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. In this article, we will explore how to recognise, solve, and apply quadratic equations using multiple methods, including factorisation, the quadratic formula, completing the square, and graphical interpretation. This comprehensive revision guide is designed to build your confidence and ensure exam success.

二次方程是 IGCSE 数学考纲中最重要的内容之一。在本文中，我们将探讨如何识别、求解和应用二次方程，涵盖因式分解法、二次公式法、配方法以及图形解释等多种方法。本复习指南旨在帮助你建立信心，确保考试成功。

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1. Standard Form of a Quadratic Equation | 二次方程的标准形式

A quadratic equation is any equation that can be written in the general form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The coefficient a is called the leading coefficient, b is the linear coefficient, and c is the constant term.

二次方程是指可以写成一般形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。系数 a 称为二次项系数，b 称为一次项系数，c 称为常数项。

• If a = 0, the equation becomes linear, not quadratic. | 如果 a = 0，方程退化为一次方程，而非二次方程。
• The highest power of x in a quadratic equation is always 2. | 二次方程中 x 的最高次幂始终为 2。
• Quadratic equations can have 0, 1, or 2 real solutions. | 二次方程可能有 0 个、1 个或 2 个实数解。

ax² + bx + c = 0 (a ≠ 0)

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2. Solving by Factorisation | 因式分解法

Factorisation is often the fastest method for solving quadratic equations when the expression can be factorised into two linear factors. The method relies on the zero product property: if the product of two factors is zero, then at least one of the factors must be zero.

因式分解法通常是解二次方程最快的方法，前提是表达式可以分解为两个一次因式。该方法依赖于零积性质：如果两个因式的乘积为零，则至少有一个因式为零。

Follow these steps: | 遵循以下步骤：

• Rearrange the equation so that one side equals zero. | 整理方程，使一边等于零。
• Factorise the quadratic expression completely. | 将二次表达式完全因式分解。
• Set each factor equal to zero and solve the resulting linear equations. | 令每个因式等于零，并求解所得的一次方程。

Example: Solve x² − 5x + 6 = 0. | 示例：解 x² − 5x + 6 = 0。

We look for two numbers that multiply to 6 and add to −5; these are −2 and −3. Thus, the factorisation is (x − 2)(x − 3) = 0. Setting each factor to zero gives x = 2 or x = 3.

我们需要找到两个数，它们的乘积为 6，和为 −5；这两个数是 −2 和 −3。因此，因式分解为 (x − 2)(x − 3) = 0。令每个因式等于零，得到 x = 2 或 x = 3。

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3. Difference of Two Squares | 平方差公式

A special type of quadratic expression is the difference of two squares: x² − p² = (x − p)(x + p). This is extremely useful for certain equations where the b term is zero.

一种特殊的二次表达式是平方差：x² − p² = (x − p)(x + p)。当 b 项为零时，这对于求解某些方程极其有用。

Example: Solve x² − 49 = 0. | 示例：解 x² − 49 = 0。

Since 49 = 7², we can write (x − 7)(x + 7) = 0. Therefore, x = 7 or x = −7.

由于 49 = 7²，我们可以写成 (x − 7)(x + 7) = 0。因此，x = 7 或 x = −7。

x² − p² = (x − p)(x + p)

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4. Perfect Square Trinomials | 完全平方式

A perfect square trinomial is a quadratic expression that can be written as (x ± p)². Recognising this pattern can significantly speed up your factorisation.

完全平方式是可以写成 (x ± p)² 的二次表达式。识别这种模式可以显著加快因式分解速度。

For example, x² + 6x + 9 = (x + 3)² because 2 × 3 = 6 and 3² = 9. Similarly, x² − 10x + 25 = (x − 5)².

例如，x² + 6x + 9 = (x + 3)²，因为 2 × 3 = 6 且 3² = 9。类似地，x² − 10x + 25 = (x − 5)²。

• Check that the constant term is a perfect square. | 检查常数项是否为完全平方数。
• Check that the middle term equals ±2 times the product of the square roots. | 检查中间项是否等于两个平方根乘积的 ±2 倍。

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5. Using the Quadratic Formula | 二次公式法

When factorisation is difficult or impossible, we can always use the quadratic formula. This formula provides the solutions for any quadratic equation in the form ax² + bx + c = 0.

