。所以 x = 3 或 x = 4。

Some quadratics have a common factor first, such as 2x² – 8x = 0. Factorise to 2x(x – 4) = 0, giving x = 0 or x = 4.

有些二次方程可以先提取公因式，例如 2x² – 8x = 0，分解为 2x(x – 4) = 0，得到 x = 0 或 x = 4。

3. Solving by Completing the Square | 配方法

Completing the square transforms a quadratic expression into the form a(x + p)² + q. This is useful because it directly reveals the turning point of a quadratic graph and can also solve the equation without factorisation.

配方法将二次表达式转化为 a(x + p)² + q 的形式。这很有用，因为可以直接看出二次图像的极点，并且无需因式分解也能解方程。

For the general equation x² + bx + c = 0, the identity is:

对于一般方程 x² + bx + c = 0，恒等式为：

x² + bx + c = (x + b/2)² – (b/2)² + c

As an example, solve x² + 6x + 2 = 0 by completing the square.

以配方法解 x² + 6x + 2 = 0 为例。

Half of 6 is 3, so we write (x + 3)² – 9 + 2 = 0, which simplifies to (x + 3)² – 7 = 0. Then (x + 3)² = 7, so x + 3 = ±√7, and thus x = -3 ± √7.

6 的一半是 3，所以我们写成 (x + 3)² – 9 + 2 = 0，即 (x + 3)² – 7 = 0。于是 (x + 3)² = 7，所以 x + 3 = ±√7，因此 x = -3 ± √7。

If the coefficient of x² is not 1, first divide the whole equation by a, then complete the square.

如果 x² 的系数不是 1，先给方程两边同时除以 a，再进行配方。

4. The Quadratic Formula | 求根公式

Every quadratic equation can be solved using the quadratic formula, which is derived from completing the square. For ax² + bx + c = 0, the solutions are given by:

所有一元二次方程都可以用求根公式求解，该公式由配方法推导而来。对于 ax² + bx + c = 0，解为：

x = (-b ± √(b² – 4ac)) / (2a)

The symbol ± indicates that there are usually two solutions: one with the plus sign and one with the minus sign.

符号 ± 表示通常有两个解：一个取加号，一个取减号。

Let us use the quadratic formula to solve 2x² + 3x – 5 = 0.

我们用求根公式解 2x² + 3x – 5 = 0。

Here a = 2, b = 3, c = -5. Substituting into the formula gives x = (-3 ± √(3² – 4×2×(-5))) / (2×2) = (-3 ± √49)/4 = (-3 ± 7)/4. Therefore x = 1 or x = -5/2.

这里 a = 2，b = 3，c = -5。代入公式得 x = (-3 ± √(3² – 4×2×(-5))) / (2×2) = (-3 ± √49)/4 = (-3 ± 7)/4。因此 x = 1 或 x = -5/2。

The quadratic formula works for all quadratic equations, even when factorisation is difficult or impossible.

求根公式适用于所有一元二次方程，即使因式分解困难或无法分解。

5. The Discriminant | 判别式

The discriminant is the expression b² – 4ac inside the square root of the quadratic formula. It determines the number and type of roots without solving the equation completely.

判别式是求根公式中根号内的表达式 b² – 4ac。它不需要完全解方程就能判断根的数量和类型。

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（两个相同的实数根）
Δ < 0	No real roots	没有实数根

For example, the equation x² – 4x + 5 = 0 has a = 1, b = -4, c = 5. Its discriminant is (-4)² – 4×1×5 = 16 – 20 = -4, which is negative, so there are no real roots.

例如，方程 x² – 4x + 5 = 0 中 a = 1，b = -4，c = 5。其判别式为 (-4)² – 4×1×5 = 16 – 20 = -4，为负数，因此没有实数根。

If Δ is a perfect square and a, b, c are rational, the quadratic can be solved by factorisation over the rational numbers.

如果 Δ 是完全平方数，且 a、b、c 都是有理数，则该二次方程可以在有理数范围内因式分解。

6. Roots, Intercepts and Turning Points | 根、截距与极点

The roots of a quadratic equation ax² + bx + c = 0 are the x-coordinates of the points where the graph of y = ax² + bx + c crosses the x-axis. Therefore, solving the equation is equivalent to finding the x-intercepts of the parabola.

一元二次方程 ax² + bx + c = 0 的根，就是函数 y = ax² + bx + c 的图像与 x 轴交点的横坐标。因此，解方程等价于求抛物线的 x 截距。

The y-intercept of the graph is simply the constant term c, because when x = 0, y = c.

图像的 y 截距就是常数项 c，因为当 x = 0 时，y = c。

The turning point, also called the vertex, is the maximum or minimum point of the parabola. Its x-coordinate is the midpoint of the two roots when they exist, but it can also be found using the axis of symmetry formula introduced in the next section.

极点也称为顶点，是抛物线的最大值或最小值点。当根存在时，其 x 坐标是两个根的中点，但也可以用下一节介绍的对称轴公式求得。

If the discriminant is positive, the graph intersects the x-axis twice. If it is zero, the graph touches the x-axis exactly once at the vertex. If it is negative, the graph never touches the x-axis.

若判别式为正，图像与 x 轴相交两次；若判别式为零，图像与 x 轴相切一次，切点就是顶点；若判别式为负，图像与 x 轴没有交点。

7. Axis of Symmetry and the Vertex | 对称轴与顶点

Every parabola has a vertical axis of symmetry that passes through its vertex. For the quadratic function y = ax² + bx + c, the equation of the axis of symmetry is:

每条抛物线都有一条穿过顶点的竖直对称轴。对于二次函数 y = ax² + bx + c，对称轴的方程为：

x = -b / (2a)

The x-coordinate of the vertex is the same value -b/(2a). To find the y-coordinate, substitute this x-value into the original equation.

顶点的 x 坐标同样是 -b/(2a)。要求 y 坐标，只需将这个 x 值代入原方程。

For example, for y = x² – 4x + 3, the axis of symmetry is x = -(-4)/(2×1) = 2. Substituting x = 2 gives y = 4 – 8 + 3 = -1. Thus the vertex is (2, -1).

例如，对于 y = x² – 4x + 3，对称轴为 x = -(-4)/(2×1) = 2。代入 x = 2 得 y = 4 – 8 + 3 = -1。因此顶点为 (2, -1)。

If a > 0, the vertex is a minimum point; if a < 0, the vertex is a maximum point.

如果 a > 0，顶点是最小值点；如果 a < 0，顶点是最大值点。

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

The graph of y = ax² + bx + c is always a parabola. The sign of a determines the direction of the opening: if a > 0, the parabola opens upward like a cup; if a < 0, it opens downward like a cap.

函数 y = ax² + bx + c 的图像始终是抛物线。a 的符号决定开口方向：a > 0 时抛物线开口向上，像杯子；a < 0 时开口向下，像帽子。

To sketch a quadratic graph, you should find:

要画二次函数图像，你应当找出：

The roots of the equation (where the graph crosses the x-axis).

方程的根（图像与 x 轴的交点）。
The y-intercept (where the graph crosses the y-axis, at x = 0).

y 截距（图像与 y 轴的交点，即 x = 0 处）。
The vertex (maximum or minimum point).

顶点（最大值或最小值点）。
The axis of symmetry, which passes through the vertex.

对称轴，它经过顶点。

For example, the graph of y = x² – 2x – 3

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Solving Quadratic Equations | 解一元二次方程