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 the IGCSE mathematics syllabus. You are expected to solve them confidently using multiple methods, interpret their solutions graphically, and apply them to real-world problems.

二次方程是 IGCSE 数学大纲中最重要的话题之一。你被期望能够自信地运用多种方法解二次方程，从图像上解释它们的解，并将它们应用于现实问题中。

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

A quadratic equation is an equation where the highest power of the variable is 2. The standard general form is:

二次方程是变量最高次数为 2 的方程。它的一般标准形式为：

ax² + bx + c = 0, a ≠ 0

Here a, b and c are constants, with a not equal to zero. If a = 0, the equation becomes linear.

这里的 a、b 和 c 是常数，且 a 不等于 0。如果 a = 0，那么这个方程就变成了线性方程。

The solutions of a quadratic equation are called the roots or solutions. Graphically, they are the x-coordinates where the parabola crosses the x-axis.

二次方程的解被称为根或解。从图像上看，它们是抛物线与 x 轴交点的 x 坐标。

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2. Standard Form and Key Features | 标准形式与关键特征

on in the form ax² + bx + c = 0. This makes factorisation and formula substitution easier.

在求解之前，一定要把方程重写为 ax² + bx + c = 0 的形式。这会让因式分解和代入公式更加容易。

• a is the coefficient of x², b is the coefficient of x, and c is the constant term.

  a 是 x² 的系数，b 是 x 的系数，c 是常数项。

• If the equation is not given in standard form, expand brackets, collect like terms, and move everything to one side of the equals sign.

  如果方程不是标准形式，需要展开括号、合并同类项，并把所有项移到等号的一侧。

• = ax² + bx + c is a parabola. When a > 0, it opens upward; when a < 0, it opens downward.

  y = ax² + bx + c 的图像是一条抛物线。当 a > 0 时开口向上；当 a < 0 时开口向下。

For example, rewrite 5x - 3 = 2x² as 2x² - 5x + 3 = 0. Here a = 2, b = -5, c = 3.

例如，将 5x - 3 = 2x² 改写为 2x² - 5x + 3 = 0。这里 a = 2，b = -5，c = 3。

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3. Solving by Factorisation | 因式分解法

Factorisation is the quickest method when the quadratic has simple integer coefficients. The idea is to express the quadratic as a product of two linear factors.

当二次方程具有简单的整数系数时，因式分解是最快捷的方法。其思想是把二次式表示为两个一次因式的乘积。

Follow these steps:

按以下步骤操作：

1. Write the equation in the form ax² + bx + c = 0.

   把方程写成 ax² + bx + c = 0 的形式。

2. Factorise the left-hand side into two brackets.

   把左边分解成两个括号的乘积。

3. Set each bracket equal to zero and solve the resulting linear equations.

   令每个括号等于零，并解出相应的一次方程。

Example: Solve x² - 5x + 6 = 0.

示例：解 x² - 5x + 6 = 0。

(x - 2)(x - 3) = 0

So x - 2 = 0 or x - 3 = 0, giving x = 2 or x = 3.

于是 x - 2 = 0 或 x - 3 = 0，得到 x = 2 或 x = 3。

Remember: the product of two numbers is zero only when at least one of them is zero.

记住：两个数的乘积为零，当且仅当至少有一个数为零。

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4. Solving by Completing the Square | 配方法

Completing the square turns any quadratic into a perfect square plus a constant. It is especially useful when factorisation is not straightforward.

配方法将任意二次式化为一个完全平方加上一个常数。当因式分解不直接时，它尤其有用。

Key identity:

关键恒等式：

x² + bx = (x + b/2)² - (b/2)²

Example: Solve x² + 6x + 8 = 0 by completing the square.

示例：用配方法解 x² + 6x + 8 = 0。

(x + 3)² - 9 + 8 = 0 → (x + 3)² = 1

Then x + 3 = ±√1 = ±1, so x = -3 ± 1. That gives x = -2 or x = -4.

因此 x + 3 = ±√1 = ±1，所以 x = -3 ± 1。得到 x = -2 或 x = -4。

For equations where a ≠ 1, first divide all terms by a before completing the square.

当 a ≠ 1 时，先对等式两边除以 a，然后再配方。

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5. The Quadratic Formula | 二次公式

The quadratic formula works for every quadratic equation. You should memorise it and know when to use it:

二次公式适用于所有二次方程。你应该牢记它，并知道何时使用：

x = (-b ± √b² - 4ac) / (2a)

his formula is derived by completing the square on the general quadratic ax² + bx + c = 0.

这个公式是通过对一般二次方程 ax² + bx + c = 0 配方而推导出来的。

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

示例：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 ± √((-4)² - 4×2×(-3))) / (2×2) = (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。

Always check whether the answer should be given in exact form or as a decimal.

始终检查题目要求答案是精确形式还是小数形式。

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6. The Discriminant | 判别式

The expression Δ = b² - 4ac under the square root is called the discriminant. It tells you how many real roots a quadratic equation has without solving it fully.

