Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, graphs, geometry, and even in everyday problem-solving. Understanding how to solve them is essential for exam success.

二次方程是IGCSE数学中最重要的话题之一。它们出现在代数、图像、几何甚至日常问题解决中。理解如何解它们是考试成功的关键。

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

A quadratic equation is a polynomial equation of degree 2. This means 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.

二次方程是次数为2的多项式方程。这意味着变量的最高次幂是2。一般形式是 ax² + bx + c = 0，其中a、b、c是常数，且a ≠ 0。

For example, 2x² + 3x − 5 = 0 is a quadratic equation, while x + 1 = 0 is not, because its highest power is 1.

例如，2x² + 3x − 5 = 0 是二次方程，而 x + 1 = 0 不是，因为它的最高次幂是1。

2. Standard Form | 标准形式

Always rewrite a quadratic equation in the standard form ax² + bx + c = 0 before solving. This helps you identify a, b, and c correctly, especially when using the quadratic formula.

在求解之前，始终将二次方程改写为标准形式 ax² + bx + c = 0。这有助于正确识别a、b、c，尤其是在使用二次公式时。

If an equation looks like 2x² = 4x + 6, subtract 4x and 6 from both sides to get 2x² − 4x − 6 = 0. Now it is ready to solve.

如果方程看起来像 2x² = 4x + 6，两边减去4x和6，得到 2x² − 4x − 6 = 0。现在就可以求解了。

3. Solving by Factorisation | 因式分解法

Factorisation is the quickest method when the quadratic can be factored into two linear factors. The idea is to write ax² + bx + c as (px + q)(rx + s) = 0.

因式分解是当二次式可以分解为两个一次因式时最快捷的方法。思路是将 ax² + bx + c 写成 (px + q)(rx + s) = 0。

Then, if the product of two factors is zero, one of them must be zero. So set each factor equal to zero and solve for x.

然后，如果两个因式的乘积为零，则其中至少一个因式必须为零。因此让每个因式等于零，然后解出x。

Example: x² − 5x + 6 = 0. Factor as (x − 2)(x − 3) = 0. Therefore x = 2 or x = 3.

例如：x² − 5x + 6 = 0。分解为 (x − 2)(x − 3) = 0。因此 x = 2 或 x = 3。

4. Solving by Completing the Square | 配方法

Completing the square turns the equation into a perfect square plus a constant. This method always works and also helps derive the quadratic formula.

配方法将方程转化为一个完全平方加上一个常数。这种方法总是有效，也帮助推导二次公式。

For x² + bx + c = 0, rewrite as (x + b/2)² − (b/2)² + c = 0. Then solve by taking square roots.

对于 x² + bx + c = 0，改写为 (x + b/2)² − (b/2)² + c = 0。然后通过开平方求解。

Example: x² + 6x − 7 = 0. Write (x + 3)² − 9 − 7 = 0, so (x + 3)² = 16. Hence x + 3 = ±4, giving x = 1 or x = −7.

例如：x² + 6x − 7 = 0。写成 (x + 3)² − 9 − 7 = 0，所以 (x + 3)² = 16。因此 x + 3 = ±4，得到 x = 1 或 x = −7。

5. The Quadratic Formula | 二次公式

For any quadratic equation ax² + bx + c = 0, the solutions are given by the quadratic formula:

对于任意二次方程 ax² + bx + c = 0，解由二次公式给出：

x = (−b ± √(b² − 4ac)) / (2a)

This formula is especially useful when factorisation is difficult or impossible. You can substitute a, b, and c directly and simplify.

当因式分解困难或不可能时，这个公式尤其有用。你可以直接代入a、b、c并化简。

Remember to calculate b² − 4ac first. This value tells you how many roots there are.

记住先计算 b² − 4ac。这个值告诉你根的数量。

6. The Discriminant | 判别式

The expression b² − 4ac is called the discriminant. It determines the nature of the roots without fully solving the equation.

表达式 b² − 4ac 称为判别式。它无需完全解方程就能确定根的性质。

If b² − 4ac > 0, there are two distinct real roots.
如果 b² − 4ac > 0，有两个不同的实数根。
If b² − 4ac = 0, there is exactly one real root (a repeated root).
如果 b² − 4ac = 0，恰好有一个实数根（重根）。
If b² − 4ac < 0, there are no real roots (complex roots).
如果 b² − 4ac < 0，没有实数根（复数根）。

For example, x² − 4x + 4 = 0 has discriminant (−4)² − 4×1×4 = 0, so it has a repeated root x = 2.

例如，x² − 4x + 4 = 0 的判别式是 (−4)² − 4×1×4 = 0，所以它有一个重根 x = 2。

7. Sum and Product of Roots | 根的和与积

For a quadratic equation ax² + bx + c = 0, the sum of the roots is −b/a and the product of the roots is c/a. This is useful for checking answers.

