Solving Quadratic Equations | 二次方程求解

📚 Solving Quadratic Equations | 二次方程求解

Quadratics are one of the most important topics in IGCSE Mathematics. A quadratic equation is a polynomial equation of degree 2, and mastering how to solve it unlocks many other areas, including graphing, coordinate geometry and practical problem solving. In this article, we will explore every key method you need, with worked examples and exam tips.

二次方程是IGCSE数学中最重要的话题之一。二次方程是次数为2的多项式方程，掌握求解二次方程的方法，将帮助你理解图像、坐标几何以及实际应用题等众多相关内容。在这篇文章中，我们将逐一讲解所有关键解法，并配有例题和考试技巧。

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

A quadratic equation is an equation that can be written in the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The variable is raised to the power 2 as its highest power, which is why we call it “quadratic”. If a = 0, the equation becomes linear, not quadratic.

二次方程是能够写成 ax² + bx + c = 0 形式的方程，其中a、b和c是常数，且 a ≠ 0。变量的最高次数是2，因此我们称之为”二次”。如果 a = 0，方程就变成了一次方程（线性方程），而不是二次方程。

Examples of quadratic equations include:

二次方程的例子包括：

x² − 4x + 3 = 0
2x² + 7x − 15 = 0
x² = 9 (which can be rewritten as x² − 9 = 0)

x² − 4x + 3 = 0
2x² + 7x − 15 = 0
x² = 9（可改写为 x² − 9 = 0）

The values of x that make the equation true are called the roots or solutions of the equation. A quadratic equation can have two distinct real roots, one repeated root, or no real roots.

使方程成立的x值称为方程的根或解。二次方程可以有两个不同的实数根、一个重根，或者没有实数根。

2. Expanding and Simplifying Quadratics | 展开与化简二次式

Before solving quadratic equations, you must be confident at expanding double brackets. The standard rule is: (px + q)(rx + s) = prx² + (ps + qr)x + qs. This is sometimes remembered by the word FOIL: First, Outer, Inner, Last.

在求解二次方程之前，你必须熟练掌握展开二项式乘积。标准规则是：(px + q)(rx + s) = prx² + (ps + qr)x + qs。有时我们用FOIL来记忆：First（首）、Outer（外）、Inner（内）、Last（尾）。

For example, expand (x + 3)(x + 5):

例如，展开 (x + 3)(x + 5)：

(x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15

Notice that 8 is the sum of 3 and 5, and 15 is the product of 3 and 5. This pattern is very useful when factoring quadratics.

注意，8是3和5的和，15是3和5的积。这个规律在因式分解时非常有用。

3. Factoring Quadratics | 二次三项式的因式分解

Factoring means writing the quadratic as a product of two brackets. For a monic quadratic x² + bx + c, look for two numbers that multiply to give c and add to give b.

因式分解就是把二次式写成两个括号相乘的形式。对于首项系数为1的二次式 x² + bx + c，需要找到两个数，使它们的乘积等于c，和等于b。

Example: factor x² − 5x + 6. We need two numbers whose product is 6 and whose sum is −5. These numbers are −2 and −3, because (−2) × (−3) = 6 and (−2) + (−3) = −5.

例如：分解 x² − 5x + 6。我们需要找两个数，乘积为6，和为−5。这两个数是−2和−3，因为(−2) × (−3) = 6，且(−2) + (−3) = −5。

x² − 5x + 6 = (x − 2)(x − 3)

For a non-monic quadratic such as 2x² + 5x + 2, you may need to use a method such as “factorising by grouping”: find two numbers with product 2 × 2 = 4 and sum 5. They are 1 and 4; then rewrite the middle term and factor in pairs.

对于首项系数不为1的二次式，例如 2x² + 5x + 2，可以使用”分组分解法”：先找两个数，乘积为 2 × 2 = 4，和为5。这两个数是1和4；然后改写中间项并分组因式分解。

2x² + 5x + 2 = 2x² + x + 4x + 2 = x(2x + 1) + 2(2x + 1) = (2x + 1)(x + 2)

4. Solving by Factoring | 因式分解法解方程

The zero product property states that if the product of two expressions is zero, then at least one of the expressions must be zero. This is the key to solving quadratic equations by factoring.

零积性质指出：如果两个表达式的乘积等于零，那么至少其中一个表达式必须等于零。这就是用因式分解法解二次方程的根据。

Example: solve x² − 5x + 6 = 0.

例如：解方程 x² − 5x + 6 = 0。

(x − 2)(x − 3) = 0

Therefore x − 2 = 0 or x − 3 = 0. Hence x = 2 or x = 3.

