Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations appear in nearly every IGCSE Mathematics paper. Understanding how to solve them efficiently and accurately is essential for top grades.

二次方程几乎出现在每份IGCSE数学试卷中。理解如何高效且准确地求解它们，是获得高分的关键。

1. Standard Form | 标准形式

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

二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 是常数，且 a ≠ 0。

For example, 2x² – 5x + 3 = 0 is a quadratic equation in standard form. Some equations need rearranging before they fit this form.

例如，2x² – 5x + 3 = 0 就是一个标准形式的二次方程。有些方程需要先整理才能变成这种形式。

The highest power of x is 2.
x 的最高次数是 2。
If a = 0, the equation becomes linear, not quadratic.
如果 a = 0，方程就变成一次方程，而不是二次方程。

2. Factorising Quadratics | 二次方程的因式分解

Factorising is the process of writing a quadratic expression as a product of two brackets. For x² + bx + c, find two numbers that multiply to give c and add to give b.

因式分解是把二次表达式写成两个括号相乘的过程。对于 x² + bx + c，需要找到两个数，它们相乘等于 c，相加等于 b。

Example: Factorise x² + 7x + 12. The numbers 3 and 4 multiply to 12 and add to 7, so x² + 7x + 12 = (x + 3)(x + 4).

例如：对 x² + 7x + 12 因式分解。数字 3 和 4 相乘得 12，相加得 7，所以 x² + 7x + 12 = (x + 3)(x + 4)。

When the coefficient of x² is not 1, the process is more involved. For ax² + bx + c, find factors of a × c that add to b, then split the middle term.

当 x² 的系数不是 1 时，过程更复杂。对于 ax² + bx + c，找到 a × c 的因子，使其相加等于 b，然后拆开中间项。

3. Solving by Factoring | 用因式分解求解

Once the quadratic is factorised, use the zero product property: if AB = 0, then A = 0 or B = 0.

一旦二次式被因式分解，就使用零乘积性质：如果 AB = 0，那么 A = 0 或 B = 0。

Solve x² – 5x + 6 = 0. Factorise to get (x – 2)(x – 3) = 0. Then x – 2 = 0 or x – 3 = 0, giving x = 2 or x = 3.

解 x² – 5x + 6 = 0。因式分解得到 (x – 2)(x – 3) = 0。于是 x – 2 = 0 或 x – 3 = 0，得到 x = 2 或 x = 3。

Always check your solutions by substituting back into the original equation.

务必把解代回原方程进行检查。

4. The Quadratic Formula | 二次公式

The quadratic formula solves any quadratic equation, even when factorising is difficult or impossible.

二次公式可以求解任何二次方程，即使因式分解困难或无法进行时也适用。

x = (−b ± √(b² − 4ac)) / 2a

This formula is derived from completing the square and is given in the Edexcel formula book, but you must know how to use it.

该公式由配方法推导而来，Edexcel 公式册中会给出，但你必须知道如何使用它。

Example: Solve 2x² + 3x − 2 = 0. Here a = 2, b = 3, c = −2. Substitute into the formula:

例如：解 2x² + 3x − 2 = 0。这里 a = 2, b = 3, c = −2。代入公式：

x = (−3 ± √(3² − 4×2×(−2))) / (2×2) = (−3 ± √25) / 4

So x = (−3 + 5)/4 = 0.5 or x = (−3 − 5)/4 = −2.

所以 x = (−3 + 5)/4 = 0.5 或 x = (−3 − 5)/4 = −2。

5. Completing the Square | 配方法

Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This is useful for finding turning points and solving equations.

配方法把 ax² + bx + c 改写为 a(x + p)² + q 的形式。这有助于找到顶点并求解方程。

For example, x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7.

例如，x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7。

To solve (x + 3)² − 7 = 0, add 7 to both sides, then take the square root:

要解 (x + 3)² − 7 = 0，两边加 7，然后取平方根：

x + 3 = ±√7, so x = −3 ± √7

6. The Discriminant | 判别式

The discriminant of a quadratic equation is the part under the square root: Δ = b² − 4ac. Its value tells us how many real roots exist.

