Solving Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 二次方程的解法：因式分解、配方法和公式法

📚 Solving Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 二次方程的解法：因式分解、配方法和公式法

A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in algebra, graphs, and problem-solving questions. In this article, we will explore three main methods for solving quadratic equations: factorising, completing the square, and using the quadratic formula. We will also look at the discriminant and the nature of roots.

二次方程是IGCSE数学中最重要的话题之一。它出现在代数、图像和应用题中。本文将探讨解二次方程的三种主要方法：因式分解法、配方法和公式法。我们还会讨论判别式和根的性质。

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

A quadratic equation is an equation where the highest power of the variable is 2. The general form is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The word ‘quadratic’ comes from the Latin word ‘quadratus’, meaning square.

二次方程是变量最高次数为2的方程。其一般形式为 ax² + bx + c = 0，其中 a、b 和 c 为常数，且 a ≠ 0。单词“quadratic”源自拉丁语“quadratus”，意为“平方”。

For example, x² + 3x – 4 = 0 and 2x² – 5x = 7 (after rearranging) are quadratic equations. A quadratic equation can have two solutions, one solution, or no real solutions.

例如，x² + 3x – 4 = 0 和 2x² – 5x = 7（整理后）都是二次方程。二次方程可以有两个解、一个解，或者没有实数解。

2. Standard Form of a Quadratic Equation | 二次方程的标准形式

To solve a quadratic equation, it is often necessary to write it in standard form: ax² + bx + c = 0. This means all terms are on one side and the other side is zero. If the equation is not in this form, rearrange it first.

解二次方程时，通常需要将其写成标准形式：ax² + bx + c = 0。这意味着所有项都在等号左边，右边为零。如果方程不是这种形式，需要先整理。

ax² + bx + c = 0

For example, the equation x² + 2x = 3 becomes x² + 2x – 3 = 0 when 3 is subtracted from both sides. Always check that a is not zero; if a = 0, it is a linear equation, not quadratic.

例如，方程 x² + 2x = 3，两边同时减去3后变为 x² + 2x – 3 = 0。始终要检查 a 不为零；如果 a = 0，则是一次方程，不是二次方程。

3. Solving by Factorising | 因式分解法

Factorising is the fastest method when the quadratic expression can be factored easily. The idea is to write ax² + bx + c as a product of two binomials, then use the zero product property: if ab = 0, then a = 0 or b = 0.

因式分解法是当二次表达式可以容易分解时最快的方法。其思路是将 ax² + bx + c 写成两个二项式的乘积，然后利用零乘积性质：如果 ab = 0，则 a = 0 或 b = 0。

For example, solve x² – 5x + 6 = 0. Factorise: (x – 2)(x – 3) = 0. Then x – 2 = 0 or x – 3 = 0, giving x = 2 or x = 3.

例如，解 x² – 5x + 6 = 0。因式分解：(x – 2)(x – 3) = 0。然后 x – 2 = 0 或 x – 3 = 0，得到 x = 2 或 x = 3。

For a quadratic with a leading coefficient not equal to 1, such as 2x² + 5x – 3 = 0, you may need to factorise as (2x – 1)(x + 3) = 0. Then 2x – 1 = 0 gives x = ½, and x + 3 = 0 gives x = -3.

对于首项系数不为1的二次方程，例如 2x² + 5x – 3 = 0，可以分解为 (2x – 1)(x + 3) = 0。于是 2x – 1 = 0 得 x = ½，x + 3 = 0 得 x = -3。

4. Difference of Two Squares | 平方差公式

A special factorisation case is the difference of two squares: a² – b² = (a – b)(a + b). This is very useful when solving equations of the form x² = k or x² – k = 0.

一个特殊的因式分解情况是平方差公式：a² – b² = (a – b)(a + b)。这在解 x² = k 或 x² – k = 0 形式的方程时非常有用。

For example, solve x² – 9 = 0. Using the difference of two squares: (x – 3)(x + 3) = 0. So x = 3 or x = -3.

例如，解 x² – 9 = 0。利用平方差公式：(x – 3)(x + 3) = 0，所以 x = 3 或 x = -3。

This method also works for expressions like 4x² – 25 = 0, which factorises to (2x – 5)(2x + 5) = 0, giving x = 5/2 or x = -5/2.

