Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, graphs, geometry, and even calculus preparation. This article explains the core methods for solving quadratics, with worked examples and exam-style tips.

二次方程是IGCSE数学中最重要的主题之一。它出现在代数、图像、几何甚至微积分预备知识中。本文讲解解二次方程的核心方法，并附有例题和考试技巧。

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

A quadratic equation is an equation in which the highest power of the variable is 2. The general form is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

二次方程是变量最高次数为2的方程。一般形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。

For example, 2x² − 5x + 3 = 0 is a quadratic equation. If a = 0, the equation becomes linear, so the condition a ≠ 0 is essential.

例如，2x² − 5x + 3 = 0 是二次方程。若 a = 0，方程变成一次方程，所以条件 a ≠ 0 是必要的。

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

The terms in a quadratic expression are named as follows:

二次表达式中的各项名称如下：

ax² is the quadratic term.

ax² 是二次项。
bx is the linear term.

bx 是一次项。
c is the constant term.

c 是常数项。

To solve a quadratic equation, you often need to rearrange it into standard form: all terms on one side, equal to zero.

解二次方程时，通常需要将其重新整理为标准形式：所有项移到一边，等于零。

Example: x² + 3x = 10 can be rewritten as x² + 3x − 10 = 0.

例如：x² + 3x = 10 可改写为 x² + 3x − 10 = 0。

3. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the quadratic factors neatly. You look for two numbers that multiply to give ac and add to give b.

当二次式能整洁地分解时，因式分解是最快的方法。你需要找两个数，它们相乘等于 ac，相加等于 b。

For a simple quadratic where a = 1, find two numbers that multiply to c and add to b.

对于 a = 1 的简单二次式，找两个数相乘等于 c，相加等于 b。

Example: Solve x² + 5x + 6 = 0. The numbers 2 and 3 multiply to 6 and add to 5, so:

例：解 x² + 5x + 6 = 0。数字 2 和 3 相乘得 6，相加得 5，所以：

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

Then set each factor equal to zero:

然后令每个因式等于零：

x + 2 = 0 or x + 3 = 0

x = −2 or x = −3

So the solutions are x = −2 and x = −3.

所以解为 x = −2 和 x = −3。

4. Solving by Completing the Square | 配方法

Completing the square rewrites the quadratic in the form a(x + p)² + q. This method is useful when factorisation is difficult, and it is also the basis for deriving the quadratic formula.

配方法将二次式改写为 a(x + p)² + q 的形式。当因式分解困难时，这个方法很有用，它也是推导二次公式的基础。

To complete the square for x² + bx + c, use:

对于 x² + bx + c 配方，使用：

x² + bx + c = (x + b/2)² − (b/2)² + c

Example: Solve x² + 6x − 7 = 0 by completing the square.

例：用配方法解 x² + 6x − 7 = 0。

Half of 6 is 3, so:

6 的一半是 3，所以：

(x + 3)² − 9 − 7 = 0

(x + 3)² − 16 = 0

Then (x + 3)² = 16, so x + 3 = ±4, giving x = 1 or x = −7.

于是 (x + 3)² = 16，所以 x + 3 = ±4，得 x = 1 或 x = −7。

5. The Quadratic Formula | 二次公式

The quadratic formula works for all quadratic equations, including those that cannot be factorised. The formula is derived from completing the square.

二次公式适用于所有二次方程，包括无法因式分解的方程。该公式由配方法推导而来。

x = (−b ± √(b² − 4ac)) / (2a)

For ax² + bx + c = 0, substitute the values of a, b and c into the formula.

对于 ax² + bx + c = 0，将 a、b、c 的值代入公式。

Example: Solve 2x² − 4x − 3 = 0 using the formula.

例：用公式解 2x² − 4x − 3 = 0。

Here a = 2, b = −4, c = −3. Substitute:

这里 a = 2，b = −4，c = −3。代入：

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

Simplify √40 = 2√10, so x = (4 ± 2√10) / 4 = (2 ± √10) / 2.

化简 √40 = 2√10，所以 x = (4 ± 2√10) / 4 = (2 ± √10) / 2。

6. Using the Discriminant | 判别式

The discriminant is the part inside the square root: Δ = b² − 4ac. It tells you how many real roots a quadratic equation has.

