Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations are among the most frequently tested topics in IGCSE Mathematics. Whether you are sitting for the Core or Extended paper, mastering the skill of solving quadratic equations is essential for success.

二次方程是IGCSE数学中最常考的内容之一。无论你参加Core还是Extended试卷，掌握解二次方程的技能都是取得好成绩的关键。

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

A quadratic equation is a polynomial equation of degree 2, written in the general form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of the variable x is 2, which gives the equation its name ‘quadratic’.

二次方程是最高次数为2的多项式方程，一般形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。变量 x 的最高次数是 2，因此得名”二次”。

Examples of quadratic equations include x² – 5x + 6 = 0, 2x² + 3x – 2 = 0, and x² = 9. Notice that in a quadratic equation, the coefficient a can never be zero, otherwise the equation becomes linear.

二次方程的例子包括 x² – 5x + 6 = 0、2x² + 3x – 2 = 0 和 x² = 9。注意在二次方程中，系数 a 不能为零，否则方程就变成一次方程了。

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2. Solving by Factorisation | 因式分解法

Factorisation is often the fastest method when the quadratic expression can be factored easily. The idea is to rewrite ax² + bx + c as a product of two linear brackets.

当二次表达式可以轻松分解时，因式分解法通常是最快的方法。其思路是将 ax² + bx + c 写成两个一次括号的乘积。

For the equation x² – 5x + 6 = 0, we look for two numbers that multiply to give 6 and add to give -5. These numbers are -2 and -3. Thus the equation factors to (x – 2)(x – 3) = 0.

对于方程 x² – 5x + 6 = 0，我们寻找两个数，乘积为 6、和为 -5。这两个数是 -2 和 -3。因此方程可分解为 (x – 2)(x – 3) = 0。

Using the zero product property, if the product of two factors is zero, then at least one factor must be zero. Hence x – 2 = 0 or x – 3 = 0, giving solutions x = 2 or x = 3.

根据零积性质，如果两个因式的乘积为零，则至少有一个因式为零。因此 x – 2 = 0 或 x – 3 = 0，解得 x = 2 或 x = 3。

When the coefficient of x² is not 1, such as in 2x² + 3x – 2 = 0, we must consider factor pairs of both the first and last coefficients. This expression factors to (2x – 1)(x + 2) = 0.

当 x² 的系数不为 1 时，例如 2x² + 3x – 2 = 0，我们需要同时考虑首项系数和常数项的因数对。该表达式分解为 (2x – 1)(x + 2) = 0。

Thus 2x – 1 = 0 or x + 2 = 0, which gives x = ½ or x = -2. Always expand your factored form to verify that you obtain the original equation.

因此 2x – 1 = 0 或 x + 2 = 0，得 x = ½ 或 x = -2。务必展开你的因式分解结果，以验证是否与原方程一致。

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3. Completing the Square | 配方法

Completing the square rewrites a quadratic expression in the form a(x + p)² + q. This method is particularly useful for finding the turning point of a quadratic graph and for solving equations that do not factorise easily.

配方法将二次表达式改写成 a(x + p)² + q 的形式。此方法在寻找二次图象的顶点以及解不易分解的方程时特别有用。

Consider x² + 6x – 7 = 0. Take half of the coefficient of x, which is 3, and write (x + 3)² = x² + 6x + 9. Since we need x² + 6x – 7, we subtract 16: (x + 3)² – 16 = 0.

考虑 x² + 6x – 7 = 0。取 x 系数的一半，即 3，写成 (x + 3)² = x² + 6x + 9。由于我们需要 x² + 6x – 7，所以减去 16：(x + 3)² – 16 = 0。

Now solve (x + 3)² = 16. Taking the square root of both sides gives x + 3 = ±4. Therefore x = 1 or x = -7.

现在解 (x + 3)² = 16。两边开平方得 x + 3 = ±4。因此 x = 1 或 x = -7。

For a general quadratic ax² + bx + c = 0, completing the square is best done after dividing all terms by a. The resulting form a(x + p)² + q reveals the minimum or maximum point at (-p, q).

对于一般二次方程 ax² + bx + c = 0，配方法最好在将所有项除以 a 之后进行。所得形式 a(x + p)² + q 直接揭示最小值或最大值点为 (-p, q)。

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4. The Quadratic Formula | 求根公式

The quadratic formula is a universal method that can solve any quadratic equation. For ax² + bx + c = 0, the solutions are given by:

求根公式是一种万能方法，能解任何二次方程。对于 ax² + bx + c = 0，解由下式给出：

x = (-b ± √(b² – 4ac)) / 2a

This formula is derived by completing the square on the general form, and it is listed in most IGCSE formula booklets. You should memorise it regardless, as it frequently appears in non-calculator papers.

这个公式通过对一般式配方推导而来，并且列在大多数IGCSE公式表中。无论如何你都应该记住它，因为它经常出现在不使用计算器的试卷中。

Let us solve 3x² – 5x – 2 = 0, where a = 3, b = -5, c = -2. Substituting into the formula: x = (5 ± √(25 – 4 × 3 × (-2))) / 6 = (5 ± √49) / 6 = (5 ± 7) / 6.

让我们解 3x² – 5x – 2 = 0，其中 a = 3，b = -5，c = -2。代入公式：x = (5 ± √(25 – 4 × 3 × (-2))) / 6 = (5 ± √49) / 6 = (5 ± 7) / 6。

Hence x = (5 + 7) / 6 = 2, or x = (5 – 7) / 6 = -⅓. Always simplify your surds and fractions fully in the final answer.

因此 x = (5 + 7) / 6 = 2，或 x = (5 – 7) / 6 = -⅓。在最终答案中务必彻底化简根式和分数。

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5. Solving Graphically | 图象解法

Quadratic equations can also be solved graphically by drawing the curve y = ax² + bx + c and reading off the x-intercepts. These points, where y = 0, correspond exactly to the real roots of the equation.

二次方程还可以通过绘图来解：画出曲线 y = ax² + bx + c，并读取其与 x 轴的交点。这些 y = 0 的点恰好对应方程的真实根。

For example, if the curve y = x² – 2x – 3 crosses the x-axis at x = -1 and x = 3, then the solutions of x² – 2x – 3 = 0 are x = -1 and x = 3. In an exam, answers from graphs are accepted with a tolerance of ±0.1 or ±0.2 units.

例如，若曲线 y = x² – 2x – 3 在 x = -1 和 x = 3 处与 x 轴相交，

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