对于二次方程 ax² + bx + c = 0，根的和是 −b/a，根的积是 c/a。这有助于检查答案。

Example: For 2x² − 8x + 6 = 0, the sum is 8/2 = 4 and the product is 6/2 = 3. If the roots are 1 and 3, their sum is 4 and product is 3, which matches.

例如：对于 2x² − 8x + 6 = 0，根的和是 8/2 = 4，积是 6/2 = 3。如果根是1和3，它们的和是4，积是3，正好吻合。

8. Solving by Using a Graph | 图像法

The roots of a quadratic equation ax² + bx + c = 0 correspond to the x-intercepts of the parabola y = ax² + bx + c. You can estimate roots by reading the graph.

二次方程 ax² + bx + c = 0 的根对应于抛物线 y = ax² + bx + c 的x截距。你可以通过读图来估计根。

If the graph does not cross the x-axis, the equation has no real roots. If it touches the x-axis at one point, there is a repeated root.

如果图像不与x轴相交，则方程没有实数根。如果它在一点处接触x轴，则有一个重根。

For example, y = x² − 4 crosses the x-axis at x = −2 and x = 2. So the equation x² − 4 = 0 has roots −2 and 2.

例如，y = x² − 4 在 x = −2 和 x = 2 处与x轴相交。因此方程 x² − 4 = 0 的根是 −2 和 2。

9. Solving Real-Life Problems | 解决实际问题

Quadratic equations often model real-world situations such as projectile motion, area problems, and profit calculations. You must translate the problem into an equation.

二次方程常用来模拟现实情境，如抛体运动、面积问题和利润计算。你必须将问题转化为方程。

Example: A rectangle has length 3 cm longer than its width. Its area is 40 cm². If the width is x, then x(x + 3) = 40. So x² + 3x − 40 = 0.

例如：一个矩形的长比宽长3厘米，面积是40平方厘米。如果宽是x，则 x(x + 3) = 40。所以 x² + 3x − 40 = 0。

Solving gives (x + 8)(x − 5) = 0, so x = 5 (since width cannot be negative). The rectangle is 5 cm by 8 cm.

解得 (x + 8)(x − 5) = 0，所以 x = 5（因为宽不能为负）。矩形是5厘米×8厘米。

10. Common Mistakes | 常见错误

Many students make avoidable errors when solving quadratics. Here are the most frequent ones:

许多学生在解二次方程时会犯可避免的错误。以下是最常见的几种：

Forgetting to set the equation to zero before factorising.
忘记在因式分解前将方程化为零。
Misapplying the quadratic formula by using the wrong signs.
错误地使用二次公式，导致符号错误。
Dividing both sides by x when x could be zero.
两边同时除以x而没有考虑x可能为零的情况。
Ignoring the ± symbol when taking square roots.
开平方时忽略±符号。

Always check your solutions by substituting them back into the original equation.

始终将解代回原方程进行验证。

11. Practice Questions | 练习题

Try these questions to test your understanding. Solve each equation or find the required value.

尝试以下问题来测试你的理解。解每个方程或求所需值。

Solve x² − 7x + 10 = 0.
解 x² − 7x + 10 = 0。
Solve 2x² + 5x − 3 = 0 using the quadratic formula.
使用二次公式解 2x² + 5x − 3 = 0。
Find the discriminant of x² − 3x + 5 = 0 and state the nature of the roots.
求 x² − 3x + 5 = 0 的判别式，并说明根的性质。
An object is thrown upward with height h = −5t² + 20t. When is the height zero?
一个物体被向上抛出，高度 h = −5t² + 20t。何时高度为零？

Answers: 1) x = 2 or 5. 2) x = 0.5 or −3. 3) Discriminant = −11, no real roots. 4) t = 0 or 4 seconds.

答案：1) x = 2 或 5。2) x = 0.5 或 −3。3) 判别式 = −11，无实数根。4) t = 0 或 4 秒。

12. Summary | 总结

Quadratic equations can be solved by factorisation, completing the square, or using the quadratic formula. The discriminant tells us the nature of the roots. Practising each method will help you choose the fastest one in the exam.

二次方程可以通过因式分解、配方或使用二次公式来求解。判别式告诉我们根的性质。练习每种方法将帮助你在考试中选择最快的方法。

Always write the equation in standard form first, show your working clearly, and check your answers. With consistent practice, you will master this topic.

始终先将方程写成标准形式，清晰地展示步骤，并检查答案。通过持续练习，你将掌握这个主题。

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Solving Quadratic Equations | 解二次方程