此方法也适用于像 4x² – 25 = 0 这样的表达式，可分解为 (2x – 5)(2x + 5) = 0，得到 x = 5/2 或 x = -5/2。

5. 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 that do not factorise easily, and it also helps in sketching graphs and finding the turning point.

配方法将二次式改写为 (x + p)² + q 的形式。这对于解不容易因式分解的方程很有用，也有助于画图像和求顶点坐标。

To complete the square for x² + bx, add and subtract (b/2)². For example, x² + 6x + 4 = 0. Take half of 6, which is 3, square it to get 9. Rewrite: (x² + 6x + 9) – 9 + 4 = 0, so (x + 3)² – 5 = 0.

对 x² + bx 配方时，加减 (b/2)²。例如，x² + 6x + 4 = 0，取6的一半得3，平方得9。改写为 (x² + 6x + 9) – 9 + 4 = 0，即 (x + 3)² – 5 = 0。

Then solve: (x + 3)² = 5. Taking square roots gives x + 3 = ±√5, so x = -3 ± √5. This gives two exact solutions.

然后解方程：(x + 3)² = 5。开平方得 x + 3 = ±√5，所以 x = -3 ± √5。这就是两个精确解。

6. The Quadratic Formula | 求根公式

The quadratic formula works for every quadratic equation, even when factorising is difficult or impossible. The formula is derived from completing the square and should be memorised.

求根公式适用于所有二次方程，即使因式分解困难或不可能时也能使用。该公式由配方法推导而来，需要牢记。

x = (−b ± √(b² − 4ac)) / (2a)

To use the formula, identify a, b and c from the standard form ax² + bx + c = 0. Then substitute them into the formula.

使用公式时，先从标准形式 ax² + bx + c = 0 中确定 a、b 和 c 的值，然后代入公式。

For example, solve x² – 4x – 5 = 0. Here a = 1, b = -4, c = -5. Substituting gives x = (4 ± √(16 + 20)) / 2 = (4 ± √36) / 2 = (4 ± 6) / 2. So x = 5 or x = -1.

例如，解 x² – 4x – 5 = 0。这里 a = 1，b = -4，c = -5。代入公式得 x = (4 ± √(16 + 20)) / 2 = (4 ± √36) / 2 = (4 ± 6) / 2，所以 x = 5 或 x = -1。

7. The Discriminant | 判别式

The discriminant is the part of the quadratic formula under the square root: b² – 4ac. It tells us how many real roots a quadratic equation has, without solving it fully.

判别式是求根公式中根号下的部分：b² – 4ac。它可以直接告诉我们二次方程有多少个实数根，而不需要完全求解。

If b² – 4ac > 0, there are two distinct real roots. If b² – 4ac = 0, there is exactly one real root (a repeated root). If b² – 4ac < 0, there are no real roots.

如果 b² – 4ac > 0，则有两个不同的实数根。如果 b² – 4ac = 0，则有一个实数根（重根）。如果 b² – 4ac < 0，则没有实数根。

For example, for x² – 6x + 9 = 0, the discriminant is (−6)² – 4 × 1 × 9 = 36 – 36 = 0, so the equation has one repeated root. Indeed, it factorises to (x – 3)² = 0.

例如，对于 x² – 6x + 9 = 0，判别式为 (−6)² – 4 × 1 × 9 = 36 – 36 = 0，所以方程有一个重根。事实上，它可分解为 (x – 3)² = 0。

8. Nature of Roots: Rational, Irrational or No Real | 根的性质：有理数、无理数或无实数根

The discriminant not only tells us the number of roots, but also the type of roots. If b² – 4ac is a perfect square (and a, b, c are rational), the roots are rational. If b² – 4ac is positive but not a perfect square, the roots are irrational.

判别式不仅告诉我们根的个数，还告诉我们根的类型。如果 b² – 4ac 是一个完全平方数（且 a、b、c 为有理数），则根为有理数。如果 b² – 4ac 为正但不是完全平方数，则根为无理数。

For example, x² – 5x + 2 = 0 has discriminant 25 – 8 = 17, which is not a perfect square, so the roots are irrational (containing √17). On the other hand, x² – 7x + 10 = 0 has discriminant 49 – 40 = 9, a perfect square, so the roots are rational (2 and 5).

