Solving Quadratic Equations: Methods and Applications | 解二次方程：方法与应用

📚 Solving Quadratic Equations: Methods and Applications | 解二次方程：方法与应用

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b and c are constants and a is not zero. This topic is one of the most important skills in IGCSE Mathematics because it appears in algebra, coordinate geometry, and real-life problems.

二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 是常数，且 a 不等于零。这是 IGCSE 数学中最重要的技能之一，因为它出现在代数、坐标几何和实际生活问题中。

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1. The Standard Form | 标准形式

Every quadratic equation should first be rearranged into the standard form ax² + bx + c = 0. This makes the coefficients clear and enables all solving methods to be applied easily.

每个二次方程都应首先整理成标准形式 ax² + bx + c = 0。这样系数就变得清晰，并且使所有求解方法都能容易地应用。

The coefficient a cannot be zero, otherwise the equation becomes linear. The constants b and c may be zero, so equations such as x² − 9 = 0 and x² − 4x = 0 are still quadratic.

系数 a 不能为零，否则方程会变成线性方程。常数 b 和 c 可以为零，因此像 x² − 9 = 0 和 x² − 4x = 0 这样的方程仍然是二次方程。

• Expand and simplify both sides first. | 首先展开并化简等号两边。
• Move every term to the left side. | 将所有项移到等号左侧。
• Write the terms in descending powers of x. | 按 x 的降幂次序排列各项。

For example, x² = 5x − 6 is written as x² − 5x + 6 = 0 before solving.

例如，x² = 5x − 6 在求解前应写成 x² − 5x + 6 = 0。

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2. Solving by Factorisation | 因式分解法

If the quadratic expression ax² + bx + c can be factorised into two brackets, then we can use the zero-product rule: if (px + q)(rx + s) = 0, then either px + q = 0 or rx + s = 0.

如果二次表达式 ax² + bx + c 可以分解成两个括号，那么我们可以使用零积规则：如果 (px + q)(rx + s) = 0，则要么 px + q = 0，要么 rx + s = 0。

Example: Solve x² − 5x + 6 = 0.

例如：解方程 x² − 5x + 6 = 0。

(x − 2)(x − 3) = 0

x = 2 or x = 3

Check by substituting each value back into the original equation. This method is quick when the factors are easy to spot.

将每个值代回原方程进行检验。当因子容易发现时，这种方法非常快捷。

For quadratics with a leading coefficient not equal to 1, such as 2x² − 5x + 3 = 0, we can factor as (2x − 3)(x − 1) = 0, giving x = 1.5 or x = 1.

对于首项系数不为 1 的二次方程，例如 2x² − 5x + 3 = 0，我们可以分解为 (2x − 3)(x − 1) = 0，得到 x = 1.5 或 x = 1。

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3. The Quadratic Formula | 二次公式

When factorisation is difficult or impossible, the quadratic formula always works for any equation in the form ax² + bx + c = 0.

当因式分解困难或无法进行时，二次公式对于任何 ax² + bx + c = 0 形式的方程都总是适用的。

x = [−b ± √(b² − 4ac)] / (2a)

To use the formula, identify the values of a, b and c. For example, in 2x² + 3x − 2 = 0, we have a = 2, b = 3, c = −2.

使用公式时，先确定 a、b 和 c 的值。例如，在 2x² + 3x − 2 = 0 中，a = 2，b = 3，c = −2。

x = [−3 ± √(3² − 4(2)(−2))] / (2 × 2) = [−3 ± √(9 + 16)] / 4 = [−3 ± 5] / 4

Therefore x = (−3 + 5)/4 = 0.5 or x = (−3 − 5)/4 = −2.

因此 x = (−3 + 5)/4 = 0.5 或 x = (−3 − 5)/4 = −2。

This formula is derived from completing the square, so it is useful to remember it as a standard tool.

这个公式是由配方法推导出来的，因此把它作为一个标准工具来记忆是很有用的。

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4. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² + q. This is useful for solving equations and for finding turning points.

配方法将二次式改写为 (x + p)² + q 的形式。这对于解方程以及寻找转向点都很有用。

Example: Solve x² + 6x + 8 = 0.

例如：解方程 x² + 6x + 8 = 0。

(x + 3)² − 9 + 8 = 0 ⇒ (x + 3)² = 1

Taking the square root of both sides gives x + 3 = ±1, so x = −2 or x = −4.

两边开平方得到 x + 3 = ±1，所以 x = −2 或 x = −4。

If the equation is written as y = (x + p)² + q, the turning point of the parabola is at (−p, q).

如果方程写成 y = (x + p)² + q，抛物线的转向点就在 (−p, q)。

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5. The Discriminant | 判别式

The discriminant Δ is the value under the square root in the formula: Δ = b² − 4ac. It tells us how many real roots a quadratic equation has.

判别式 Δ 是公式中根号下的值：Δ = b² − 4ac。它告诉我们一个二次方程有多少个实数根。

Value of Δ | Δ 的值	Nature of roots | 根的性质
Δ > 0	Two distinct real roots | 两个不同的实数根
Δ = 0	One repeated real root | 一个重复的实数根
Δ < 0	No real roots | 没有实数根

For example, x² + 2x + 5 = 0 has Δ = 2² − 4(1)(5) = 4 − 20 = −16, so it has no real roots.

例如，x² + 2x + 5 =

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