当因式分解困难或无法进行时，我们始终可以使用二次公式。该公式为任何形式为 ax² + bx + c = 0 的二次方程提供解的表达式。

x = (−b ± √(b² − 4ac)) / 2a

To use the formula, substitute the values of a, b, and c carefully, then simplify. The symbol ± means you must compute both x with plus and x with minus.

使用该公式时，请仔细代入 a、b、c 的值，然后化简。符号 ± 表示需要分别计算取加号和取减号时 x 的值。

Example: Solve 2x² + 3x − 2 = 0. | 示例：解 2x² + 3x − 2 = 0。

Here a = 2, b = 3, c = −2. Substituting into the formula gives:

这里 a = 2，b = 3，c = −2。代入公式得到：

x = (−3 ± √(9 + 16)) / 4 = (−3 ± 5) / 4

Thus x = 0.5 or x = −2. | 因此 x = 0.5 或 x = −2。

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6. The Discriminant | 判别式

The expression b² − 4ac inside the square root is called the discriminant. It tells us the nature of the roots without solving the full equation.

根号内的表达式 b² − 4ac 称为判别式。它告诉我们根的性质，而无需完整解方程。

Discriminant (Δ) | 判别式	Nature of roots | 根的性质
Δ > 0 and a perfect square | Δ > 0 且为完全平方数	Two rational, distinct roots | 两个不相等的有理根
Δ > 0 but not a perfect square | Δ > 0 但不是完全平方数	Two irrational, distinct roots | 两个不相等的无理根
Δ = 0	One repeated (double) root | 一个重根
Δ < 0	No real roots | 无实数根

Example: Determine the nature of the roots of x² − 4x + 4 = 0. Since Δ = 16 − 16 = 0, the equation has one repeated root.

示例：判断 x² − 4x + 4 = 0 的根的性质。由于 Δ = 16 − 16 = 0，该方程有一个重根。

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7. Completing the Square | 配方法

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

配方法是一种将 ax² + bx + c 改写为 a(x + p)² + q 形式的技术。该方法在求顶点坐标或因式分解法失效时非常有用。

For x² + bx, add (b/2)² and subtract it to maintain equality:

对于 x² + bx，加上 (b/2)² 并减去（b/2)² 以保持等式平衡：

x² + bx = (x + b/2)² − (b/2)²

Example: Solve x² + 6x + 2 = 0 by completing the square. | 示例：用配方法解 x² + 6x + 2 = 0。

(x + 3)² − 9 + 2 = 0, so (x + 3)² = 7. Therefore x = −3 ± √7.

(x + 3)² − 9 + 2 = 0，因此 (x + 3)² = 7。所以 x = −3 ± √7。

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8. Graphical Interpretation | 图形解释

A quadratic function y = ax² + bx + c produces a parabola. The roots of the equation ax² + bx + c = 0 correspond to the x-intercepts of the parabola, where the graph crosses the x-axis.

二次函数 y = ax² + bx + c 的图象是一条抛物线。方程 ax² + bx + c = 0 的根对应于抛物线与 x 轴的交点。

• Two distinct roots: parabola crosses the x-axis at two points. | 两个不等根：抛物线与 x 轴交于两点。
• One repeated root: parabola touches the x-axis at one point (the vertex). | 一个重根：抛物线在顶点处与 x 轴相切。
• No real roots: parabola does not intersect the x-axis at all. | 无实数根：抛物线与 x 轴没有交点。

The vertex (turning point) of the parabola y = ax² + bx + c occurs at x = −b/2a. Substituting this value gives the y-coordinate of the vertex.

抛物线 y = ax² + bx + c 的顶点出现在 x = −b/2a 处。将该值代入即可得到顶点的 y 坐标。

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9. Solving Quadratic Inequalities | 解二次不等式

Quadratic inequalities appear frequently in IGCSE exams. To solve an inequality such as x² − 5x + 6 < 0, first solve the corresponding equation, then test intervals on a number line.

