Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in almost every exam paper, either as a direct solving question or as a tool for solving harder problems in algebra, geometry and mensuration. In this article, we will cover every method you need: factorisation, completing the square, the quadratic formula, the discriminant, and the graphical approach.

二次方程是 IGCSE 数学中最重要的话题之一。它几乎出现在每份试卷中，要么直接要求解方程，要么作为解决代数、几何和测量中更复杂问题的工具。在本文中，我们将涵盖你需要的一切方法：因式分解法、配方法、求根公式、判别式以及图像法。

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

A quadratic equation is an equation that can be written in the standard form:

一元二次方程是能够写成标准形式的方程：

ax² + bx + c = 0, where a ≠ 0

Here, x is the unknown variable, and a, b and c are constants. The highest power of x is 2, which is why the equation is called “quadratic”. If a = 0, the equation is no longer quadratic but linear.

其中 x 是未知数，a、b、c 是常数。x 的最高次数是 2，因此这个方程被称为”二次”方程。如果 a = 0，方程就不再是二次的，而是一次方程。

Examples of quadratic equations include:

二次方程的例子包括：

x² − 7x + 10 = 0
2x² + 3x − 5 = 0
x² = 16 (which can be rewritten as x² − 16 = 0)

A quadratic equation normally has two solutions, which are also called roots. These roots may be real and different, real and equal, or not real, depending on the value of the discriminant.

二次方程通常有两个解，也称为根。这些根可能是两个不同的实数、两个相等的实数，或者不是实数，具体取决于判别式的值。

2. Expanding and Factorising Quadratics | 展开与因式分解二次式

Before solving, you must be comfortable with factorising. Factorising is the reverse process of expanding brackets. For example:

在求解之前，你必须熟练进行因式分解。因式分解是展开括号的逆过程。例如：

(x + 2)(x + 3) = x² + 5x + 6

To factorise the expression x² + 5x + 6, we look for two numbers that multiply to give 6 and add to give 5. The numbers 2 and 3 satisfy both conditions: 2 × 3 = 6 and 2 + 3 = 5. Therefore:

要对 x² + 5x + 6 进行因式分解，我们寻找两个数，它们相乘得 6，相加得 5。数字 2 和 3 同时满足这两个条件：2 × 3 = 6 且 2 + 3 = 5。因此：

x² + 5x + 6 = (x + 2)(x + 3)

When the coefficient of x² is not 1, the process is slightly harder. For example, to factorise 2x² + 7x + 3, we multiply the coefficient of x² (2) by the constant term (3) to get 6. We then find two numbers that multiply to 6 and add to 7, which are 1 and 6. We rewrite the middle term:

当 x² 的系数不为 1 时，过程稍微复杂一些。例如，要对 2x² + 7x + 3 进行因式分解，我们将 x² 的系数（2）与常数项（3）相乘得到 6。然后我们找到两个数相乘得 6、相加得 7，即 1 和 6。我们改写中间项：

2x² + 7x + 3 = 2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (x + 3)(2x + 1)

This method is called factoring by grouping. Practise it until it becomes automatic, as it is the fastest way to solve simple quadratics.

这种方法称为分组分解法。多加练习直到它变得自动熟练，因为这是求解简单二次方程最快的方法。

3. Solving by Factorisation | 用因式分解法求解

Once a quadratic equation is factorised, solving it relies on the zero product property: if the product of two expressions is zero, then at least one of them must be zero.

一旦二次方程被因式分解，求解就依赖零乘积性质：如果两个表达式的乘积为零，那么其中至少有一个必须为零。

Solve: x² − 7x + 10 = 0

求解：x² − 7x + 10 = 0

(x − 2)(x − 5) = 0

So either x − 2 = 0 or x − 5 = 0. Hence x = 2 or x = 5. These are the two roots of the equation.

因此要么 x − 2 = 0，要么 x − 5 = 0。所以 x = 2 或 x = 5。这就是方程的两个根。

Let us look at another example where the coefficient of x² is greater than 1. Solve 2x² + 7x + 3 = 0.

再看一个 x² 系数大于 1 的例子。求解 2x² + 7x + 3 = 0。

(x + 3)(2x + 1) = 0

Therefore x + 3 = 0 or 2x + 1 = 0, giving x = −3 or x = −½.