判别式是根号内的部分：Δ = b² − 4ac。它告诉我们二次方程有多少个实数根。

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

若 Δ > 0，有两个不同的实数根。
If Δ = 0, there is exactly one repeated real root.

若 Δ = 0，有一个重复的实数根。
If Δ < 0, there are no real roots.

若 Δ < 0，没有实数根。

Example: For 3x² + 2x + 1 = 0, Δ = 4 − 12 = −8, so there are no real roots.

例：对于 3x² + 2x + 1 = 0，Δ = 4 − 12 = −8，所以没有实数根。

Exam questions often ask you to find the value of k for which an equation has equal roots. Set Δ = 0 and solve for k.

考试题常要求你找使方程有等根的 k 值。令 Δ = 0 并解出 k。

7. Solving Quadratics with Fractions or Decimals | 处理分数和小数

If a quadratic equation contains fractions, multiply every term by the common denominator to clear them.

如果二次方程含有分数，将所有项乘以公分母以消除分数。

Example: Solve x²/2 + x/3 − 1 = 0. The common denominator is 6.

例：解 x²/2 + x/3 − 1 = 0。公分母为 6。

6 × (x²/2) + 6 × (x/3) − 6 × 1 = 0

3x² + 2x − 6 = 0

Then solve using the quadratic formula because this quadratic does not factorise easily.

然后用二次公式求解，因为这个二次式不易因式分解。

For decimal coefficients, you can multiply by a power of 10 to turn them into integers.

对于小数系数，可以乘以10的幂，将其化为整数。

8. Word Problems with Quadratics | 二次方程应用题

Word problems often produce quadratic equations. You need to define a variable, set up an equation from the given information, solve it, and check whether the answers make sense in the context.

应用题通常产生二次方程。你需要设变量，根据给定信息建立方程，解方程，并检查答案在情境中是否合理。

Example: The area of a rectangle is 40 cm². Its length is 3 cm longer than its width. Find the width.

例：矩形的面积为 40 cm²。长比宽多 3 cm。求宽。

Let width = w, length = w + 3. Then w(w + 3) = 40.

设宽为 w，长为 w + 3。于是 w(w + 3) = 40。

w² + 3w − 40 = 0

Factorise: (w + 8)(w − 5) = 0, so w = −8 or w = 5. Since width cannot be negative, the width is 5 cm.

因式分解：(w + 8)(w − 5) = 0，所以 w = −8 或 w = 5。由于宽度不能为负，宽为 5 cm。

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

A quadratic function y = ax² + bx + c has a parabolic graph. The roots of the equation ax² + bx + c = 0 are the x-intercepts of the graph.

二次函数 y = ax² + bx + c 的图像是抛物线。方程 ax² + bx + c = 0 的根就是图像与 x 轴的交点。

If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point.

若 a > 0，抛物线开口向上，有最小值点。若 a < 0，开口向下，有最大值点。

The vertex (turning point) can be found by completing the square or using x = −b / (2a).

顶点（转向点）可通过配方或 x = −b / (2a) 求得。

Example: For y = x² − 4x + 3, completing the square gives y = (x − 2)² − 1, so the vertex is at (2, −1).

例：对于 y = x² − 4x + 3，配方得 y = (x − 2)² − 1，所以顶点为 (2, −1)。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

Many students lose marks by forgetting to rearrange the equation into standard form before applying the formula or factorisation.

许多学生因在应用公式或因式分解之前忘记将方程整理为标准形式而失分。

Always check that a ≠ 0, and be careful with negative signs when substituting into the quadratic formula.

始终检查 a ≠ 0，并且在代入二次公式时小心负号。

Another common error is forgetting the ± symbol when taking square roots.

另一个常见错误是在取平方根时忘记 ± 符号。

When solving word problems, reject answers that do not make sense in context, such as negative lengths or time.

在解应用题时，拒绝在情境中不合理的答案，如负数长度或时间。

Exam tip: If a quadratic factorises, factorise first. If not, use the formula. The discriminant is useful for checking how many roots to expect.

考试技巧：如果二次式能因式分解，先分解。如果不能，用公式。判别式有助于检查预期有多少个根。

Remember to show all steps clearly to earn method marks even if a calculation error occurs.

记住要清晰地写出所有步骤，这样即使出现计算错误也能获得方法分。

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Solving Quadratic Equations | 解二次方程