因此 x − 2 = 0 或 x − 3 = 0，所以 x = 2 或 x = 3。

Sometimes a quadratic equation must first be rearranged so that all terms are on one side and the other side is zero. For example, x² = 3x becomes x² − 3x = 0, then factor x(x − 3) = 0. The roots are x = 0 or x = 3. Do not divide by x, because you would lose the root x = 0.

有时需要先将二次方程整理，使所有项在等号一边，另一边为0。例如，x² = 3x 要化为 x² − 3x = 0，再分解因式得 x(x − 3) = 0。根为 x = 0 或 x = 3。切记不要两边同除以x，否则会丢失 x = 0 这个根。

5. The Quadratic Formula | 求根公式

When a quadratic cannot be factored easily, we use the quadratic formula. For ax² + bx + c = 0, the solutions are given by:

当二次方程难以因式分解时，我们使用求根公式。对于 ax² + bx + c = 0，解为：

x = (−b ± √(b² − 4ac)) / (2a)

Example: solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, c = −3.

例如：解方程 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。

x = (4 ± √((−4)² − 4 × 2 × (−3))) / (2 × 2) = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4

Simplifying, √40 = 2√10, so x = (4 ± 2√10) / 4 = 1 ± (√10)/2. You should usually give answers in exact surd form unless the question asks for decimals.

化简得 √40 = 2√10，因此 x = (4 ± 2√10) / 4 = 1 ± (√10)/2。除非题目要求小数，否则通常要保留精确的根式形式。

6. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x + p)² + q. This is especially useful for finding turning points of graphs and solving equations without using the quadratic formula.

配方法将二次式改写为 a(x + p)² + q 的形式。这种方法在求图像顶点以及解方程时非常有用，且无需使用求根公式。

For a monic quadratic x² + bx + c, we use: x² + bx = (x + b/2)² − (b/2)².

对于首项系数为1的二次式 x² + bx + c，我们使用：x² + bx = (x + b/2)² − (b/2)²。

Example: solve x² + 6x − 2 = 0 by completing the square.

例如：用配方法解方程 x² + 6x − 2 = 0。

x² + 6x = (x + 3)² − 9, so (x + 3)² − 9 − 2 = 0

(x + 3)² = 11, hence x + 3 = ±√11, so x = −3 ±√11

Remember to take both the positive and negative square roots when solving.

记住：求解时要同时取正负平方根。

7. The Discriminant | 判别式

The expression under the square root in the quadratic formula is called the discriminant: Δ = b² − 4ac. It tells us the nature of the roots without solving the equation.

求根公式中根号内的表达式称为判别式：Δ = b² − 4ac。它无需解方程就能告诉我们根的性质。

If Δ > 0, there are two distinct real roots.
If Δ = 0, there is one repeated real root.
If Δ < 0, there are no real roots.

如果 Δ > 0，有两个不同的实数根。
如果 Δ = 0，有一个重根。
如果 Δ < 0，没有实数根。

For example, for the equation x² − 4x + 5 = 0, the discriminant is (−4)² − 4 × 1 × 5 = 16 − 20 = −4. Since −4 < 0, the equation has no real roots. This means the graph of y = x² − 4x + 5 does not cross the x-axis.

例如，对于方程 x² − 4x + 5 = 0，判别式为 (−4)² − 4 × 1 × 5 = 16 − 20 = −4。因为 −4 < 0，方程没有实数根。这意味着 y = x² − 4x + 5 的图像与x轴没有交点。

8. Quadratic Graphs and Roots | 二次函数图像与根

The graph of y = ax² + bx + c is a parabola. When a > 0, the parabola opens upwards; when a < 0, it opens downwards. The roots of the equation ax² + bx + c = 0 are the x-coordinates where the graph crosses the x-axis.

函数 y = ax² + bx + c 的图像是抛物线。当 a > 0 时，抛物线开口向上；当 a < 0 时，开口向下。方程 ax² + bx + c = 0 的根就是图像与x轴交点的x坐标。

The axis of symmetry and the turning point can be found using the formula x = −b/(2a), or from completed square form y = a(x + p)² + q, where the turning point is (−p, q).

对称轴和顶点可以用公式 x = −b/(2a) 求出，也可以从配方式 y = a(x + p)² + q 中直接得到，顶点坐标为 (−p, q)。

Example: the turning point of y = (x − 3)² + 2 is (3, 2). Because the minimum value is 2 and it never goes below 2, this graph has no real roots.

