📚 Solving Quadratic Equations: Factorisation, Completing the Square and the Quadratic Formula | 解二次方程：因式分解、配方法与求根公式
Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, graphs, geometry and real-life modelling, so mastering how to solve them by factorisation, completing the square and the quadratic formula is essential for exam success. This article explains the key methods, the discriminant and common mistakes, with clear examples and exam tips.
二次方程是 IGCSE 数学中最重要的主题之一。它出现在代数、图像、几何和实际建模中，因此掌握因式分解法、配方法和求根公式这几种解法对考试成功至关重要。本文讲解关键方法、判别式和常见错误，并配有清晰的示例和考试技巧。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is a polynomial equation in which the highest power of the variable is 2. Its standard form is written as ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. If a were 0, the equation would become linear, not quadratic.
二次方程是未知量最高次数为 2 的多项式方程。它的标准形式写作 ax² + bx + c = 0，其中 a、b、c 为常数，并且 a ≠ 0。如果 a 为 0，方程就变成一次方程，而不是二次方程。
ax² + bx + c = 0, a ≠ 0
ax² + bx + c = 0，a ≠ 0
The values of x that make the equation true are called the roots or solutions. A quadratic equation can have two real roots, one repeated real root or no real roots, depending on the discriminant.
使方程成立的 x 值称为根或解。根据判别式的不同，二次方程可以有两个实数根、一个重复的实数根或没有实数根。
2. Standard Form and Key Terms | 标准形式与关键术语
Before solving any quadratic equation, always rearrange it into standard form. This means collecting all terms on one side and setting the other side equal to zero. For example, 3x² = 5x - 2 should be rewritten as 3x² - 5x + 2 = 0.
在解任何二次方程之前，一定要先将其整理成标准形式。这意味着把所有项移到一边，让另一边等于 0。例如 3x² = 5x - 2 应先改写为 3x² - 5x + 2 = 0。
The coefficient a is called the quadratic coefficient, b is the linear coefficient, and c is the constant term. Identifying these values correctly is especially important when using the quadratic formula.
系数 a 称为二次项系数，b 称为一次项系数，c 称为常数项。正确识别这些数值在使用求根公式时尤其重要。
For example, in 2x² - 7x + 3 = 0 a = 2, b = -7 and c = 3. If the equation is not equal to zero, these values may be wrong.
例如，在 2x² - 7x + 3 = 0 中，a = 2，b = -7，c = 3。如果方程右边不是 0，这些数值就可能出错。
3. Solving by Factorisation | 因式分解法
Factorisation is often the fastest method when the quadratic expression can be written as a product of two binomials. For example, to solve x² + 5x + 6 = 0, look for two numbers that multiply to give +6 and add to give +5. These numbers are +2 and +3, so the equation becomes (x + 2)(x + 3) = 0.
当二次式可以写成两个一次因式的乘积时，因式分解法通常是最快的方法。例如，解 x² + 5x + 6 = 0，要找两个数，它们的积为 +6，和为 +5。这两个数是 +2 和 +3，因此方程变为 (x + 2)(x + 3) = 0。
x² + 5x + 6 = 0 → (x + 2)(x + 3) = 0
x² + 5x + 6 = 0 → (x + 2)(x + 3) = 0
x² + 5x + 6. If the middle term or constant term does not match, the factors are incorrect.
务必通过展开来检查因式分解：(x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6。如果中间项或常数项不匹配，说明因式分解有误。
4. The Zero Product Property | 零积性质
The reason factorisation works is the zero product property: if two factors multiply to give zero, then at least one of the factors must be zero. This allows us to split the equation into two simpler linear equations.
因式分解法之所以成立，是因为零积性质：如果两个因式的乘积为 0，那么至少有一个因式必须为 0。这使我们可以把原方程拆成两个更简单的一次方程。
From (x + 2)(x + 3) = 0, we set each factor equal to zero:
由 (x + 2)(x + 3) = 0，我们令每个因式等于 0：
x + 2 = 0 or x + 3 = 0 → x = -2 or x = -3
x + 2 = 0 或 x + 3 = 0 → x = -2 或 x = -3
ution. For example, from x(x - 4) = 0, do not divide by x; instead set x = 0 or x - 4 = 0.
不要将方程两边同时除以含有 x 的因式，因为这样可能会丢解。例如，对于 x(x - 4) = 0，不要除以 x；而应令 x = 0 或 x - 4 = 0。
5. Solving by Completing the Square | 配方法
Completing the square is useful when a quadratic cannot be factorised easily. The goal is to rewrite the quadratic expression in the form (x + p)² + q = 0, then solve by taking square roots.
当二次式不容易因式分解时，配方法非常有用。其目标是把二次式改写成 (x + p)² + q = 0 的形式，然后通过开平方来求解。
For example, solve x² + 6x + 2 = 0. First move the constant term: x² + 6x = -2. Then add (6 ÷ 2)² = 9 to both sides to create a perfect square.
