Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations appear frequently in IGCSE Mathematics. Learning to solve them correctly is essential for success in exams. This guide covers all the standard methods with clear examples.

在 IGCSE 数学中，二次方程频繁出现。学会正确解二次方程是考试取得好成绩的关键。本指南将用清晰的例子讲解所有标准解法。

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

A quadratic equation is a polynomial equation of degree 2. Its general form is:

二次方程是次数为 2 的多项式方程，其一般形式为：

ax² + bx + c = 0, a ≠ 0

Here x is the variable, and a, b and c are constants. The coefficient a must not be zero, otherwise the equation becomes linear.

其中 x 是未知数，a、b、c 是常数。系数 a 不能为零，否则方程就变成了一次方程。

Examples of quadratic equations include x² – 5x + 6 = 0 and 2x² + 3x – 1 = 0.

二次方程的例子包括 x² – 5x + 6 = 0 和 2x² + 3x – 1 = 0。

2. Factorising Quadratics | 因式分解二次式

Before solving by factorisation, you must know how to factorise a quadratic expression. For a monic quadratic x² + bx + c, look for two numbers that multiply to give c and add to give b.

在使用因式分解求解之前，你必须学会如何分解二次式。对于首项系数为 1 的二次式 x² + bx + c，找两个数，它们相乘等于 c，相加等于 b。

For example, factorise x² – 5x + 6. The two numbers are -2 and -3, because (-2) × (-3) = 6 and (-2) + (-3) = -5.

例如，分解 x² – 5x + 6。两个数是 -2 和 -3，因为 (-2) × (-3) = 6，且 (-2) + (-3) = -5。

x² – 5x + 6 = (x – 2)(x – 3)

For a non-monic quadratic like ax² + bx + c, you may need to use the multiplication cross method or check common factors first.

对于形如 ax² + bx + c 的非首一二次式，你可能需要使用十字相乘法，或先提取公因式。

3. Solving by Factorisation | 用因式分解求解

The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. This is the key to solving factorised quadratics.

零因子性质指出：如果两个因式的乘积为零，那么至少有一个因式为零。这是解因式分解后二次方程的关键。

To solve (x – 2)(x – 3) = 0, set each factor equal to zero:

要解 (x – 2)(x – 3) = 0，令每个因式等于零：

(x – 2) = 0 or (x – 3) = 0

So x = 2 or x = 3. Always write the two solutions clearly.

所以 x = 2 或 x = 3。务必清晰地写出两个解。

If the quadratic is not already equal to zero, rearrange it first. For example, x² = 4x becomes x² – 4x = 0.

如果二次式还没有等于零，先移项整理。例如，x² = 4x 应变为 x² – 4x = 0。

4. The Quadratic Formula | 二次公式

When factorisation is difficult or impossible, you can use the quadratic formula. For any quadratic equation ax² + bx + c = 0, the solutions are given by:

当因式分解困难或无法分解时，你可以使用二次公式。对于任意二次方程 ax² + bx + c = 0，解由以下公式给出：

x = (-b ± √(b² – 4ac)) / (2a)

Substitute the values of a, b and c carefully. Pay attention to negative signs.

代入 a、b、c 的值时要仔细，尤其要注意负号。

Example: Solve 2x² + 3x – 1 = 0. Here a = 2, b = 3, c = -1.

例如：解 2x² + 3x – 1 = 0。此时 a = 2，b = 3，c = -1。

x = (-3 ± √(9 + 8)) / 4 = (-3 ± √17) / 4

Therefore the exact solutions are (-3 + √17)/4 and (-3 – √17)/4.

因此精确解为 (-3 + √17)/4 和 (-3 – √17)/4。

5. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² + q. This method is useful for solving equations without factorisation and for understanding graph transformations.

配方法将二次式改写成 (x + p)² + q 的形式。这种方法在不用因式分解时求解方程很有用，也有助于理解图像变换。

Start with x² + bx. To complete the square, add and subtract (b/2)². For example, x² + 6x becomes (x + 3)² – 9.

从 x² + bx 开始。配方时，加上并减去 (b/2)²。例如，x² + 6x 变成 (x + 3)² – 9。

For a full quadratic x² + 6x + 2 = 0, rewrite as (x + 3)² – 9 + 2 = 0, so (x + 3)² = 7.

对于完整的二次方程 x² + 6x + 2 = 0，改写为 (x + 3)² – 9 + 2 = 0，所以 (x + 3)² = 7。

x + 3 = ±√7, so x = -3 ± √7

This method is especially useful when a is not 1, but it takes more steps.

当 a 不为 1 时，这个方法尤其有用，但步骤更多。

6. The Discriminant | 判别式

The discriminant is the expression under the square root in the quadratic formula: Δ = b² – 4ac. It tells us the number and type of solutions without solving the full equation.

