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 paper, either as standalone questions or as tools inside geometry, sequences, and word problems. Mastering the three main methods — factorisation, completing the square, and the quadratic formula — is essential for full marks.

二次方程几乎出现在每一份 IGCSE 数学试卷中，既可能单独设题，也常见于几何、数列和应用题中。熟练掌握因式分解法、配方法和求根公式这三大方法，是拿到满分的关键。

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

A quadratic equation is any equation that can be written in the standard form:

二次方程是指可以写成以下标准形式的方程：

ax² + bx + c = 0, where a ≠ 0

Here a, b, and c are constants, and x is the unknown variable. The highest power of x is 2, which gives the equation its name. If a = 0, the equation becomes linear, not quadratic.

其中 a、b、c 为常数，x 为未知数。方程中 x 的最高次数为 2，因此称为二次方程。若 a = 0，方程就变成一次方程，不再是二次方程。

Examples | 例如：
x² − 4x + 3 = 0
2x² + x − 6 = 0
−x² + 9 = 0

2. Solving by Factorisation | 因式分解法

Factorisation works when the quadratic can be written as a product of two linear brackets. The principle is simple: if two numbers multiply to give zero, then one of them must be zero.

因式分解适用于二次式可以写成两个一次括号相乘的情况。其原理很简单：如果两个数相乘等于零，那么其中一个数必然为零。

For example, solve x² + 5x + 6 = 0. We look for two numbers that multiply to 6 and add to 5. Those numbers are 2 and 3, so:

例如，解 x² + 5x + 6 = 0。我们需要找两个数，相乘得 6，相加得 5。这两个数是 2 和 3，因此：

(x + 2)(x + 3) = 0

x + 2 = 0 or x + 3 = 0

x = −2 or x = −3

Always check your solutions by substituting them back into the original equation. This catches sign errors quickly.

务必把解代回原方程进行验算，这样能快速发现符号错误。

3. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² + q. This method works for every quadratic, including those that do not factorise neatly.

配方法将二次式改写为 (x + p)² + q 的形式。这一方法适用于所有二次方程，包括那些无法直接因式分解的方程。

Take x² − 6x + 4 = 0. Half of −6 is −3, so we use (x − 3)² which expands to x² − 6x + 9. To keep the expression equal, we subtract 9:

以 x² − 6x + 4 = 0 为例。−6 的一半是 −3，因此使用 (x − 3)²，展开后得到 x² − 6x + 9。为保持等式不变，需减去 9：

(x − 3)² − 9 + 4 = 0

(x − 3)² − 5 = 0

Then isolate the square and take the square root of both sides:

接着分离平方项，并对两边开平方：

(x − 3)² = 5

x − 3 = ±√5

x = 3 ± √5

Remember: the square root of a positive number has two values, one positive and one negative. That is why ± appears.

注意：正数开平方有两个值，一正一负，因此会出现 ±。

4. The Quadratic Formula | 求根公式

The quadratic formula solves any quadratic equation directly. It is the fastest method when factorisation is difficult or impossible, and it is provided in most IGCSE formula booklets.

求根公式可以直接求解任何二次方程。当因式分解困难或无法进行时，它是最快的方法，而且大多数 IGCSE 公式表中都会提供该公式。

For ax² + bx + c = 0:

对于 ax² + bx + c = 0：

x = (−b ± √(b² − 4ac)) / 2a

Let us solve 2x² − 7x + 3 = 0. Here a = 2, b = −7, c = 3. Substitute carefully:

我们来解 2x² − 7x + 3 = 0。这里 a = 2，b = −7，c = 3。代入时务必仔细：

x = (7 ± √((−7)² − 4 × 2 × 3)) / (2 × 2)

x = (7 ± √(49 − 24)) / 4

x = (7 ± √25) / 4 = (7 ± 5) / 4

So x = 3 or x = 0.5. Notice that when b is negative, the formula gives −b as a positive number. This is a common source of errors.

因此 x = 3 或 x = 0.5。注意当 b 为负数时，公式中的 −b 变为正数，这是常见的出错点。

5. The Discriminant | 判别式

The discriminant is the part of the formula under the square root: Δ = b² − 4ac. It tells us how many real roots an equation has without solving it.

判别式是求根公式中根号内的部分：Δ = b² − 4ac。它可以在不解方程的情况下告诉我们方程实根的个数。

Δ value \| 判别式取值	Number of real roots \| 实根个数
Δ > 0	Two distinct real roots \| 两个不相等实根
Δ = 0	One repeated real root \| 一个二重实根
Δ < 0	No real roots \| 无实根

For example, x² + 2x + 5 = 0 has Δ = 2² − 4 × 1 × 5 = 4 − 20 = −16, which is negative. Therefore the equation has no real roots; its graph never touches the x-axis.

例如，x² + 2x + 5 = 0 的判别式 Δ = 2² − 4 × 1 × 5 = 4 − 20 = −16，为负数。因此该方程没有实根，它的图像与 x 轴永不相交。

6. Roots and the Graph | 根与图像的关系

The graph of y = x² + 2x − 3 is a U-shaped parabola. The roots of the equation x² + 2x − 3 = 0 are exactly the x-coordinates where the graph cuts the x-axis.

y = x² + 2x − 3 的图像是一条开口向上的抛物线。方程 x² + 2x − 3 = 0 的根正是该图像与 x 轴交点的 x 坐标。

Factorising gives (x + 3)(x − 1) = 0, so the roots are x = −3 and x = 1. The graph crosses the x-axis at (−3, 0) and (1, 0).

