Solving Quadratic Equations: Methods, Discriminant and Exam Tips | 解一元二次方程：方法、判别式与考试技巧

📚 Solving Quadratic Equations: Methods, Discriminant and Exam Tips | 解一元二次方程：方法、判别式与考试技巧

Quadratic equations appear in almost every IGCSE Mathematics paper, whether as a standalone question or hidden inside a word problem, a graph question, or a coordinate geometry problem. Mastering this topic is essential for securing top grades.

一元二次方程几乎出现在每一份 IGCSE 数学试卷中，无论是独立题目，还是隐藏在应用题、函数图像或坐标几何问题里。掌握这一主题是冲击高分的关键。

1. What Is a Quadratic Equation | 什么是一元二次方程

A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b and c are constants, and a ≠ 0. The highest power of the variable x is 2, which is why it is called “quadratic”.

一元二次方程是可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。变量 x 的最高次数为 2，因此称为”二次”方程。

Examples of quadratic equations include x² − 5x + 6 = 0, 2x² + 3x − 2 = 0, and x² = 9. Notice that the equation must be rearranged into the standard form before applying most solution methods.

例如，x² − 5x + 6 = 0、2x² + 3x − 2 = 0 以及 x² = 9 都是一元二次方程。注意，在运用大多数解法之前，需要先将方程整理成标准形式。

If a = 0, the equation becomes linear, not quadratic.

如果 a = 0，方程将变为一次方程，而不是二次方程。
Every quadratic equation has at most two real roots.

每个一元二次方程至多有两个实数根。

2. The Factorisation Method | 因式分解法

Factorisation is usually the fastest method when the quadratic has simple integer coefficients. We rewrite the left-hand side as a product of two brackets and then set each bracket equal to zero.

当二次项的系数为简单整数时，因式分解通常是最快的方法。我们将等号左边改写为两个括号的乘积，然后令每个括号分别等于零。

For example, to solve x² − 5x + 6 = 0, we look for two numbers that multiply to give +6 and add to give −5. The numbers are −2 and −3, so we write (x − 2)(x − 3) = 0. Therefore x = 2 or x = 3.

例如，解 x² − 5x + 6 = 0 时，我们需要找到两个数，它们相乘得 +6，相加得 −5。这两个数是 −2 和 −3，因此可以写成 (x − 2)(x − 3) = 0，所以 x = 2 或 x = 3。

x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 or x = 3

For coefficients where a ≠ 1, such as 2x² + 7x + 3 = 0, we find factors of a × c = 6 that add to b = 7. The numbers 6 and 1 work, so we split the middle term: 2x² + 6x + x + 3 = 0, then factor by grouping.

当 a ≠ 1 时，例如 2x² + 7x + 3 = 0，我们先找出 a × c = 6 的因数，使其相加等于 b = 7。数字 6 和 1 满足条件，因此我们拆分为：2x² + 6x + x + 3 = 0，再进行分组因式分解。

2x² + 6x + x + 3 = 0 → 2x(x + 3) + 1(x + 3) = 0 → (2x + 1)(x + 3) = 0

This gives x = −½ or x = −3. Always expand your brackets to check your answer before moving on.

由此得到 x = −½ 或 x = −3。继续解题前，务必展开括号来检验答案是否正确。

3. The Quadratic Formula | 求根公式法

When factorisation is difficult or impossible, we use the quadratic formula. It works for every quadratic equation, provided the discriminant is not negative.

当因式分解困难或无法进行时，我们使用求根公式。只要判别式不为负数，该公式适用于所有一元二次方程。

x = (−b ± √(b² − 4ac)) / 2a

This formula is given on your IGCSE formula sheet, but you must know how to apply it correctly. Substitute the values of a, b and c carefully, then simplify the square root.

这个公式会出现在 IGCSE 公式表中，但你必须知道如何正确应用。仔细代入 a、b、c 的值，然后化简根号部分。

For example, solve 2x² + 3x − 2 = 0. Here a = 2, b = 3 and c = −2. Substituting into the formula gives x = (−3 ± √(9 + 16)) / 4 = (−3 ± 5) / 4. Therefore x = ½ or x = −2.

例如，解 2x² + 3x − 2 = 0。这里 a = 2，b = 3，c = −2。代入公式得 x = (−3 ± √(9 + 16)) / 4 = (−3 ± 5) / 4，因此 x = ½ 或 x = −2。

Here b² − 4ac = 25, which is a perfect square, so the roots are rational.

这里 b² − 4ac = 25，是一个完全平方数，因此根为有理数。
If the discriminant is not a perfect square, leave the answer in surd form.

如果判别式不是完全平方数，请将答案保留为根式形式。

4. Completing the Square | 配方法

Completing the square is another algebraic method. It rewrites the quadratic in the form p(x + q)² + r = 0. This form is especially useful for finding turning points and solving equations that cannot factorise.