例如，x² – 5x + 2 = 0 的判别式为 25 – 8 = 17，不是完全平方数，所以根是无理数（含 √17）。而 x² – 7x + 10 = 0 的判别式为 49 – 40 = 9，是完全平方数，所以根是有理数（2和5）。

9. Solving Quadratic Equations by Graphing | 利用图像解二次方程

A quadratic equation ax² + bx + c = 0 can be solved graphically by drawing the curve y = ax² + bx + c and finding the x-coordinates of the points where the curve crosses the x-axis. These x-intercepts are the roots of the equation.

二次方程 ax² + bx + c = 0 可以通过绘制曲线 y = ax² + bx + c，并找到曲线与x轴交点的横坐标来求解。这些x轴截距就是方程的根。

If the curve touches the x-axis at one point, the equation has a repeated root. If it does not cross the x-axis, the equation has no real roots. This graphical view matches the discriminant.

如果曲线与x轴相切于一点，则方程有一个重根。如果曲线不与x轴相交，则方程没有实数根。这种图像理解与判别式的结果一致。

For instance, the graph of y = x² – 2x – 3 crosses the x-axis at x = -1 and x = 3, confirming that x² – 2x – 3 = 0 has solutions -1 and 3.

例如，y = x² – 2x – 3 的图像在 x = -1 和 x = 3 处穿过x轴，这验证了 x² – 2x – 3 = 0 的解为 -1 和 3。

10. Word Problems Involving Quadratics | 二次方程应用题

Many real-world problems lead to quadratic equations. For example, the area of a rectangle is given by length × width. If the width is x, the length is x + 3, and the area is 10, then x(x + 3) = 10, which becomes x² + 3x – 10 = 0.

许多实际问题会转化为二次方程。例如，矩形的面积为长乘宽。如果宽为 x，长为 x + 3，面积为10，则 x(x + 3) = 10，即 x² + 3x – 10 = 0。

Factorising gives (x + 5)(x – 2) = 0, so x = -5 or x = 2. Since a width cannot be negative, the width is 2 and the length is 5.

因式分解得 (x + 5)(x – 2) = 0，所以 x = -5 或 x = 2。因为宽度不能为负，所以宽为2，长为5。

In projectile motion problems, the height h of an object after t seconds might be h = -5t² + 20t + 1. Setting h = 0 and solving gives the time when the object hits the ground.

在抛体运动问题中，物体在t秒后的高度可能为 h = -5t² + 20t + 1。令 h = 0 并求解，可以得到物体落地的时间。

11. Common Mistakes to Avoid | 应避免的常见错误

Forgetting to rearrange the equation to standard form before solving. 解方程前忘记将方程整理成标准形式。
Dividing both sides by x when x = 0 is a possible solution, which loses a root. 当 x = 0 可能是解时，两边除以 x，导致丢根。
Misplacing signs when substituting a, b and c into the quadratic formula. 将 a、b、c 代入求根公式时符号出错。
Ignoring the ± sign when taking square roots. 开平方时忽略 ± 符号。
Assuming that every quadratic has two distinct real roots. 假设所有二次方程都有两个不同的实数根。
Thinking that x² = 9 gives only x = 3. 认为 x² = 9 只有 x = 3 一个解。

Being careful with these points will help you avoid losing marks in exams. Always check your solutions by substituting them back into the original equation.

注意这些细节可以帮助你在考试中避免失分。一定要将解代回原方程进行验证。

12. Practice and Summary | 练习与总结

Let us summarise the three methods. Factorising is best for simple equations with rational roots. Completing the square is useful when the coefficient of x is even and you need the turning point. The quadratic formula always works, especially when roots are irrational or the equation is complicated.

我们来总结三种方法。因式分解法适用于根为有理数的简单方程。配方法在x的系数为偶数且需要求顶点时很有用。求根公式始终有效，尤其是根为无理数或方程复杂时。

Try solving these practice problems: (a) x² – 3x – 10 = 0; (b) 2x² + 4x – 6 = 0; (c) x² – 2x – 6 = 0. Check your answers using the discriminant.

尝试求解以下练习题：(a) x² – 3x – 10 = 0；(b) 2x² + 4x – 6 = 0；(c) x² – 2x – 6 = 0。并用判别式检查你的答案。

For further revision, practise past paper questions and always write down the method you are using. This builds confidence and speed.

为了进一步复习，多练习历年真题，并写出你所用的方法。这样可以增强信心，提高速度。

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Solving Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 二次方程的解法：因式分解、配方法和公式法