二次方程的判别式是根号下的部分：Δ = b² − 4ac。它的值告诉我们存在多少个实根。

Value of b² − 4ac	Number of real roots	Graphical meaning
Positive	Two distinct roots	Cuts the x-axis twice
Zero	One repeated root	Touches the x-axis once
Negative	No real roots	Does not meet the x-axis

判别式的值

实根数量

图形意义

Positive | 正值; Zero | 零; Negative | 负值。这些表格行对应为：两个不同的根/一个重根/无实根；与x轴相交两次/相切一次/不相交。

7. Graphical Solutions | 图形解法

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.

二次方程 ax² + bx + c = 0 的根，是抛物线 y = ax² + bx + c 与 x 轴交点的横坐标。

To solve a quadratic graphically, draw the curve accurately and read off the x-intercepts. Solutions may be approximate.

用图形法解二次方程时，需要准确画出曲线并读出 x 截距。解可能是近似值。

You can also solve equations like x² + 2x − 8 = 0 by plotting y = x² + 2x − 8 and finding where it crosses the x-axis.

你也可以通过绘制 y = x² + 2x − 8 并找到它与 x 轴的交点，来解像 x² + 2x − 8 = 0 这样的方程。

8. Word Problems | 应用题

Many mark-worthy questions involve setting up a quadratic equation from a real-world context. Define your variable carefully.

许多分值较高的问题需要从实际情境中建立二次方程。要仔细定义未知数。

Example: The length of a rectangle is 3 cm longer than its width, and its area is 40 cm². Let width = x, then length = x + 3. Therefore x(x + 3) = 40.

例如：一个矩形的长比宽多 3 cm，面积为 40 cm²。设宽为 x，则长为 x + 3。因此 x(x + 3) = 40。

x² + 3x − 40 = 0

Factorise: (x + 8)(x − 5) = 0. Since width is positive, x = 5. The length is 8 cm.

因式分解：(x + 8)(x − 5) = 0。因为宽为正数，所以 x = 5。长为 8 cm。

9. Roots and Coefficients | 根与系数

For a quadratic equation x² + bx + c = 0 with roots α and β, the sum of roots is α + β = −b and the product is αβ = c.

对于二次方程 x² + bx + c = 0，若根为 α 和 β，则根之和 α + β = −b，根之积 αβ = c。

This relationship is useful for finding unknown coefficients without solving the equation.

这个关系可用于不解方程而求出未知系数。

For example, if one root of x² + kx + 12 = 0 is 3, then the other root is 12 ÷ 3 = 4. The sum of roots is 3 + 4 = 7, so −k = 7, meaning k = −7.

例如，如果 x² + kx + 12 = 0 的一个根是 3，那么另一个根是 12 ÷ 3 = 4。根之和为 3 + 4 = 7，所以 −k = 7，即 k = −7。

10. Practice Questions | 练习

Work through these questions to test your understanding. Answer each one fully before checking.

完成以下问题来测试你的理解。请先完整作答，再核对。

1. Solve x² − 9x + 20 = 0.
1. 解 x² − 9x + 20 = 0。
2. Solve 3x² + 5x − 2 = 0 using the quadratic formula.
2. 用二次公式解 3x² + 5x − 2 = 0。
3. Write x² − 8x + 3 in the form (x − p)² + q.
3. 将 x² − 8x + 3 写成 (x − p)² + q 的形式。
4. For 4x² + kx + 9 = 0 to have one repeated root, find the possible values of k.
4. 要使 4x² + kx + 9 = 0 有一个重根，求 k 的可能值。

11. Common Mistakes | 常见错误

Here are frequent errors that lose marks in exams:

以下是在考试中常见的丢分错误：

Forgetting to rearrange into standard form before applying the formula.
在使用公式前忘记整理成标准形式。
Dropping the ± sign when taking square roots.
取平方根时漏掉 ± 号。
Misinterpreting the discriminant as positive, zero, or negative.
错误判断判别式的正、零或负。
Not checking solutions in word problems to ensure they make sense.
在应用题中不检查解是否合理。

12. Summary | 总结

You now have four main methods for solving quadratic equations: factorising, the quadratic formula, completing the square, and graphical interpretation. Choose the most efficient method for each question.

你现在掌握了四种解二次方程的主要方法：因式分解、二次公式、配方法和图形解法。根据每道题选择最有效的方法。

Remember to always check the discriminant, verify your solutions, and read word problems carefully. Regular practice is the key to success in IGCSE Mathematics.

记住要始终检查判别式、验证解，并仔细阅读应用题。定期练习是IGCSE数学成功的关键。

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Solving Quadratic Equations | 解二次方程