因此 x + 3 = 0 或 2x + 1 = 0，得到 x = −3 或 x = −½。

Always rearrange the equation into the standard form ax² + bx + c = 0 before factorising. If the equation is given as x² = 5x, do not divide both sides by x, because this loses the root x = 0. Instead, write it as x² − 5x = 0 and factorise to get x(x − 5) = 0.

在因式分解之前，一定要把方程整理成标准形式 ax² + bx + c = 0。如果方程以 x² = 5x 形式给出，不要在两边同时除以 x，因为这会丢失根 x = 0。相反，把它写成 x² − 5x = 0，然后因式分解得到 x(x − 5) = 0。

4. Completing the Square | 配方法

Completing the square is a technique that rewrites a quadratic expression in the form a(x + p)² + q. This is useful not only for solving equations but also for finding the turning point of a parabola.

配方法是一种将二次表达式改写为 a(x + p)² + q 形式的技巧。这不仅适用于解方程，也适用于寻找抛物线的顶点。

For the expression x² + 6x + 2, we look at the coefficient of x, which is 6. Half of 6 is 3, and we write:

对于表达式 x² + 6x + 2，我们看 x 的系数 6。6 的一半是 3，我们写出：

x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7

To solve the equation x² + 6x + 2 = 0, we now have:

要求解方程 x² + 6x + 2 = 0，我们现在有：

(x + 3)² − 7 = 0

(x + 3)² = 7

x + 3 = ±√7

x = −3 ± √7

So the two solutions are x = −3 + √7 and x = −3 − √7. Remember to include the ± sign when taking the square root of both sides.

所以两个解是 x = −3 + √7 和 x = −3 − √7。记住在两边同时开平方时一定要带上正负号（±）。

When the coefficient of x² is not 1, first factor out that coefficient from the x² and x terms. For example:

当 x² 的系数不是 1 时，先从 x² 项和 x 项中提出该系数。例如：

2x² + 8x + 5 = 2(x² + 4x) + 5 = 2[(x + 2)² − 4] + 5 = 2(x + 2)² − 8 + 5 = 2(x + 2)² − 3

5. The Quadratic Formula | 求根公式

Every quadratic equation ax² + bx + c = 0 can be solved using the quadratic formula. This formula works for all quadratics, including those that cannot be factorised easily.

每一个一元二次方程 ax² + bx + c = 0 都可以用求根公式来求解。这个公式适用于所有二次方程，包括那些不容易因式分解的方程。

x = (−b ± √(b² − 4ac)) / 2a

This formula is given in many IGCSE formula sheets, but you must know how to apply it correctly. Let us solve 2x² + 3x − 5 = 0 using the formula.

这个公式在许多 IGCSE 公式表中都会给出，但你必须知道如何正确应用它。让我们用公式求解 2x² + 3x − 5 = 0。

Here a = 2, b = 3 and c = −5. Substitute these values:

这里 a = 2，b = 3，c = −5。代入这些值：

x = (−3 ± √(3² − 4 × 2 × (−5))) / (2 × 2)

x = (−3 ± √(9 + 40)) / 4

x = (−3 ± √49) / 4 = (−3 ± 7) / 4

So x = (−3 + 7) / 4 = 4 / 4 = 1, or x = (−3 − 7) / 4 = −10 / 4 = −2.5. The two roots are x = 1 and x = −2.5.

因此 x = (−3 + 7) / 4 = 4 / 4 = 1，或 x = (−3 − 7) / 4 = −10 / 4 = −2.5。两个根是 x = 1 和 x = −2.5。

Be very careful with negative signs when substituting c into the formula. A common mistake is to forget that b² − 4ac involves subtracting a negative number, which becomes addition.

将 c 代入公式时，要特别注意负号。一个常见错误是忘记 b² − 4ac 中减去一个负数会变成加法。

6. The Discriminant | 判别式

The expression b² − 4ac inside the square root is called the discriminant. It tells us the nature of the roots without actually solving the equation. Let Δ represent the discriminant.

平方根里面的表达式 b² − 4ac 称为判别式。它无需真正解方程就能告诉我们根的性质。用 Δ 表示判别式。

Discriminant Δ	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 \| 一个相等的实数根（重根）
Δ < 0	No real roots; two complex roots \| 无实数根；两个复数根

For example, for the equation x² − 4x + 4 = 0, a = 1, b = −4, c = 4, so Δ = (−4)² − 4(1)(4) = 16 − 16 = 0. This equation has exactly one repeated root. Indeed, x² − 4x + 4 = (x − 2)², so x = 2 is the only root.