根号下的表达式 Δ = b² - 4ac 被称为判别式。不必完全求解，它就能告诉你二次方程有多少个实根。

• If Δ > 0: two distinct real roots.

  如果 Δ > 0：有两个不同的实根。

• If Δ = 0: one repeated real root (the parabola touches the x-axis).

  如果 Δ = 0：有一个重根（抛物线与 x 轴相切）。

• If Δ < 0: no real roots (the parabola does not touch the x-axis).

  如果 Δ < 0：没有实根（抛物线与 x 轴不相交）。

For x² - 4x + 4 = 0, Δ = (-4)² - 4×1×4 = 16 - 16 = 0, so there is exactly one root, x = 2.

示例：对于 x² - 4x + 4 = 0，Δ = (-4)² - 4×1×4 = 16 - 16 = 0，因此只有一个根，x = 2。

The discriminant is also useful when solving problems involving tangents and line intersections.

判别式在解决切线和直线与曲线交点的问题时也很有用。

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7. Solving Word Problems | 解应用题

Many exam questions describe a real situation and require you to form a quadratic equation. Read carefully and define your variable first.

许多考试题目描述现实情境，并要求你建立二次方程。请仔细阅读，首先定义变量。

Example: The length of a rectangle is 3 cm more than its width. Its area is 28 cm². Find the width.

示例：一个矩形的长比宽多 3 cm。它的面积为 28 cm²。求宽。

Let width = x. Then length = x + 3. Area: x(x + 3) = 28.

设宽为 x。则长为 x + 3。面积：x(x + 3) = 28。

x² + 3x - 28 = 0 → (x + 7)(x - 4) = 0

So x = -7 or x = 4. Since width cannot be negative, the width is 4 cm.

因此 x = -7 或 x = 4。由于宽不能为负，所以宽为 4 cm。

Always interpret your solutions in the context of the problem and reject any unreasonable ones.

始终在题目情境中解释你的解，并舍弃不合理的解。

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8. Common Mistakes to Avoid | 常见错误

Many students lose marks unnecessarily because of small errors. Be careful with the following:

许多学生因为小错误而失分。请特别注意以下几点：

• Forgetting to rearrange the equation to standard form before factorising.

  在因式分解前忘记将方程整理成标准形式。

• Sign errors when substituting into the quadratic formula, especially with negative b.

  代入二次公式时出现符号错误，尤其是当 b 为负数时。

• Forgetting that the equation has two solutions, except when Δ = 0.

  忘记方程有两个解，除非 Δ = 0。

• Dividing both sides by a factor containing x, which loses a root.

  两边同时除以含有 x 的因式，导致失去一个根。

• Ignoring a negative root in a geometry context.

  在几何情境中忽略负根。

Check your answers by substituting them back into the original equation.

通过把答案代回原方程来检查是否正确。

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9. Practice Tips | 练习建议

To master quadratic equations, you need repeated, focused practice. Here are some effective strategies:

要掌握二次方程，你需要反复、专注地练习。以下是一些有效策略：

• Start each question by identifying the method: factorise, complete the square, or use the formula.

  每道题开始时先确定方法：因式分解、配方，还是使用公式。

• Practise factorising harder expressions like 6x² - x - 2.

  练习分解更复杂的式子，如 6x² - x - 2。

• Work on completing the square for expressions with fractional coefficients.

  练习对含有分数系数的式子进行配方。

• Use past paper questions to see how quadratics are tested in IGCSE.

  使用真题来了解 IGCSE 中二次方程如何考查。

• Time yourself to improve speed and accuracy.

  给自己计时，提高速度和准确性。

Consistent practice is the key to avoiding careless errors under exam pressure.

持续练习是避免考试压力下粗心错误的关键。

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10. Exam Strategies | 考试策略

In the IGCSE exam, quadratic questions can appear in both non-calculator and calculator papers. Adjust your approach accordingly.

在 IGCSE 考试中，二次方程问题可能出现在非计算器试卷和计算器试卷中。请相应调整你的方法。

Recommendations:

建议如下：

• On a non-calculator paper, try factorisation first. If it does not factor neatly, use completing the square.

  在非计算器试卷上，优先尝试因式分解。如果无法整洁分解，则使用配方法。

• On a calculator paper, the quadratic formula is often the safest choice, especially for awkward coefficients.

  在计算器试卷上，二次公式通常是最安全的选择，尤其是系数较复杂时。

• Read the question carefully: do they ask for exact values or 3 decimal places?

  仔细读题：题目要求精确值还是保留三位小数？

• Show all working clearly. Even if your final answer is wrong, you can earn method marks.

  清晰地展示所有步骤。即使最终答案错误，你也能获得方法分。

• Always interpret the discriminant when a question asks about the nature of roots.

  当问题问及根的性质时，一定要使用判别式。

By mastering these techniques, you will approach every quadratic equation with confidence.

通过掌握这些技巧，你将自信地应对每道二次方程题。

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