IGCSE Maths: Solving Quadratic Equations | IGCSE 数学：解二次方程

📚 IGCSE Maths: Solving Quadratic Equations | IGCSE 数学：解二次方程

Quadratic equations appear throughout the IGCSE syllabus, from algebraic manipulation to graphs and real-world problems. A quadratic equation contains a squared term as the highest power, and it can be solved by factorising, completing the square, or using the quadratic formula. This article explains each method step by step, highlights common errors, and shows how the discriminant helps you predict the nature of the roots.

二次方程在 IGCSE 大纲中随处可见，从代数运算到函数图像和实际应用问题都会涉及。二次方程的最高次项是平方项，可以通过因式分解、配方法或二次公式来求解。本文逐步讲解每种方法，指出常见错误，并说明判别式如何帮助你判断根的性质。

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

A quadratic equation is an equation that can be written in the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The expression on the left is a polynomial of degree 2. If a = 0, the equation becomes linear, so the condition a ≠ 0 is essential.

二次方程是可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 是常数，且 a ≠ 0。左边是一个二次多项式。若 a = 0，方程就变成一次方程，因此 a ≠ 0 这个条件至关重要。

Examples include x² − 5x + 6 = 0, 2x² + 3x − 2 = 0 and 4x² − 9 = 0. The solutions are also called the roots of the equation, and a quadratic equation can have two real roots, one repeated root, or no real roots.

例如 x² − 5x + 6 = 0、2x² + 3x − 2 = 0 和 4x² − 9 = 0。方程的解也叫做方程的根。二次方程可能有两个实根、一个重根，或没有实根。

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

Before solving, rearrange the equation so that one side equals 0 and the terms are usually written in descending powers of x. For example, 3x² = 7x − 2 becomes 3x² − 7x + 2 = 0.

在求解之前，要先把方程整理成一边等于 0，并通常按 x 的降幂排列。例如 3x² = 7x − 2 应写成 3x² − 7x + 2 = 0。

This standard form makes it easy to identify a, b and c. In 3x² − 7x + 2 = 0, we have a = 3, b = −7 and c = 2. Always include the sign of each coefficient.

这种标准形式便于确定 a、b 和 c。在 3x² − 7x + 2 = 0 中，a = 3，b = −7，c = 2。务必带上每个系数的符号。

3. Solving by Factorising | 因式分解法

If a quadratic expression can be written as a product of two brackets, you can use the zero-product rule: if p × q = 0, then p = 0 or q = 0. This gives two simple linear equations.

如果二次式可以写成两个括号的乘积，就可以使用零乘积性质：若 p × q = 0，则 p = 0 或 q = 0。这样得到两个简单的一次方程。

For x² − 5x + 6 = 0, factorise to (x − 2)(x − 3) = 0. Then x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3.

对于 x² − 5x + 6 = 0，因式分解为 (x − 2)(x − 3) = 0。然后 x − 2 = 0 或 x − 3 = 0，所以 x = 2 或 x = 3。

For equations where a ≠ 1, such as 2x² + 3x − 2 = 0, factorise carefully to (2x − 1)(x + 2) = 0, giving x = ½ or x = −2.

当 a ≠ 1 时，如 2x² + 3x − 2 = 0，要仔细分解为 (2x − 1)(x + 2) = 0，得到 x = ½ 或 x = −2。

4. Difference of Two Squares | 平方差

A special case is a² − b² = (a − b)(a + b). For example, x² − 16 = 0 can be written as (x − 4)(x + 4) = 0, so x = 4 or x = −4.

特殊情形是 a² − b² = (a − b)(a + b)。例如 x² − 16 = 0 可以写成 (x − 4)(x + 4) = 0，因此 x = 4 或 x = −4。

This pattern is useful when there is no x term. Equations like 9x² − 25 = 0 factorise to (3x − 5)(3x + 5) = 0, giving x = 5/3 or x = −5/3.

当方程没有 x 项时，这种模式很有用。如 9x² − 25 = 0 可分解为 (3x − 5)(3x + 5) = 0，得到 x = 5/3 或 x = −5/3。

5. Completing the Square | 配方法

Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. For x² + 6x + 1 = 0, write (x + 3)² − 9 + 1 = 0, so (x + 3)² = 8. Then x + 3 = ±√8, so x = −3 ± 2√2.

配方法把 ax² + bx + c 改写成 a(x + p)² + q 的形式。对于 x² + 6x + 1 = 0，写作 (x + 3)² − 9 + 1 = 0，即 (x + 3)² = 8。于是 x + 3 = ±√8，因此 x = −3 ± 2√2。

This method is especially important because it works for every quadratic, even when factorising is difficult or impossible. It also leads directly to the quadratic formula.

这种方法尤其重要，因为它适用于所有二次方程，即使因式分解困难或无法分解时也可用。它也直接导出二次公式。

6. The Quadratic Formula | 二次公式

The quadratic formula gives the solutions of ax² + bx + c = 0 as:

二次公式给出 ax² + bx + c = 0 的解：

x = (−b ± √(b² − 4ac)) / 2a

Always substitute a, b and c with their signs. For 2x² + 3x − 2 = 0, a = 2, b = 3, c = −2, so x = (−3 ± √(9 + 16)) / 4 = (−3 ± 5) / 4, giving x = ½ or x = −2.

代入 a、b、c 时一定要带上符号。对于 2x² + 3x − 2 = 0，a = 2，b = 3，c = −2，所以 x = (−3 ± √(9 + 16)) / 4 = (−3 ± 5) / 4，得到 x = ½ 或 x = −2。

7. Discriminant and Nature of Roots | 判别式与根的性质

The expression D = b² − 4ac inside the square root is called the discriminant. It tells you the nature of the roots without solving the equation.

平方根内的式子 D = b² − 4ac 叫做判别式。它可以在不解方程的情况下判断根的性质。

If D > 0, there are two distinct real roots. / 若 D > 0，方程有两个不同的实根。
If D = 0, there is one repeated real root. / 若 D = 0，方程有一个重根。
If D < 0, there are no real roots. / 若 D < 0，方程没有实根。

For example, x² + 2x + 5 = 0 has D = 4 − 20 = −16, so it has no real roots. This is useful for questions that ask you to find the number of solutions.

例如 x² + 2x + 5 = 0 的 D = 4 − 20 = −16，所以它没有实根。这在考查解的个数的题目中非常有用。

8. Graphical Interpretation | 图像解释

The solutions of ax² + bx + c = 0 are the x-coordinates where the graph y = ax² + bx + c crosses the x-axis. If the graph does not cross the x-axis, the equation has no real roots.

方程 ax² + bx + c = 0 的解就是函数 y = ax² + bx + c 的图像与 x 轴交点的 x 坐标。如果图像不与 x 轴相交，则方程没有实根。

A repeated root occurs when the graph touches the x-axis at one point, often called a tangent to the axis. Knowing the shape of a quadratic graph helps check whether your answers are reasonable.

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IGCSE Maths: Solving Quadratic Equations | IGCSE 数学：解二次方程