Mastering Quadratic Equations | 掌握二次方程

📚 Mastering Quadratic Equations | 掌握二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, graphs, measurement and real-world modelling, so a confident grasp of the main solution methods is essential for exam success.

二次方程是 IGCSE 数学中最重要的主题之一。它们出现在代数、图像、测量和现实建模中，因此熟练掌握主要求解方法对于考试成功至关重要。

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

hat can be written in the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

二次方程是可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数，且 a ≠ 0。

The highest power of the unknown x is 2, which is why it is called a quadratic equation.

未知数 x 的最高次数是 2，因此它被称为二次方程。

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2. Standard Form and Key Terms | 标准形式与关键术语

c = 0, a is the coefficient of x², b is the coefficient of x, and c is the constant term.

在标准形式 ax² + bx + c = 0 中，a 是 x² 的系数，b 是 x 的系数，c 是常数项。

The solutions of a quadratic equation are called roots. A quadratic equation can have two distinct real roots, one repeated real root, or no real roots.

二次方程的解称为根。二次方程可能有两个不同的实数根、一个重根或没有实数根。

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3. Solving by Factorisation | 因式分解法

e product is c and whose sum is b. These numbers are then used to write the quadratic as a product of two brackets.

当 a = 1 时，我们寻找两个数，它们的积为 c、和为 b。然后用这两个数将二次式写成两个括号的乘积。

he numbers 2 and 3 multiply to 6 and add to 5, so the equation becomes (x + 2)(x + 3) = 0.

例题：解 x² + 5x + 6 = 0。2 和 3 的积为 6、和为 5，所以方程变为 (x + 2)(x + 3) = 0。

By the zero product property, x + 2 = 0 or x + 3 = 0, giving x = −2 or x = −3.

根据零乘积性质，x + 2 = 0 或 x + 3 = 0，因此 x = −2 或 x = −3。

When a is not equal to 1, use the ac method: find two numbers whose product is ac and whose sum is b, then split the middle term and factor by grouping. For example, 2x² + 7x + 3 becomes (2x + 1)(x + 3).

当 a 不等于 1 时，使用 ac 方法：找到两个数，积为 ac、和为 b，然后拆分中间项并分组分解。例如，2x² + 7x + 3 可化为 (2x + 1)(x + 3)。

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4. Solving by Completing the Square | 配方法

Completing the square uses the identity x² + bx = (x + b/2)² − (b/2)².

配方法使用恒等式 x² + bx = (x + b/2)² − (b/2)²。

Example: Solve x² + 6x + 4 = 0. First rewrite the left side by completing the square:

例题：解 x² + 6x + 4 = 0。首先通过配方法改写左边：

x² + 6x = (x + 3)² − 9

So x² + 6x + 4 = 0 becomes (x + 3)² − 5 = 0, hence (x + 3)² = 5.

因此 x² + 6x + 4 = 0 变为 (x + 3)² − 5 = 0，所以 (x + 3)² = 5。

Taking square roots gives x + 3 = ±√5, so x = −3 ± √5.

开平方得到 x + 3 = ±√5，所以 x = −3 ± √5。

This method is particularly useful when the quadratic expression does not factorise easily. If a ≠ 1, divide the whole equation by a first.

当二次式不容易因式分解时，这种方法尤其有用。如果 a ≠ 1，先将整个方程除以 a。

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5. The Quadratic Formula | 二次公式

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

二次公式直接给出 ax² + bx + c = 0 的解：

x = (−b ± √b² − 4ac) / 2a

This formula is obtained by completing the square on the general quadratic equation. It works for all quadratic equations, including those that factorise and those that do not.

该公式是通过对一般二次方程进行配方法得到的。它适用于所有二次方程，包括可以因式分解和不能因式分解的方程。

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

he square root, b² − 4ac, is called the discriminant and is often written as Δ = b² − 4ac.

平方根内的表达式 b² − 4ac 称为判别式，通常记作 Δ = b² − 4ac。

It tells us the nature of the roots without solving the equation fully.

它能在不完全求解方程的情况下告诉我们根的性质。

Δ > 0	Two distinct real roots | 两个不同的实数根
Δ = 0	One repeated real root | 一个重根
Δ < 0	No real roots | 没有实数根

Checking the discriminant before solving can save time and help you decide which method to use.

在求解前先检查判别式可以节省时间，并帮助你决定使用哪种方法。

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7. Graphs of Quadratic Functions | 二次函数图像

. If a > 0, the parabola opens upwards and has a minimum turning point. If a < 0, it opens downwards and has a maximum turning point.

y = ax² + bx + c 的图像是抛物线。如果 a > 0，抛物线开口向上并具有最低点；如果 a < 0，抛物线开口向下并具有最高点。

The x-coordinate of the turning point is x = −b / 2a.

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