Solving Quadratic Equations by Factorisation, Completing the Square and the Quadratic Formula | 解二次方程：因式分解、配方法与求根公式

📚 Solving Quadratic Equations by Factorisation, Completing the Square and the Quadratic Formula | 解二次方程：因式分解、配方法与求根公式

In IGCSE Mathematics, quadratic equations appear in many contexts, from projectile problems to area calculations. A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. There are three main algebraic methods for solving such equations: factorisation, completing the square, and the quadratic formula. Mastering all three gives you flexibility in exams and helps you check your work.

在 IGCSE 数学中，二次方程广泛出现于抛体运动、面积计算等情境。二次方程是可以写成标准形式 ax² + bx + c = 0 的任何方程，其中 a、b、c 为常数且 a ≠ 0。解这类方程有三种主要代数方法：因式分解法、配方法和求根公式法。熟练掌握这三种方法会让你在考试中灵活选择，并帮助你检验答案。

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

A quadratic equation is a polynomial equation of degree 2. Its highest power of the unknown is 2. The most general form is ax² + bx + c = 0, where x is the unknown, and a, b, c are real numbers with a ≠ 0. If a were 0, the equation would become linear, not quadratic.

二次方程是次数为 2 的多项式方程。未知数的最高次幂是 2。最一般的形式是 ax² + bx + c = 0，其中 x 是未知数，a、b、c 是实数且 a ≠ 0。如果 a = 0，方程就变成一次方程，而不是二次方程。

Some examples include x² − 5x + 6 = 0, 3x² + 2x − 1 = 0 and 4x² − 9 = 0. Notice that not all quadratic equations have every term; a quadratic can have no x term (b = 0) or no constant term (c = 0). The graph of a quadratic equation is called a parabola, and solving the equation tells us where the parabola crosses the x-axis.

示例包括 x² − 5x + 6 = 0、3x² + 2x − 1 = 0 和 4x² − 9 = 0。注意，并不是所有二次方程都包含每一项；二次方程可以没有 x 项（b = 0），也可以没有常数项（c = 0）。二次方程的图像叫做抛物线，解方程就是求抛物线与 x 轴的交点。

2. Standard Form ax² + bx + c = 0 | 标准形式 ax² + bx + c = 0

Before solving, always rearrange the equation into standard form. Move all terms to one side so the other side equals zero. Expand brackets if necessary, and collect like terms. For example, x² + 3x = 4 becomes x² + 3x − 4 = 0.

解题前，一定要先把方程化为标准形式。把所有项移到一边，使另一边等于零。必要时展开括号并合并同类项。例如，x² + 3x = 4 要化为 x² + 3x − 4 = 0。

Key steps for writing standard form:

写成标准形式的关键步骤：

Expand all brackets first. | 先展开所有括号。
Move all terms to the left-hand side. | 把所有项移到等号左边。
Write terms in descending powers of x. | 按 x 的降幂排列各项。
Set the right-hand side equal to 0. | 让等号右边等于 0。

Standard form is essential because factorisation, completing the square and the quadratic formula all require the equation to be set equal to zero. If the equation is not in standard form, you may accidentally miss a solution or apply a formula incorrectly.

标准形式至关重要，因为因式分解法、配方法和求根公式法都要求方程等于零。如果方程不是标准形式，你可能会漏掉解或错误地套用公式。

3. Solving by Factorisation | 因式分解法

Factorisation is often the fastest method when the quadratic expression can be written as a product of two linear factors. The key principle is the zero product property: if two factors multiply to give zero, then at least one of them must be zero. That is, if A × B = 0, then A = 0 or B = 0.

当二次式可以写成两个一次因式的乘积时，因式分解法通常是最快的。其核心原理是零乘积性质：如果两个因式相乘为零，那么其中至少有一个因式为零。也就是说，如果 A × B = 0，那么 A = 0 或 B = 0。

For simple quadratics with a = 1, find two numbers whose sum equals b and whose product equals c. For example, to factorise x² + 7x + 12, we need two numbers with sum 7 and product 12: they are 3 and 4. Thus x² + 7x + 12 = (x + 3)(x + 4). Setting each factor to zero gives x = −3 or x = −4.