因式分解得到 (x + 3)(x − 1) = 0，因此根为 x = −3 和 x = 1。图像在 (−3, 0) 和 (1, 0) 两点处穿过 x 轴。

The x-coordinate of the vertex (turning point) of a parabola is the midpoint of the two roots:

抛物线顶点（转向点）的 x 坐标是两个根的中点：

x = (−3 + 1) / 2 = −1

Substitute back to find the y-coordinate: y = (−1)² + 2(−1) − 3 = −4. If the discriminant is negative, the parabola does not intersect the x-axis at all.

代回求得 y 坐标：y = (−1)² + 2(−1) − 3 = −4。如果判别式为负，抛物线就与 x 轴完全没有交点。

7. Solving Word Problems | 实际应用题

Quadratic equations often appear in mensuration and number problems. The key is to define the unknown carefully and translate each sentence into algebra.

二次方程常出现在度量几何和数字类应用题中。关键是仔细设未知数，并把每句话转成代数关系。

A rectangle has length 3 cm longer than its width, and its area is 28 cm². Let the width be w. Then length is w + 3, so:

一个矩形的长比宽长 3 cm，面积为 28 cm²。设宽为 w，则长为 w + 3，于是：

w(w + 3) = 28

w² + 3w − 28 = 0

Factorising gives (w + 7)(w − 4) = 0, so w = −7 or w = 4. A width cannot be negative, so w = 4 cm and the length is 7 cm.

因式分解得 (w + 7)(w − 4) = 0，所以 w = −7 或 w = 4。宽度不能为负，因此 w = 4 cm，长为 7 cm。

Always reject negative or nonsensical solutions in context, and state the final answer with units.

在应用题中务必舍去负值或无意义的解，并带上单位写出最终答案。

8. Common Mistakes | 常见错误

Many marks are lost in quadratic questions due to small but avoidable errors. Here are the most frequent ones:

许多分数都因为细小但可以避免的错误而丢掉。以下是最常见的问题：

Not rearranging to ax² + bx + c = 0 before factorising | 因式分解前没有把方程整理成 ax² + bx + c = 0 的形式

For example, x² = 3x + 4 must become x² − 3x − 4 = 0 first. | 例如，x² = 3x + 4 必须先变成 x² − 3x − 4 = 0。
Losing the negative sign when substituting into the formula | 代入求根公式时弄丢负号

Write b = −7 explicitly, not just 7. | 要明确写出 b = −7，而不是只写 7。
Forgetting the ± when taking square roots | 开平方时忘记 ±

Every positive square root yields two solutions. | 每个正数开平方都会产生两个解。
Dividing by a variable that could be zero | 除以可能为零的变量

Never cancel x from both sides of x² = 2x, because x = 0 is a solution. | 绝不能从 x² = 2x 两边同时约去 x，因为 x = 0 是一个解。

9. Exam Tactics | 考试技巧

Choosing the right method saves time and reduces errors. For a straightforward exam question, apply this rule of thumb:

选择合适的方法可以节省时间并减少错误。面对常规考题，可参考以下原则：

If the coefficient of x² is 1 and the numbers are small, try factorisation first. | 若 x² 系数为 1 且数字较小，先尝试因式分解。
If the equation cannot be factorised easily, use the quadratic formula. It always works. | 若不易因式分解，直接使用求根公式，它始终有效。
If the question asks for a point on the graph, such as the turning point, use completing the square. | 若题目要求图像上的点（如顶点），则使用配方法。
Show every substitution step when using the formula. Even if your answer is wrong, method marks are awarded. | 使用公式时写出每一步代入过程。即使答案错误，过程分仍然可以拿到。

10. Practice Questions | 练习

Try these questions on your own before checking the answers below.

请先独立完成以下题目，再核对下方的答案。

1. Solve x² − 7x + 10 = 0. | 解方程 x² − 7x + 10 = 0。
2. Solve 3x² + 5x − 2 = 0 using the quadratic formula. | 用求根公式解 3x² + 5x − 2 = 0。
3. Find the value of k such that x² + 6x + k = 0 has exactly one real root. | 求 k 的值，使 x² + 6x + k = 0 恰有一个实根。
4. Write x² + 8x + 3 in the form (x + p)² + q. | 把 x² + 8x + 3 写成 (x + p)² + q 的形式。

Answers: 1. x = 2 or x = 5. 2. x = 1/3 or x = −2. 3. k = 9. 4. (x + 4)² − 13.

答案：1. x = 2 或 x = 5。2. x = 1/3 或 x = −2。3. k = 9。4. (x + 4)² − 13。

Remember: a methodical approach — rearrange, choose the right tool, solve, and check — will earn you consistent marks on every quadratic question.

请记住：有条理的解题流程——整理、选对方法、求解、验算——能帮助你在每道二次函数题目中稳定得分。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Δ value \| 判别式取值	Number of real roots \| 实根个数
Δ > 0	Two distinct real roots \| 两个不相等实根
Δ = 0	One repeated real root \| 一个二重实根
Δ < 0	No real roots \| 无实根

Solving Quadratic Equations | 解一元二次方程