配方法是另一种代数解法。它将二次方程改写为 p(x + q)² + r = 0 的形式。这种形式特别适用于求顶点坐标以及解无法因式分解的方程。

For x² + 6x + 1 = 0, we start with x² + 6x and add the square of half of 6, which is 9, then adjust: (x + 3)² − 9 + 1 = 0, so (x + 3)² = 8.

对于 x² + 6x + 1 = 0，我们先看 x² + 6x，加上 6 的一半的平方，即 9，然后调整： (x + 3)² − 9 + 1 = 0，所以 (x + 3)² = 8。

x² + 6x + 1 = 0 → (x + 3)² − 8 = 0 → (x + 3)² = 8

Taking square roots gives x + 3 = ±√8 = ±2√2, so x = −3 ± 2√2. Remember to include both the positive and negative square roots.

两边开平方得 x + 3 = ±√8 = ±2√2，因此 x = −3 ± 2√2。记住要同时取正负两个平方根。

5. The Graphical Method | 图像解法

The roots of a quadratic equation ax² + bx + c = 0 are the x-coordinates where the curve y = ax² + bx + c crosses the x-axis. Reading these points from a graph gives approximate solutions.

一元二次方程 ax² + bx + c = 0 的根，就是曲线 y = ax² + bx + c 与 x 轴交点的横坐标。从图像上读出这些交点即可得到近似解。

If the curve cuts the x-axis at two points, there are two distinct real roots. If it just touches the axis, there is one repeated root. If it never touches the axis, there are no real roots.

如果曲线与 x 轴有两个交点，则方程有两个不同的实数根；如果曲线恰好与 x 轴相切，则有一个重根；如果曲线永不接触 x 轴，则没有实数根。

You may also be asked to solve a more complicated equation such as x² − 2x − 1 = 0 by drawing the line y = x² − 2x − 1 and reading the x-intercepts. The graph should be drawn accurately on graph paper, and you should clearly label all intercepts.

你也可能被要求通过绘制 y = x² − 2x − 1 的图像并读取 x 轴交点来解方程 x² − 2x − 1 = 0。应在坐标纸上精确作图，并清楚标注所有交点。

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

The expression b² − 4ac is called the discriminant, often denoted by the symbol Δ. It tells us about the nature of the roots without actually solving the equation.

表达式 b² − 4ac 被称为判别式，常用符号 Δ 表示。无需实际解方程，它就能告诉我们根的性质。

Value of Δ	Nature of roots	判别式取值	根的性质
Δ > 0 and perfect square	Two distinct rational roots	Δ > 0 且为完全平方数	两个不同的有理数根
Δ > 0 and not a perfect square	Two distinct irrational roots	Δ > 0 但不是完全平方数	两个不同的无理数根
Δ = 0	One repeated real root	Δ = 0	一个重根
Δ < 0	No real roots	Δ < 0	没有实数根

For example, in the equation 3x² − 4x + 5 = 0, we have Δ = 16 − 60 = −44 < 0, so the equation has no real roots.

例如，方程 3x² − 4x + 5 = 0 中，Δ = 16 − 60 = −44 < 0，因此该方程没有实数根。

7. Sum and Product of Roots | 根与系数的关系

For a quadratic equation ax² + bx + c = 0 with roots α and β, two useful relationships connect the roots to the coefficients: α + β = −b/a and αβ = c/a.

对于根为 α 和 β 的一元二次方程 ax² + bx + c = 0，有两个重要的关系式将根与系数联系起来：α + β = −b/a 以及 αβ = c/a。

These relationships allow you to find the sum and product of the roots without solving the equation, which is a common exam question.

利用这些关系，无需解方程就能求出两根之和与两根之积，这是常见的考试题型。

For instance, for x² − 7x + 12 = 0, we directly see that α + β = 7 and αβ = 12. Indeed, the roots are 3 and 4, which confirm both relationships.

例如，对于 x² − 7x + 12 = 0，可直接得出 α + β = 7，αβ = 12。事实上，方程的根为 3 和 4，与这两个关系完全吻合。

You may also be asked to form a new quadratic equation with given roots. If the roots are p and q, then the equation is x² − (p + q)x + pq = 0.

你也可能被要求根据给定的根构造一个新的二次方程。如果根为 p 和 q，则方程为 x² − (p + q)x + pq = 0。

8. Solving Word Problems | 应用题解法

Many IGCSE problems place quadratics in real-world contexts, such as area, projectile motion, or number puzzles. The key is to translate the words into an equation carefully and define your variable clearly.