对于 a = 1 的简单二次式，找出两个数，使它们的和等于 b，乘积等于 c。例如，要分解 x² + 7x + 12，我们需要两个数，和为 7，积为 12：它们就是 3 和 4。因此 x² + 7x + 12 = (x + 3)(x + 4)。令每个因式等于零，得到 x = −3 或 x = −4。

When a ≠ 1, factorisation requires more care. Use trial and error or the ac method. For 2x² + 5x + 2, multiply a and c: 2 × 2 = 4. We need two numbers whose sum is 5 and product is 4: 1 and 4. Split the middle term: 2x² + x + 4x + 2 = x(2x + 1) + 2(2x + 1) = (x + 2)(2x + 1). Then solve (x + 2)(2x + 1) = 0 to get x = −2 or x = −1/2.

当 a ≠ 1 时，因式分解需要更细心。可以使用试错法或 ac 法。对于 2x² + 5x + 2，先把 a 和 c 相乘：2 × 2 = 4。我们需要两个数，和为 5，积为 4：它们是 1 和 4。将中间项拆开：2x² + x + 4x + 2 = x(2x + 1) + 2(2x + 1) = (x + 2)(2x + 1)。然后解 (x + 2)(2x + 1) = 0，得到 x = −2 或 x = −1/2。

4. Worked Example: Factorisation | 例题：因式分解法

Solve x² + 5x + 6 = 0 by factorisation.

用因式分解法解方程 x² + 5x + 6 = 0。

Find two numbers whose product is 6 and sum is 5: 2 and 3. So x² + 5x + 6 = (x + 2)(x + 3). The equation becomes (x + 2)(x + 3) = 0. Set each factor to zero: x + 2 = 0 gives x = −2; x + 3 = 0 gives x = −3.

找出乘积为 6，和为 5 的两个数：2 和 3。因此 x² + 5x + 6 = (x + 2)(x + 3)。方程变为 (x + 2)(x + 3) = 0。令每个因式为零：x + 2 = 0 得 x = −2；x + 3 = 0 得 x = −3。

Solutions: x = −3 or x = −2

解为：x = −3 或 x = −2

Always check by substituting back into the original equation. For x = −2: (−2)² + 5(−2) + 6 = 4 − 10 + 6 = 0. For x = −3: (−3)² + 5(−3) + 6 = 9 − 15 + 6 = 0. Both values satisfy the equation.

一定要把解代回原方程检验。当 x = −2 时：(−2)² + 5(−2) + 6 = 4 − 10 + 6 = 0。当 x = −3 时：(−3)² + 5(−3) + 6 = 9 − 15 + 6 = 0。两个值都满足方程。

5. Completing the Square | 配方法

Completing the square rewrites the quadratic expression as a perfect square plus or minus a constant. This method always works, even when the quadratic cannot be factorised using integers. It is also the method used to derive the quadratic formula and to find the vertex of a parabola.

配方法把二次式改写成一个完全平方加上或减去一个常数。这种方法始终有效，即使二次式不能用整数因式分解。它也是推导求根公式和求抛物线顶点的方法。

For a quadratic with leading coefficient 1, x² + bx + c can be written as (x + b/2)² − (b/2)² + c. If the leading coefficient is not 1, first divide every term by a. Then complete the square on the resulting expression.

对于首项系数为 1 的二次式，x² + bx + c 可以写成 (x + b/2)² − (b/2)² + c。如果首项系数不是 1，先把每一项都除以 a，然后再对得到的式子进行配方。

x² + bx + c = (x + b/2)² − (b/2)² + c

The steps are: move the constant term to the right side, add (b/2)² to both sides, write the left side as a perfect square, then take the square root of both sides. Remember to include both positive and negative square roots.

配方法的步骤是：把常数项移到右边，两边都加上 (b/2)²，把左边写成完全平方，然后对两边开平方。记得开平方时要同时取正负两个平方根。

6. Worked Example: Completing the Square | 例题：配方法

Solve x² + 6x + 5 = 0 by completing the square.

用配方法解方程 x² + 6x + 5 = 0。

Move the constant to the right: x² + 6x = −5. Add (6/2)² = 9 to both sides: x² + 6x + 9 = −5 + 9. Write the left side as a square: (x + 3)² = 4. Take the square root of both sides: x + 3 =

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Solving Quadratic Equations by Factorisation, Completing the Square and the Quadratic Formula | 解二次方程：因式分解、配方法与求根公式