例如，解 x² + 6x + 2 = 0。先把常数项移到右边：x² + 6x = -2。然后在两边同时加上 (6 ÷ 2)² = 9，构成完全平方。
x² + 6x + 9 = -2 + 9 → (x + 3)² = 7
x² + 6x + 9 = -2 + 9 → (x + 3)² = 7
Now take the square root of both sides, remembering to include both the positive and negative square roots:
现在对两边开平方，并记得要同时取正平方根和负平方根：
x + 3 = ±√7 → x = -3 ± √7
x + 3 = ±√7 → x = -3 ± √7
Completing the square is also the method used to derive the quadratic formula, so it is worth understanding fully.
配方法也是推导求根公式所使用的方法，因此值得完全理解。
6. Deriving the Quadratic Formula | 求根公式的推导
The quadratic formula gives the solutions of ax² + bx + c = 0 directly. It is derived by completing the square on the general quadratic equation. The result is shown below.
求根公式可以直接给出 ax² + bx + c = 0 的解。它是对一般形式的二次方程进行配方后推导出来的。公式如下所示。
x = (-b ± √b² - 4ac) / (2a)
x = (-b ± √b² - 4ac) / (2a)
at there are usually two solutions: one using the plus sign and one using the minus sign. The entire expression -b ± √b² - 4ac is divided by 2a, so be careful with the denominator.
在该公式中，符号 ± 表示通常有两个解：一个使用加号，另一个使用减号。整个表达式 -b ± √b² - 4ac 都除以 2a，因此要注意分母。
For 2x² - 7x + 3 = 0, substitute a = 2, b = -7, c = 3. The formula becomes:
对于 2x² - 7x + 3 = 0，代入 a = 2，b = -7，c = 3。公式变为：
x = (7 ± √49 - 24) / 4 = (7 ± √25) / 4 = (7 ± 5) / 4
x = (7 ± √49 - 24) / 4 = (7 ± √25) / 4 = (7 ± 5) / 4
2/4 = 1/2. The formula works for any quadratic, including those that cannot be factorised.
由此得到两个根 x = 12/4 = 3 和 x = 2/4 = 1/2。该公式适用于任何二次方程，包括无法因式分解的方程。
7. Using the Quadratic Formula | 使用求根公式
To use the quadratic formula correctly, follow these steps. First rewrite the equation in standard form. Second identify a, b and c. Third substitute them into the formula. Finally simplify the square root and divide carefully.
要正确使用求根公式，请遵循以下步骤。首先把方程改写为标准形式。第二，识别 a、b、c 的值。第三，把它们代入公式。最后化简平方根并仔细进行除法。
For example, solve 3x² - 4x - 2 = 0. Here a = 3, b = -4, c = -2. Substituting gives:
例如，解 3x² - 4x - 2 = 0。这里 a = 3，b = -4，c = -2。代入得到：
x = (4 ± √16 + 24) / 6 = (4 ± √40) / 6 = (4 ± 2√10) / 6
x = (4 ± √16 + 24) / 6 = (4 ± √40) / 6 = (4 ± 2√10) / 6
Simplify this to x = (2 ± √10) / 3. If the question asks for decimal answers, use a calculator to evaluate the two roots separately.
化简后得到 x = (2 ± √10) / 3。如果题目要求保留小数，可以用计算器分别求出两个根的近似值。
Always give exact answers in simplified surd form unless the question says otherwise. This is a key skill at IGCSE level.
除非题目另有说明，否则始终给出最简根式形式下的精确答案。这是 IGCSE 阶段的一项关键技能。
8. Discriminant and Nature of Roots | 判别式与根的性质
The discriminant is the part of the quadratic formula under the square root: Δ = b² - 4ac. It tells us the nature of the roots without solving the equation fully.
判别式是求根公式中平方根下的部分：Δ = b² - 4ac。它可以在不完全解出方程的情况下告诉我们根的性质。
- Δ > 0: two distinct real roots. Δ > 0：两个不同的实数根。
- Δ = 0: one repeated real root. Δ = 0：一个重复的实数根。
- Δ < 0: no real roots, but two complex roots. Δ < 0：没有实数根，但有两个复数根。
For example, in x² - 5x + 7 = 0, the discriminant is (-5)² - 4(1)(7) = 25 - 28 = -3. Since Δ < 0, this equation has no real solutions, and its graph does not cross the x-axis.
例如，在 x² - 5x + 7 = 0 中，判别式为 (-5)² - 4(1)(7) = 25 - 28 = -3。由于 Δ < 0，此方程没有实数解，其图像也不与 x 轴相交。
Exam questions often ask you to find the value of k for which a quadratic has equal roots. In that case, set b² - 4ac = 0 and solve for k.