许多 IGCSE 题目将二次方程置于实际情境中，如面积问题、抛体运动或数字谜题。关键在于仔细将文字转化为方程，并清楚定义变量。

For example, the length of a rectangle is 3 cm longer than its width, and its area is 40 cm². If the width is x, then the length is x + 3, so x(x + 3) = 40.

例如，一个长方形的长比宽长 3 厘米，面积为 40 平方厘米。设宽为 x，则长为 x + 3，因此 x(x + 3) = 40。

x² + 3x − 40 = 0 → (x + 8)(x − 5) = 0 → x = 5 (since x > 0)

The negative root x = −8 is rejected because width cannot be negative. Always check that your final answer makes sense in the original context.

负根 x = −8 应被舍去，因为宽不可能为负数。始终检查最终答案在原始情境中是否合理。

Define the unknown variable first.

先定义未知变量。
Write down the equation from the given conditions.

根据已知条件列出方程。
Solve, then review any invalid roots.

求解，然后排除不合理的根。

9. Common Mistakes to Avoid | 常见错误提醒

Students often lose marks on quadratic questions for avoidable reasons. A few classic pitfalls are listed below.

学生常常因为一些可以避免的原因在二次方程题目中失分。下面列出几个典型陷阱。

First, forgetting to rearrange the equation into the standard form ax² + bx + c = 0 before identifying a, b and c. For example, 2x² = 3x + 5 must become 2x² − 3x − 5 = 0 first.

首先，在识别 a、b、c 之前忘记将方程整理成标准形式 ax² + bx + c = 0。例如，2x² = 3x + 5 必须先化为 2x² − 3x − 5 = 0。

Second, losing the negative sign when substituting b into the quadratic formula. Writing −b instead of −(−3) wrongly changes the answer.

其次，将 b 代入求根公式时丢失负号。把 −b 错写成 −(−3) 会完全改变答案。

Third, dividing both sides by x to simplify x² = 5x. This destroys one root: x = 0 is lost. Instead, rearrange to x² − 5x = 0 and factorise.

第三，为了简化 x² = 5x 而直接两边除以 x。这会丢失一个根 x = 0。正确做法是移项得 x² − 5x = 0，然后因式分解。

Fourth, giving only one root with a ± sign incorrectly. Always write both solutions, and if asked for exact answers, leave surds in simplified form.

第四，只给出一个根或因 ± 符号使用不当。务必写出两个解，如果要求精确答案，根式要化为最简形式。

10. Exam Tips and Practice Strategy | 考试技巧与练习策略

When you see a quadratic equation in an exam, first check whether it factorises quickly. If the numbers are small and simple, factorisation saves time. If not, use the quadratic formula or complete the square.

在考试中遇到二次方程时，首先检查是否可以快速因式分解。如果数字较小且简单，因式分解最省时间。如果不能，则使用求根公式或配方法。

Write down each step clearly. Even if your final answer is wrong, method marks can be awarded for correct substitution or factorisation. In addition, show that you checked the discriminant when asked about the nature of roots.

清晰写出每一步。即使最终答案有误，正确的代入或因式分解步骤仍可获得步骤分。此外，当被问及根的性质时，应展示判别式的计算过程。

For graph questions, use a sharp pencil and plot at least seven points to ensure the curve is accurate. Label the x-intercepts, the y-intercept and the turning point.

对于图像题，请使用削尖的铅笔，至少描绘七个点以确保曲线准确。标注 x 轴交点、y 轴交点以及顶点。

Practise factorising every quadratic in one sitting.

一次性集中练习因式分解所有二次方程。
Memorise the quadratic formula, but do not rely on it alone.

牢记求根公式，但不要只依赖它。
Review past papers and identify which method is tested most often.

复习历年真题，找出最常考查的方法。

11. Summary and Final Advice | 总结与建议

Quadratic equations can be solved by factorisation, the quadratic formula, completing the square, or drawing a graph. The discriminant tells us how many real roots exist, and the sum and product formulas help link roots to coefficients.

一元二次方程可以通过因式分解、求根公式、配方法或作图来求解。判别式告诉我们实数根的个数，根与系数关系则将根与系数联系起来。

In the exam, read the question carefully: do you want exact answers or decimal approximations? Should you leave your answer in surd form? Does the context require rejecting a negative root?

考试时请仔细审题：需要精确答案还是小数近似值？答案是否要保留根式形式？根据实际情境是否需要舍去负根？

With consistent practice, solving quadratic equations will become quick and automatic. Use past exam questions to build confidence, and always check your answers by substituting them back into the original equation.

通过持续练习，解一元二次方程将变得快速而熟练。利用历年真题建立信心，并始终将答案代回原方程检验。

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Solving Quadratic Equations: Methods, Discriminant and Exam Tips | 解一元二次方程：方法、判别式与考试技巧