例如：y = (x − 3)² + 2 的顶点是 (3, 2)。因为最小值为2，函数值永远不会低于2，所以这个图像没有实数根。

9. Modelling Real-World Problems | 实际应用题建模

Quadratic equations often arise in geometry and physics. For example, the area of a rectangle can be expressed as a quadratic expression, and projectile motion is described by a quadratic equation in time.

二次方程经常出现在几何和物理实际问题中。例如，矩形的面积可以用二次式表示，抛体运动也可以用关于时间的二次方程来描述。

Example: the length of a rectangle is 4 cm longer than its width, and its area is 60 cm². Find its width.

例如：一个矩形的长比宽多4厘米，面积为60平方厘米。求它的宽。

Let the width be x cm. Then the length is (x + 4) cm. The area is x(x + 4) = 60, so x² + 4x − 60 = 0.

设宽为x厘米，则长为 (x + 4) 厘米。面积为 x(x + 4) = 60，因此 x² + 4x − 60 = 0。

x² + 4x − 60 = (x + 10)(x − 6) = 0, so x = −10 or x = 6

Since width cannot be negative, x = 6. The rectangle is 6 cm by 10 cm. Always check whether a root is sensible in the context.

因为宽不能为负数，所以 x = 6。这个矩形宽6厘米，长10厘米。一定要检查根在实际情况中是否合理。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

Many students lose marks on quadratic equations because of small but avoidable errors. Here are the most common ones to watch out for.

很多学生在二次方程题上丢分，往往是因为一些细小但可以避免的错误。以下是最常见的几个易错点。

Not rearranging to zero: Before factoring, always make sure the equation is in the form ax² + bx + c = 0.
Dropping the ± sign: When taking square roots, always include both positive and negative values.
Losing roots: Do not divide both sides by x or by an expression containing x unless you are sure it is never zero.
Miscalculating the discriminant: Be careful with negative signs when substituting into b² − 4ac.
Giving answers to the wrong accuracy: Always check whether the question asks for exact values or rounded decimals.

未将方程整理为零：因式分解前，务必确保方程形式为 ax² + bx + c = 0。
丢失 ± 号：开平方时，一定要同时取正负值。
丢失根：不要两边同时除以x或含有x的表达式，除非你能确定它不为零。
判别式计算错误：代入 b² − 4ac 时，要特别注意负号。
答案精度不符合要求：注意题目要求精确值还是保留小数。

Always check your answers by substituting them back into the original equation. This simple step can save valuable marks.

解完后一定要把答案代回原方程检验。这个简单的步骤能为你保住宝贵的分数。

11. Practice Questions | 巩固练习

Try these exam-style questions by yourself before looking at the answers.

先自己尝试以下考试风格的题目，再查看答案。

1. Solve x² − 7x + 10 = 0.

2. Solve 3x² + 5x − 2 = 0.

3. Find the discriminant of 2x² − 3x + 4 = 0 and state the number of real roots.

4. Write x² − 8x + 3 in the form (x − p)² + q.

5. A right-angled triangle has hypotenuse 13 cm and one leg is 7 cm longer than the other. Find the length of the shorter leg.

1. 解方程 x² − 7x + 10 = 0。

2. 解方程 3x² + 5x − 2 = 0。

3. 求 2x² − 3x + 4 = 0 的判别式，并说明实数根的个数。

4. 将 x² − 8x + 3 写成 (x − p)² + q 的形式。

5. 一个直角三角形斜边长为13厘米，其中一条直角边比另一条长7厘米。求较短直角边的长度。

Question \| 题号	Answer \| 答案
1	x = 2 or 5
2	x = 1/3 or −2
3	Δ = 9 − 24 = −15; no real roots
4	(x − 4)² − 13
5	5 cm

12. Conclusion | 总结

Solving quadratic equations is a core skill for IGCSE Mathematics. You should be comfortable with factoring, using the quadratic formula, completing the square and understanding the discriminant. Practice each method until it becomes automatic, and always check your answers.

求解二次方程是IGCSE数学的核心技能。你应该熟练掌握因式分解、求根公式、配方法以及判别式的运用。反复练习每种方法，直到熟练自如，并养成检验答案的好习惯。

Remember: read the question carefully, choose the most appropriate method, and show all your working clearly. Marks are awarded for accurate method as well as correct answers.

记住：仔细审题，选择最合适的解法，并清晰写出每一步过程。评分不仅看最终答案，也会给方法步骤分。

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Solving Quadratic Equations | 二次方程求解