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. You will use them in algebra, geometry, and real-world problem solving.

二次方程是IGCSE数学中最重要的主题之一。在代数、几何以及实际生活问题解决中都会用到它们。

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

A quadratic equation is a polynomial equation of degree 2. This means the highest power of the variable is 2. The simplest examples include equations like x² = 9 or x² + 5x + 6 = 0.

二次方程是次数为2的多项式方程，这意味着变量的最高次幂是2。最简单的例子包括像x² = 9或x² + 5x + 6 = 0这样的方程。

Not every expression with x² is a quadratic equation. A quadratic equation always contains an equals sign and is true only for certain values of the variable, called solutions or roots.

并非所有含有x²的表达式都是二次方程。二次方程一定包含等号，并且仅对变量的某些值成立，这些值称为解或根。

2. The Standard Form | 标准形式

Every quadratic equation can be written in the standard form:

每个二次方程都可以写成标准形式：

ax² + bx + c = 0

Here a, b and c are constants, with a ≠ 0. If a = 0, the equation becomes linear, not quadratic. The value of a also determines whether the graph is a U shape or an inverted U shape.

这里a、b、c是常数，且a ≠ 0。如果a = 0，方程就变成线性的，而不是二次的。a的值还决定图形是U形还是倒U形。

For example, 3x² − 2x + 7 = 0 has a = 3, b = −2 and c = 7. Always rearrange the equation into this form before applying any solving method.

例如，3x² − 2x + 7 = 0具有a = 3，b = −2，c = 7。在应用任何求解方法之前，一定要先把方程整理成这种标准形式。

3. Solving by Factorising | 因式分解法

Factorising is often the quickest method when the quadratic has simple integer roots. To solve a quadratic equation by factorising, write the left-hand side as a product of two brackets.

当二次方程有简单的整数根时，因式分解通常是最快的方法。要通过因式分解解二次方程，请将左边写成两个括号的乘积。

For example, solve x² + 5x + 6 = 0. We need two numbers that multiply to give 6 and add to give 5. These numbers are 2 and 3, so the equation becomes:

例如，解x² + 5x + 6 = 0。我们需要两个数，相乘得6，相加得5。这两个数是2和3，因此方程变为：

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

If the product of two expressions is zero, then one of them must be zero. So x + 2 = 0 or x + 3 = 0, giving x = −2 or x = −3.

如果两个表达式的乘积为零，那么其中至少一个必须为零。所以x + 2 = 0或x + 3 = 0，得到x = −2或x = −3。

4. The Quadratic Formula | 二次求根公式

When factorising is difficult or impossible, you can always use the quadratic formula. This formula works for any quadratic equation in standard form ax² + bx + c = 0:

当因式分解困难或不可能时，你总是可以使用二次求根公式。这个公式适用于任何标准形式ax² + bx + c = 0的二次方程：

x = (−b ± √(b² − 4ac)) / (2a)

To use the formula, substitute the values of a, b and c carefully. The symbol ± means two solutions: one with a plus sign and one with a minus sign.

要使用该公式，请仔细代入a、b和c的值。符号±表示两个解：一个用加号，一个用减号。

For example, solve 2x² + 3x − 5 = 0. Here a = 2, b = 3, c = −5. Put these into the formula to find x = 1 or x = −2.5.

例如，解2x² + 3x − 5 = 0。这里a = 2，b = 3，c = −5。将这些值代入公式，得到x = 1或x = −2.5。

5. Completing the Square | 配方法

Completing the square is another algebraic method that also helps you find the turning point of a quadratic graph. The idea is to rewrite the expression in the form p(x + q)² + r.

配方法也是一种代数方法，同时还能帮助你找到二次图像的顶点。其思想是将表达式改写成p(x + q)² + r的形式。

Start with x² + bx. Add and subtract (b/2)² to create a perfect square. For example, for x² + 6x, note that (x + 3)² = x² + 6x + 9, so we write:

从x² + bx开始。加上并减去(b/2)²，以构成完全平方。例如，对于x² + 6x，注意(x + 3)² = x² + 6x + 9，因此我们写成：

x² + 6x = (x + 3)² − 9

You can solve an equation such as x² + 6x + 2 = 0 by rewriting it as (x + 3)² = 7 and then taking square roots.

你可以通过将x² + 6x + 2 = 0改写成(x + 3)² = 7，然后开平方来解这个方程。

6. Solving Equations with No Linear Term | 不含一次项的方程

If the equation has the form ax² + c = 0, you can solve it directly without factorising or using the formula. First isolate x², then take the square root of both sides.

如果方程具有ax² + c = 0的形式，你可以直接求解，而无需因式分解或使用公式。首先孤立x²，然后对方程两边开平方。

For example, solve 3x² − 12 = 0. Add 12 to both sides to get 3x² = 12. Divide by 3 to get x² = 4. Then x = ±2.

例如，解3x² − 12 = 0。两边加12得到3x² = 12。除以3得到x² = 4。然后x = ±2。

Remember that the square root of a positive number always gives two values: one positive and one negative. If x² equals a negative number, there are no real solutions.

记住，一个正数的平方根总是给出两个值：一个正数和一个负数。如果x²等于一个负数，则没有实数解。

7. The Discriminant | 判别式

The expression b² − 4ac inside the quadratic formula is called the discriminant. It tells us how many real solutions a quadratic equation has without solving it fully.