判别式是二次公式根号内的表达式：Δ = b² – 4ac。它无需解完整方程即可告诉我们解的数量和类型。

If Δ > 0, there are two distinct real roots.

如果 Δ > 0，方程有两个不相等的实数根。
If Δ = 0, there is exactly one real root (a repeated root).

如果 Δ = 0，方程有一个实数根（重根）。
If Δ < 0, there are no real roots (two complex roots).

如果 Δ < 0，方程没有实数根（有两个复数根）。

For example, for x² – 4x + 4 = 0, Δ = 16 – 16 = 0, so it has a repeated root x = 2.

例如，对于 x² – 4x + 4 = 0，Δ = 16 – 16 = 0，所以它有重根 x = 2。

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

The graph of a quadratic function y = ax² + bx + c is a parabola. If a > 0, it opens upwards; if a < 0, it opens downwards.

二次函数 y = ax² + bx + c 的图像是一条抛物线。若 a > 0，抛物线开口向上；若 a < 0，开口向下。

The x-coordinate of the vertex is given by x = -b/(2a). The y-coordinate can be found by substituting this x-value into the equation.

顶点的 x 坐标由 x = -b/(2a) 给出。将 x 值代入原方程即可得到 y 坐标。

The solutions of the equation ax² + bx + c = 0 correspond to the x-intercepts of the graph. If the discriminant is negative, the graph does not cross the x-axis.

方程 ax² + bx + c = 0 的解对应图像与 x 轴的交点。若判别式为负，图像不与 x 轴相交。

The axis of symmetry is the vertical line through the vertex, with equation x = -b/(2a).

对称轴是经过顶点的竖直直线，其方程为 x = -b/(2a)。

8. Solving Word Problems | 解应用题

Many real-world problems can be modelled by quadratic equations, such as area problems, projectile motion and number puzzles.

许多实际问题可以用二次方程建模，例如面积问题、抛体运动和数字谜题。

Example: The area of a rectangle is 24 cm². Its length is 2 cm more than its width. Find the width.

例子：一个矩形的面积是 24 平方厘米。它的长比宽多 2 厘米。求宽。

Let the width be x cm. Then the length is (x + 2) cm. The area equation is x(x + 2) = 24.

设宽为 x 厘米，则长为 (x + 2) 厘米。面积方程为 x(x + 2) = 24。

x² + 2x – 24 = 0 → (x + 6)(x – 4) = 0

The solutions are x = -6 or x = 4. Since width cannot be negative, the width is 4 cm.

解为 x = -6 或 x = 4。由于宽度不能为负，所以宽为 4 厘米。

Always check that your final answer makes sense in the context of the problem.

始终检查最终答案在问题情境中是否合理。

9. Common Mistakes | 常见错误

Students often make small errors that lead to wrong answers. Here are the most frequent ones.

学生常常犯一些导致答案错误的小失误。以下是最常见的几种。

Do not divide both sides by x unless you know x ≠ 0. This loses the solution x = 0.

除非知道 x ≠ 0，否则不要两边同除以 x。这会丢失解 x = 0。
Remember that (x + 2)² ≠ x² + 4. It is x² + 4x + 4.

记住 (x + 2)² 不等于 x² + 4，而是 x² + 4x + 4。
When using the quadratic formula, pay special attention to negative values of b and c.

使用二次公式时，要特别注意 b 和 c 为负数的情况。
Always rearrange the equation so that zero appears on one side before factorising.

在进行因式分解之前，务必整理方程，使一边为零。

Avoiding these mistakes will improve both your accuracy and your confidence.

避免这些错误可以提高你的准确性和信心。

10. Practice Questions | 练习题

Try solving the following questions on your own before checking the answers.

先自己尝试解以下问题，再核对答案。

Solve x² – 7x + 10 = 0.

解方程 x² – 7x + 10 = 0。
Solve 3x² + 5x – 2 = 0 using the quadratic formula.

用二次公式解 3x² + 5x – 2 = 0。
Find the value of k for which x² + kx + 9 = 0 has exactly one root.

求 k 的值，使 x² + kx + 9 = 0 只有一个根。
Solve (x – 1)(x + 4) = 6.

解方程 (x – 1)(x + 4) = 6。

Answers: 1) x = 2 or x = 5. 2) x = 1/3 or x = -2. 3) k = ±6. 4) x = 2 or x = -5.

答案：1) x = 2 或 x = 5。2) x = 1/3 或 x = -2。3) k = ±6。4) x = 2 或 x = -5。

Review any questions you answered incorrectly before moving to the next topic.

在下一次主题之前，请复习你答错的题目。

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Solving Quadratic Equations | 解二次方程