Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. You will meet them in algebra, coordinate geometry, and many real-world problems. Learning to solve them confidently is essential for exam success.

一元二次方程是 IGCSE 数学中最重要的话题之一。在代数、坐标几何以及许多现实问题中，你都会遇到它们。学会熟练求解二次方程，是考试取得好成绩的关键。

1. What is a Quadratic Equation? | 什么是一元二次方程

A quadratic equation is an equation of degree 2, which means 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.

一元二次方程是次数为 2 的方程，也就是说变量的最高次数是 2。它的一般形式是 ax² + bx + c = 0，其中 a、b、c 是常数，且 a ≠ 0。

For example, x² − 5x + 6 = 0 and 2x² + 3x − 1 = 0 are quadratic equations. The equation x³ + 2x = 0 is not quadratic because its highest power is 3. In IGCSE Mathematics, the variable is usually x, but it can be any letter. Quadratic equations may appear in factorised form, expanded form, or as part of word problems.

例如，x² − 5x + 6 = 0 和 2x² + 3x − 1 = 0 都是一元二次方程。而 x³ + 2x = 0 不是二次方程，因为它的最高次数是 3。在 IGCSE 数学中，变量通常是 x，但也可以是其他字母。二次方程可能以因式分解形式、展开形式出现，也可能藏在应用题里。

2. Checking Solutions by Substitution | 代入检验解

A solution of a quadratic equation is a value of x that makes the equation true. You can check your answer by substituting it back into the original equation. For example, if x = 2 is a solution of x² − 5x + 6 = 0, then 2² − 5(2) + 6 = 4 − 10 + 6 = 0, so it works.

一元二次方程的解是使方程成立的 x 值。你可以把答案代回原方程来检验。例如，如果 x = 2 是 x² − 5x + 6 = 0 的一个解，那么 2² − 5(2) + 6 = 4 − 10 + 6 = 0，因此它成立。

Substitution is especially useful when you have used the quadratic formula and want to avoid arithmetic mistakes. If the left side does not equal zero after substitution, then the value is not a solution. Use a calculator to check values quickly, but always show the substitution step in your written working.

当你使用求根公式求解并希望避免计算错误时，代入检验尤其有用。如果代入后左边不等于零，那么这个值就不是方程的解。可以用计算器快速检验数值，但在书面作答中要写出代入步骤。

3. Factorising Quadratics (a = 1) | 因式分解二次式（a = 1）

When the coefficient of x² is 1, we look for two numbers whose product is c and whose sum is b. For a quadratic x² + bx + c = 0, if it factorises as (x + p)(x + q) = 0, then p + q = b and pq = c.

当 x² 的系数为 1 时，我们要找两个数，使它们的积为 c，和为 b。对于二次方程 x² + bx + c = 0，若能因式分解为 (x + p)(x + q) = 0，则 p + q = b，pq = c。

Example: Solve x² − 5x + 6 = 0. We need two numbers with product 6 and sum −5. These are −2 and −3, so the equation becomes (x − 2)(x − 3) = 0. Therefore x = 2 or x = 3.

例：解 x² − 5x + 6 = 0。我们需要找积为 6、和为 −5 的两个数。它们是 −2 和 −3，所以原方程化为 (x − 2)(x − 3) = 0。因此 x = 2 或 x = 3。

If c is positive, both numbers have the same sign. If c is negative, the two numbers have opposite signs. Remember: if the product of two factors is zero, at least one factor must be zero. This is called the zero-product property.

如果 c 是正数，那么两个数同号；如果 c 是负数，那么两个数异号。记住：如果两个因式的乘积为零，那么至少有一个因式为零。这叫做“零积性质”。

4. Factorising Quadratics (a > 1) | 因式分解二次式（a > 1）

When the coefficient of x² is not 1, factorising requires more care. For example, solve 2x² + 7x + 3 = 0. We look for two numbers whose product is 2 × 3 = 6 and whose sum is 7. The numbers are 1 and 6, so rewrite the middle term: 2x² + x + 6x + 3 = 0.

当 x² 的系数不是 1 时，因式分解需要更仔细。例如，解 2x² + 7x + 3 = 0。我们要找两个数，使它们的积为 2 × 3 = 6，和为 7。这两个数是 1 和 6，所以改写中间项：2x² + x + 6x + 3 = 0。

Then factor by grouping: x(2x + 1) + 3(2x + 1) = 0, so (2x + 1)(x + 3) = 0. Hence x = −½ or x = −3.

然后分组分解：x(2x + 1) + 3(2x + 1) = 0，因此 (2x + 1)(x + 3) = 0。所以 x = −½ 或 x = −3。

If the coefficient of x² is negative, factor out −1 first. Always check whether the equation can be simplified first. If all terms have a common factor, divide by it before factorising. With practice, you can often spot the correct binomial pair directly.

如果 x² 的系数是负数，先提出 −1。始终先检查方程能否化简。如果所有项都有公因数，先除以公因数再进行因式分解。多加练习后，你通常可以直接看出正确的二项式因式。

5. Difference of Two Squares | 平方差公式

Some quadratics have no middle term and can be written as a² − b². The identity is a² − b² = (a − b)(a + b). For example, x² − 9 = 0 can be written as (x − 3)(x + 3) = 0, so x = 3 or x = −3.

有些二次式没有中间项，可以写成 a² − b² 的形式。恒等式为 a² − b² = (a − b)(a + b)。例如，x² − 9 = 0 可写成 (x − 3)(x + 3) = 0，所以 x = 3 或 x = −3。

This also works for expressions like 4x² − 25 = 0. Since 4x² = (2x)² and 25 = 5², we get (2x − 5)(2x + 5) = 0, so x = 5/2 or x = −5/2.

这个方法同样适用于 4x² − 25 = 0。因为 4x² = (2x)²，25 = 5²，所以得到 (2x − 5)(2x + 5) = 0，故 x = 5/2 或 x = −5/2。

This identity is also useful when manipulating algebraic fractions or simplifying expressions later in the course. It works exactly the same way if the squared term

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