二次不等式在 IGCSE 考试中经常出现。要解形如 x² − 5x + 6 < 0 的不等式，首先解对应的方程，然后在数轴上测试各个区间。

For x² − 5x + 6 = 0, we already found x = 2 or x = 3. The parabola opens upwards, so it is negative between the roots. Therefore the solution is 2 < x < 3.

对于 x² − 5x + 6 = 0，我们已求出 x = 2 或 x = 3。抛物线开口向上，因此在两根之间函数值为负。所以解集为 2 < x < 3。

• If the sign is < or ≤, the solution is between the roots (when a > 0). | 如果符号是 < 或 ≤，当 a > 0 时解在两根之间。
• If the sign is > or ≥, the solution is outside the roots (when a > 0). | 如果符号是 > 或 ≥，当 a > 0 时解在两根之外。

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

Quadratic equations frequently appear in real-world problems. Common scenarios include area calculations, projectile motion, and number problems.

二次方程经常出现在实际应用题中。常见情境包括面积计算、抛体运动和数字问题。

Example: A rectangle has length (x + 5) cm and width (x − 1) cm. Its area is 36 cm². Find x.

示例：一个矩形的长为 (x + 5) cm，宽为 (x − 1) cm，面积为 36 cm²。求 x。

We set up the equation (x + 5)(x − 1) = 36, which expands to x² + 4x − 5 = 36, or x² + 4x − 41 = 0. Using the quadratic formula gives x ≈ 4.32 or x ≈ −9.32. Since x represents a positive dimension, we take x ≈ 4.32.

我们建立方程 (x + 5)(x − 1) = 36，展开得 x² + 4x − 5 = 36，即 x² + 4x − 41 = 0。使用二次公式可得 x ≈ 4.32 或 x ≈ −9.32。由于 x 表示正尺寸，我们取 x ≈ 4.32。

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11. Common Mistakes to Avoid | 常见错误提醒

Many students lose marks on quadratic equations due to avoidable errors. Here are the most common pitfalls:

许多学生因为可避免的错误在二次方程上失分。以下是常见的陷阱：

• Forgetting to rearrange the equation to equal zero before factorising. | 因式分解前忘记将方程整理成等于零的形式。
• Sign errors when substituting negative values into the quadratic formula. | 将负值代入二次公式时出现符号错误。
• Dividing both sides by x when x could be zero, losing a solution. | 在 x 可能为零时两边同除以 x，从而丢失一个解。
• Incorrectly stating the range of a quadratic inequality. | 错误描述二次不等式的解集范围。
• Forgetting that a quadratic can have two solutions. | 忘记二次方程可能有两个解。

Careful algebraic manipulation and always checking your answers by substitution will prevent most of these errors.

仔细进行代数运算，并始终通过代入来检验答案，可以避免大多数此类错误。

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12. Exam Tips and Summary | 考试技巧与总结

In the IGCSE Edexcel exam, quadratic equations can appear in Paper 1 (non-calculator) and Paper 2 (calculator). You must be comfortable with all methods of solving quadratics.

在 IGCSE Edexcel 考试中，二次方程可能出现在 Paper 1（不能使用计算器）和 Paper 2（可以使用计算器）中。你必须熟练掌握所有解二次方程的方法。

• Always identify a, b, and c correctly before applying the formula. | 在使用公式前，务必正确识别 a、b、c。
• Check whether the discriminants is positive, zero, or negative early in the problem. | 在解题初期先判断判别式为正、零还是负。
• In non-calculator papers, look for simple factorisation first. | 在无计算器的试卷中，优先寻找简单的因式分解。
• When the question asks for answers to a certain number of decimal places, use the formula or your calculator directly. | 当题目要求保留特定小数位数时，直接用公式或计算器求解。

a ≠ 0, ax² + bx + c = 0 → x = (−b ± √(b² − 4ac)) / 2a

Mastering quadratic equations opens the door to advanced topics like the quadratic function, completing the square, and coordinate geometry. Keep practising with past papers to build fluency and speed. Good luck in your revision!

熟练掌握二次方程将为学习二次函数、配方法和坐标几何等高级内容打开大门。坚持用历年真题练习，提高熟练度和速度。祝复习顺利，考试成功！

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