例如，对于方程 x² − 4x + 4 = 0，a = 1，b = −4，c = 4，所以 Δ = (−4)² − 4(1)(4) = 16 − 16 = 0。这个方程恰好有一个重根。实际上，x² − 4x + 4 = (x − 2)²，所以 x = 2 是唯一的根。

For the equation x² + x + 1 = 0, Δ = 1² − 4(1)(1) = 1 − 4 = −3, which is negative. This means the parabola does not cross the x-axis at all, and the equation has no real solutions.

对于方程 x² + x + 1 = 0，Δ = 1² − 4(1)(1) = 1 − 4 = −3，为负数。这意味着抛物线根本不与 x 轴相交，这个方程没有实数解。

Exam questions often ask you to find the value of k such that a quadratic equation has equal roots. Set the discriminant equal to zero and solve for k.

考试题经常要求你找出使二次方程有等根的 k 值。令判别式等于零，然后解出 k 即可。

7. Solving Quadratic Equations by Graphs | 用图像法解二次方程

The graph of a quadratic function y = ax² + bx + c is a smooth curve called a parabola. If a > 0, the parabola opens upwards like a “U”. If a < 0, it opens downwards like an "n".

二次函数 y = ax² + bx + c 的图像是一条称为抛物线的光滑曲线。如果 a > 0，抛物线开口朝上，像一个”U”形。如果 a < 0，开口朝下，像一个"n"形。

The solutions of the equation ax² + bx + c = 0 correspond to the x-coordinates of the points where the graph crosses the x-axis. These points are called the x-intercepts or roots.

方程 ax² + bx + c = 0 的解对应于图像与 x 轴交点的 x 坐标。这些点称为 x 截距或根。

If the graph touches the x-axis at exactly one point, the equation has one repeated root. If the graph never touches the x-axis, the equation has no real roots. This agrees with the discriminant, because the discriminant determines whether the parabola crosses the axis, touches it, or misses it entirely.

如果图像恰好与 x 轴相切于一点，则方程有一个重根。如果图像从未接触 x 轴，则方程没有实数根。这与判别式一致，因为判别式决定了抛物线与 x 轴相交、相切还是完全不相交。

To solve a quadratic equation graphically, plot the curve y = ax² + bx + c and read off the x-values where the curve intersects the x-axis. You can also solve an equation like x² + 3x − 1 = 0 by plotting y = x² + 3x − 1 and finding its intercepts, or by drawing the line y = 0. In more advanced questions, you may need to draw a second curve or line and find intersection points.

要用图像法求解二次方程，绘制曲线 y = ax² + bx + c，然后读出曲线与 x 轴交点的 x 值。你也可以通过绘制 y = x² + 3x − 1 并找到其截距来求解 x² + 3x − 1 = 0，或者绘制直线 y = 0。在更高级的问题中，你可能需要绘制第二条曲线或直线并找到交点。

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

Many real-world situations can be modelled by quadratic equations. The most common types of IGCSE word problems involve area, consecutive numbers and projectile motion.

许多现实世界的情境可以用二次方程来建模。IGCSE 中最常见的应用题类型涉及面积、连续整数和抛体运动。

Example: A rectangle has length (x + 3) cm and width x cm. Its area is 70 cm². Find x.

示例：一个矩形的长为 (x + 3) cm，宽为 x cm。它的面积为 70 cm²。求 x。

We write the area equation:

我们写出面积方程：

x(x + 3) = 70

x² + 3x − 70 = 0

(x + 10)(x − 7) = 0

So x = −10 or x = 7. Since a length cannot be negative, the only valid solution is x = 7. The rectangle has width 7 cm and length 10 cm.

因此 x = −10 或 x = 7。由于长度不能为负，唯一有效的解是 x = 7。该矩形的宽为 7 cm，长为 10 cm。

When solving word problems, always check whether both roots are valid in the context of the problem. Negative lengths, times or counts are almost always rejected. Also, remember to include the correct units in your final answer.