考试题经常要求找出使二次方程具有相等实根的 k 值。在这种情况下，令 b² - 4ac = 0，然后解出 k。
9. Graphical Interpretation | 图像解释
The graph of y = ax² + bx + c is a parabola. The solutions of ax² + bx + c = 0 are the x-coordinates where the parabola intersects the x-axis. These points are called x-intercepts or roots.
y = ax² + bx + c 的图像是一条抛物线。ax² + bx + c = 0 的解就是抛物线与 x 轴交点的 x 坐标。这些点称为 x 轴截距或根。
If the discriminant is positive, the parabola crosses the x-axis twice. If it is zero, the parabola touches the x-axis at one point. If it is negative, the parabola is entirely above or below the x-axis.
如果判别式为正，抛物线与 x 轴相交两次。如果判别式为零，抛物线在一点处与 x 轴相切。如果判别式为负，抛物线完全位于 x 轴上方或下方。
For example, x² - 4x + 3 = (x - 1)(x - 3) has roots x = 1 and x = 3, so its graph crosses the x-axis at (1, 0) and (3, 0).
例如，x² - 4x + 3 = (x - 1)(x - 3) 的根为 x = 1 和 x = 3，因此它的图像在 (1, 0) 和 (3, 0) 处与 x 轴相交。
Understanding the link between roots and graphs is very useful for solving inequalities and for sketching parabolas.
理解根与图像之间的联系对解不等式和绘制抛物线草图非常有用。
10. Choosing the Best Method | 选择最佳方法
The table below summarises when to use each method. Choosing efficiently can save time in exams and reduce arithmetic mistakes.
下表总结了每种方法在何种情况下使用。高效选择方法可以节省考试时间并减少计算错误。
| Method | 方法 | Best for | 适用情况 |
| Factorisation | 因式分解法 | Simple quadratics with integer roots | 具有整数根的简单二次方程 |
| Completing the square | 配方法 | Finding vertex form, proving results | 求顶点式、证明结论 |
| Quadratic formula | 求根公式 | Any quadratic, especially non-factorable | 任何二次方程，尤其是不能因式分解的 |
If you are not sure which method to use, the quadratic formula is always a reliable choice. However, remember to simplify your final answer completely.
如果不确定使用哪种方法，求根公式始终是一个可靠的选择。但是，请记住要彻底化简最终答案。
For IGCSE exams, you should practise all three methods so you can switch between them confidently and check your answers using another method.
在 IGCSE 考试中，你应该练习全部三种方法，这样你就能自信地在它们之间切换，并用另一种方法检查答案。
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
One common mistake is forgetting to set the equation equal to zero before factorising. Another is dividing by a variable factor and losing a root. Also be careful with negative signs when substituting into the quadratic formula.
一个常见错误是在因式分解前忘记把方程右边写成 0。另一个错误是除以含有变量的因式而丢根。此外，在代入求根公式时要特别小心负号。
When completing the square, do not forget to add the same value to both sides. The value added is always (b ÷ 2)² when the coefficient of x² is 1.
配方法中，不要忘记在两边同时加上同一个数。当 x² 的系数为 1 时，加上的数始终是 (b ÷ 2)²。
Always check your roots by substituting them back into the original equation. This takes a few seconds and can catch sign errors or arithmetic slips.
始终把根代回原方程进行检查。这只需要几秒钟，却能发现符号错误或计算失误。
In exact answer questions, leave your answer in surd form rather than rounding too early. Rounding early can lead to inaccurate final answers.
在要求精确答案的题目中，应保留根式形式，而不要过早取近似值。过早四舍五入可能导致最终答案不准确。
12. Worked Practice Example | 典型例题精讲
Let us solve 2x² - 8x + 6 = 0 using all three methods to see how they agree. First take out the common factor: 2(x² - 4x + 3) = 0, so x² - 4x + 3 = 0. Factorising gives (x - 1)(x - 3) = 0, so x = 1 or x = 3.
让我们用三种方法解 2x² - 8x + 6 = 0，看看它们如何得到一致结果。首先提出公因数：2(x² - 4x + 3) = 0，所以 x² - 4x + 3 = 0。因式分解得到 (x - 1)(x - 3) = 0，所以 x = 1 或 x = 3。
Using completing the square on x² - 4x + 3 = 0: rewrite as x² - 4x = -3, add (4 ÷ 2)² = 4 to both sides, giving (x - 2)² = 1. Therefore x - 2 = ±1, so x = 3 or x = 1.
对 x² - 4x + 3 = 0 使用配方法：改写为 x² - 4x = -3，两边加上 (4 ÷ 2)² = 4，得到 (x - 2)² = 1。因此 x - 2 = ±1，所以 x = 3 或 x = 1。
Using the quadratic formula with a = 2, b = -8, c = 6 gives x = (8 ± √64 - 48) / 4 = (8 ± √16) / 4 = (8 ± 4) / 4. This again gives x = 3 and x = 1.
使用求根公式，代入 a = 2，b = -8，c = 6，得到 x = (8 ± √64 - 48) / 4 = (8 ± √16) / 4 = (8 ± 4) / 4。同样得到 x = 3 和 x = 1。
Having three methods that produce the same roots is a powerful way to build confidence and check your work in the exam.
三种方法得到相同的根，这是建立信心并在考试中检查答案的有力方式。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图