二次求根公式中的表达式b² − 4ac称为判别式。它无需完整求解就能告诉我们一个二次方程有多少个实数解。

If b² − 4ac > 0, there are two distinct real solutions. If b² − 4ac = 0, there is exactly one repeated real solution. If b² − 4ac < 0, there are no real solutions.

如果b² − 4ac > 0，则有两个不同的实数解。如果b² − 4ac = 0，则只有一个重根。如果b² − 4ac < 0，则没有实数解。

Δ = b² − 4ac

For example, the equation x² + 2x + 5 = 0 has Δ = 2² − 4 × 1 × 5 = −16, so it has no real solutions.

例如，方程x² + 2x + 5 = 0的Δ = 2² − 4 × 1 × 5 = −16，因此它没有实数解。

8. Roots and the Graph | 根与图像

The solutions of a quadratic equation correspond to the x-intercepts of the parabola y = ax² + bx + c. If a quadratic equation has two real roots, the graph crosses the x-axis at two points.

二次方程的解对应于抛物线y = ax² + bx + c的x轴截距。如果二次方程有两个实数根，则图像在两点处与x轴相交。

When the discriminant is zero, the graph touches the x-axis at exactly one point, called the vertex. When the discriminant is negative, the graph never touches the x-axis.

当判别式为零时，图像与x轴恰好相切于一点，称为顶点。当判别式为负时，图像与x轴不相交。

The axis of symmetry of the parabola lies halfway between the two roots, or at x = −b/(2a). This is also the x-coordinate of the turning point.

抛物线的对称轴位于两个根的正中间，或者为x = −b/(2a)。这也是顶点的x坐标。

9. Setting Up Quadratic Equations from Word Problems | 从应用题建立二次方程

Many real-life problems can be modelled by quadratic equations. For example, the area of a rectangle might be given in terms of an unknown length, producing a quadratic equation.

许多实际问题可以用二次方程建模。例如，矩形的面积可能由未知长度表示，从而产生二次方程。

Suppose a rectangle has length (x + 3) cm and width (x − 1) cm, and its area is 21 cm². Then (x + 3)(x − 1) = 21, which expands to x² + 2x − 3 = 21, so x² + 2x − 24 = 0.

假设一个矩形的长为(x + 3)厘米，宽为(x − 1)厘米，面积为21平方厘米。那么(x + 3)(x − 1) = 21，展开得x² + 2x − 3 = 21，即x² + 2x − 24 = 0。

Solving this gives x = 4 or x = −6. Since a length cannot be negative, we discard x = −6 and keep x = 4.

解这个方程得到x = 4或x = −6。由于长度不能为负，我们舍去x = −6，保留x = 4。

10. Common Mistakes | 常见错误

One common mistake is forgetting to rearrange the equation into standard form before using the formula. Always write ax² + bx + c = 0 first, and identify a, b and c correctly, including negative signs.

一个常见错误是在使用公式前忘记将方程整理成标准形式。务必先写出ax² + bx + c = 0，并正确识别a、b和c，包括负号。

Another mistake is forgetting the ± sign when taking square roots. For example, from x² = 16, you must write x = ±4, not just x = 4.

另一个错误是在开平方时忘记±符号。例如，由x² = 16，你必须写x = ±4，而不仅仅是x = 4。

When factorising, be careful with signs. For x² − 5x + 6, the factors are (x − 2)(x − 3), not (x + 2)(x + 3). Always expand your answer to check.

因式分解时，要小心符号。对于x² − 5x + 6，因式是(x − 2)(x − 3)，而不是(x + 2)(x + 3)。始终展开你的答案来检查。

11. Choosing the Best Method | 选择最佳方法

Sometimes you can choose between factorising, the quadratic formula, or completing the square. Factorising is fastest for simple integer roots, but it is not always possible.

有时你可以在因式分解、二次求根公式或配方法之间进行选择。对于简单的整数根，因式分解最快，但并非总是可行。

Use the quadratic formula when the coefficients are large or the roots are irrational. Use completing the square when you also need to find the maximum or minimum point of the graph.

当系数较大或根是无理数时，使用二次求根公式。当你还需要找到图像的最大值或最小值点时，使用配方法。

Always check the discriminant first. If it is a perfect square, factorising over integers may be possible. This saves time in the exam.

总是先检查判别式。如果它是一个完全平方数，则可能可以用整数因式分解。这能在考试中节省时间。

12. Summary and Exam Tips | 总结与考试技巧

Quadratic equations are a key part of the IGCSE Mathematics syllabus. You should be comfortable with the standard form, all three solving methods, and the discriminant.

二次方程是IGCSE数学大纲的关键部分。你应该熟悉标准形式、三种求解方法以及判别式。

In the exam, read the question carefully to see which method is expected. Sometimes the problem tells you to factorise or to use the quadratic formula. Show all steps clearly to earn full marks.

在考试中，仔细阅读题目，看看期望使用哪种方法。有时问题会告诉你因式分解或使用二次求根公式。请清晰地写出所有步骤，以获得满分。

Practice regularly with a variety of equations, including those with fractions and surds. The more fluent you become, the more confident you will feel on exam day.

定期练习各类方程，包括带有分数和根式的方程。你越熟练，考试时就越有信心。

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Solving Quadratic Equations | 解二次方程