在解决应用题时，始终检查两个根在问题情境中是否都有效。负数长度、时间或数量几乎总是要被舍弃的。同时，记得在最终答案中包含正确的单位。

9. Common Exam Mistakes | 常见考试错误

Students frequently lose marks on quadratic questions for avoidable reasons. Here are some of the most common pitfalls that you should watch out for in the exam.

学生经常因为一些可避免的原因在二次方程题目上失分。以下是一些最常见的陷阱，你在考试中应该注意。

Forgetting to rearrange the equation into standard form before factorising or using the formula.
忘记在因式分解或使用公式之前，将方程整理成标准形式。
Dividing both sides by x, which loses the root x = 0.
两边同时除以 x，导致丢失根 x = 0。
Forgetting the ± sign when taking square roots in the completed-square method.
在配方法中开平方时忘记正负号（±）。
Substituting the wrong sign for c in the quadratic formula, especially when c is negative.
在求根公式中代入错误的 c 符号，特别是当 c 为负数时。
Miscalculating the discriminant, for example writing b² − 4ac as b² + 4ac.
计算判别式出错，例如把 b² − 4ac 写成 b² + 4ac。
Not giving answers to the required degree of accuracy, such as 3 significant figures or 2 decimal places.
没有按照要求给出精确度，例如保留 3 位有效数字或 2 位小数。

Always check your answers by substituting them back into the original equation. This simple step can catch many errors and is an excellent habit for the exam.

始终通过把答案代回原方程来检查。这个简单的步骤能发现很多错误，是考试中极好的习惯。

10. Practice Questions | 练习题目

Now it is time to test your understanding. Try these questions without looking at the answers first, then check your working carefully.

现在是时候测试你的理解了。先不要看答案尝试这些题目，然后仔细检查你的解题过程。

Question 1: Solve x² − 5x − 14 = 0 by factorisation.

题目 1：用因式分解法求解 x² − 5x − 14 = 0。

Answer: x = 7 or x = −2

答案：x = 7 或 x = −2

Question 2: Use the quadratic formula to solve 3x² + 6x − 2 = 0. Give your answers correct to 2 decimal places.

题目 2：用求根公式求解 3x² + 6x − 2 = 0。答案保留 2 位小数。

x = (−6 ± √(36 + 24)) / 6 = (−6 ± √60) / 6

x ≈ 0.29 or x ≈ −2.29

Question 3: Find the value of k such that kx² + 6x + 3 = 0 has exactly one real root.

题目 3：求 k 的值，使得 kx² + 6x + 3 = 0 恰好有一个实数根。

Δ = 36 − 12k = 0, so k = 3

Δ = 36 − 12k = 0，所以 k = 3

Question 4: A square has its sides increased by 2 cm, and its area becomes 81 cm². Find the original side length.

题目 4：一个正方形的边长增加 2 cm 后，面积为 81 cm²。求原来的边长。

(x + 2)² = 81, so x + 2 = ±9, giving x = 7 or x = −11. The answer is 7 cm.

(x + 2)² = 81，所以 x + 2 = ±9，得到 x = 7 或 x = −11。答案为 7 cm。

11. Summary and Revision Tips | 总结与复习建议

You now have four tools for solving quadratic equations: factorisation is fastest for simple equations; completing the square is useful for equations that cannot be factorised and for finding turning points; the quadratic formula always works with any quadratic; and graphing provides a visual understanding of the roots.

你现在有四种工具来求解二次方程：因式分解法对于简单方程最快；配方法适用于无法因式分解的方程，并且有助于找顶点；求根公式对任何二次方程都有效；图像法提供了对根的直观理解。

For your revision, make a one-page summary sheet with the quadratic formula, the discriminant rules, and one worked example of each method. Practise at least ten mixed questions under timed conditions, paying special attention to accuracy with negative numbers and surds.

在复习时，做一张一页的总结表，写上求根公式、判别式规则，以及每种方法的一个完整示例。在限时条件下至少练习十道混合题，特别关注负数和根式计算的准确性。

Remember that quadratic equations are not just an isolated topic; they link to functions, graphs, inequalities and coordinate geometry. A solid understanding here will support your performance across the whole IGCSE Mathematics paper.

请记住，二次方程不仅仅是一个孤立的主题；它与函数、图像、不等式和坐标几何紧密相连。在这里打下扎实的基础将帮助你在整个 IGCSE 数学试卷中表现出色。

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Solving Quadratic Equations | 解一元二次方程