Mastering Quadratic Equations for IGCSE Mathematics | IGCSE数学掌握二次方程

📚 Mastering Quadratic Equations for IGCSE Mathematics | IGCSE数学掌握二次方程

Quadratic equations are one of the most important algebra topics in IGCSE Mathematics. They appear in solving equations, sketching graphs, and modelling real-life problems. A strong understanding of factorising, completing the square, and the quadratic formula will help you answer both straightforward and high-level exam questions.

二次方程是 IGCSE 数学代数部分最重要的主题之一。它在解方程、绘制图像和建立实际问题模型中经常出现。扎实掌握因式分解、配方法和二次公式，能帮助你解决考试中从基础到高阶的各类问题。

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

A quadratic equation is a polynomial equation where the highest power of the variable is 2. The general form includes an x² term, an x term, and a constant term.

二次方程是变量最高次数为 2 的多项式方程。它的一般形式包含 x² 项、x 项和常数项。

The most common form is ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. If a were equal to 0, the equation would become linear rather than quadratic.

最常见的形式是 ax² + bx + c = 0，其中 a、b、c 是常数，且 a ≠ 0。如果 a 等于 0，方程就变成一次方程而不是二次方程。

= 0 or 2x² = 8x + 10. You should always rearrange them into standard form before solving.

在 IGCSE 考试中，题目给出的方程往往不是标准形式，例如 (x + 2)(x − 3) = 0 或 2x² = 8x + 10。解题前应先把方程整理成标准形式。

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2. Standard Form and Coefficients | 标准形式与系数

The standard form of a quadratic equation is ax² + bx + c = 0. The coefficients a, b, and c are real numbers; a is called the leading coefficient and cannot be zero.

二次方程的标准形式是 ax² + bx + c = 0。系数 a、b、c 是实数；a 被称为首项系数，且不能为零。

For example, in 3x² − 5x + 2 = 0 nd c = 2. Identifying a, b, and c correctly is essential when using the quadratic formula.

例如，在 3x² − 5x + 2 = 0 中，系数为 a = 3、b = −5、c = 2。正确识别 a、b、c 对使用二次公式至关重要。

Sometimes equations need to be rearranged. For example, x² = 4x − 7 becomes x² − 4x + 7 = 0 after moving all terms to one side.

有些方程需要先移项。例如 x² = 4x − 7 把所有项移到一边后变为 x² − 4x + 7 = 0。

When rearranging, try to keep the x² coefficient positive. Multiplying both sides by −1 is often useful before applying any method.

移项时，尽量保持 x² 的系数为正数。在应用任何解法之前，将等式两边乘以 −1 通常是很有用的。

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3. Solving by Factorising | 因式分解法求解

Factorising a quadratic means writing it as a product of two linear brackets. For example, x² + 5x + 6 = (x + 2)(x + 3).

因式分解二次式就是把二次式写成两个一次括号的乘积。例如 x² + 5x + 6 = (x + 2)(x + 3)。

To factorise x² + bx + c, find two numbers that multiply to c and add to b. Here, 2 and 3 multiply to 6 and add to 5.

对于 x² + bx + c 形式，要找两个数，使它们相乘得到 c，相加得到 b。这里 2 和 3 相乘为 6，相加为 5。

For quadratics with a leading coefficient greater than 1, such as 2x² + 7x + 3, you can split the middle term or use trial and error. The factorised form is (2x + 1)(x + 3).

对于首项系数大于 1 的二次式，如 2x² + 7x + 3，可以拆分中项或通过尝试法分解。因式分解结果为 (2x + 1)(x + 3)。

Another example is x² − x − 12. The numbers −4 and 3 multiply to −12 and add to −1, so the factorised form is (x − 4)(x + 3).

另一个例子是 x² − x − 12。−4 和 3 相乘得 −12，相加得 −1，所以因式分解结果为 (x − 4)(x + 3)。

• Always check your brackets by expanding. / 展开括号检查你的因式分解是否正确。
• Look for common factors first, such as 2x² + 4x = 2x(x + 2). / 先寻找公因式，例如 2x² + 4x = 2x(x + 2)。

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4. Zero Product Property | 零乘积性质

Once a quadratic is factorised, you can solve it using the zero product property: if the product of two expressions is zero, then at least one of the expressions must be zero.

二次式因式分解后，可以利用零乘积性质求解：如果两个式子的乘积为零，那么至少其中一个式子必须为零。

For example, if (x + 2)(x + 3) = 0, then either x + 2 = 0 or x + 3 = 0. This gives the solutions x = −2 or x = −3.

例如，若 (x + 2)(x + 3) = 0，则 x + 2 = 0 或 x + 3 = 0。因此解为 x = −2 或 x = −3。

Always write both solutions separately. A quadratic equation can have up to two real solutions.

务必分别写出两个解。一个二次方程最多可以有两个实数解。

If a factor has a coefficient, solve using inverse operations. For example, from (2x + 1)(x + 3) = 0, you get 2x + 1 = 0 or x + 3 = 0, so x = −½ or x = −3.

如果因式中含有系数，要用逆运算求解。例如由 (2x + 1)(x + 3) = 0 可得 2x + 1 = 0 或 x + 3 = 0，所以 x = −½ 或 x = −3。

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5. Solving by Completing the Square | 配方法求解

Completing the square transforms a quadratic into the form (x + p)² = q. This allows you to solve by taking square roots.

配方法将二次方程转化为 (x + p)² = q 的形式，这样就可以通过开平方来求解。

To complete the square for x² + 6x + 4 = 0, first write x² + 6x as (x + 3)² − 9. Then the equation becomes (x + 3)² − 9 + 4 = 0 or (x + 3)² = 5.

对方程 x² + 6x + 4 = 0 配方：先把 x² + 6x 写成 (x + 3)² − 9。方程变为 (x + 3)² − 9 + 4 = 0，即 (x + 3)² = 5。

Taking square roots gives x + 3 = ±√5, so x = −3 ± √5. This method is especially useful when factorising is difficult.

开平方得到 x + 3 = ±√5，因此 x = −3 ± √5。当

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