Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations appear in many areas of IGCSE Mathematics, from algebra to graphs and problem solving. Mastering the different methods for solving them is essential for exam success.

二次方程在IGCSE数学中无处不在，从代数到图像，再到实际应用题。掌握不同的解法对于考试成功至关重要。

1. The Standard Form | 标准形式

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where x is the unknown and a, b and c are constants, with a ≠ 0.

二次方程是指能写成 ax² + bx + c = 0 形式的方程，其中x是未知数，a、b、c为常数，且a ≠ 0。

The coefficient a cannot be zero because the equation would then become linear. Here are some examples:

系数a不能为零，否则方程就变成一次方程了。下面是一些例子：

Equation 方程	a	b	c
3x² – 5x + 2 = 0	3	-5	2
x² = 4	1	0	-4
2x(x – 1) = 0	2	-2	0

In the second example, b = 0 because there is no x term. In the third example, we must expand the brackets before identifying a, b and c.

在第二个例子中，因为不含x项，所以b = 0。在第三个例子中，我们需要先展开括号，再确定a、b、c的值。

2. Expanding Brackets | 展开括号

Before solving a quadratic equation, we sometimes need to expand and simplify algebraic expressions. The product of two linear factors is a quadratic expression.

在解二次方程之前，我们有时需要展开并化简代数表达式。两个一次因式的乘积是一个二次表达式。

For example: (x + 3)(x – 2) = x² – 2x + 3x – 6 = x² + x – 6.

例如： (x + 3)(x – 2) = x² – 2x + 3x – 6 = x² + x – 6。

Use the distributive law (often remembered as FOIL): multiply the First terms, Outer terms, Inner terms and Last terms.

使用乘法分配律（通常记为FOIL）：依次相乘首项、外项、内项和末项。

When a square of a binomial appears, remember: (x + 5)² = x² + 10x + 25.

当出现二项式的平方时，记住： (x + 5)² = x² + 10x + 25。

3. Solving by Factorising | 因式分解法

If a quadratic expression can be factorised into two linear factors, the equation is quick to solve. Follow these steps:

如果一个二次表达式可以分解为两个一次因式，那么这个方程很容易求解。步骤如下：

Write the equation in the form ax² + bx + c = 0.

将方程写成 ax² + bx + c = 0 的形式。
Factorise the left-hand side into two brackets.

将左边分解成两个括号。
Set each bracket equal to zero and solve the resulting linear equations.

令每个括号等于零，然后解所得的一次方程。

Example: Solve x² – 5x + 6 = 0. This factorises as (x – 2)(x – 3) = 0. So x – 2 = 0 or x – 3 = 0, giving x = 2 or x = 3.

例如：解 x² – 5x + 6 = 0。因式分解得 (x – 2)(x – 3) = 0。因此 x – 2 = 0 或 x – 3 = 0，得到 x = 2 或 x = 3。

Check your solutions by substituting them back into the original equation.

将解代回原方程进行检验。

4. The Quadratic Formula | 公式法

When factorisation is difficult or impossible, use the quadratic formula. For ax² + bx + c = 0, the solutions are:

当因式分解困难或无法分解时，可使用求根公式。对于 ax² + bx + c = 0，解为：

x = (-b ± √(b² – 4ac)) / (2a)

Steps to apply the formula:

应用公式的步骤：

Identify a, b and c carefully, including their signs.

仔细确定a、b、c，包括它们的符号。
Substitute the values into the formula.

将这些值代入公式。
Simplify the numerator and denominator.

化简分子与分母。

Example: Solve 2x² + 3x – 5 = 0. Here a = 2, b = 3 and c = -5. The formula gives x = (-3 ± √(9 + 40)) / 4 = (-3 ± 7) / 4. So x = 1 or x = -5/2.

例如：解 2x² + 3x – 5 = 0。这里 a = 2，b = 3，c = -5。公式给出 x = (-3 ± √(9 + 40)) / 4 = (-3 ± 7) / 4。所以 x = 1 或 x = -5/2。

5. The Discriminant | 判别式

The expression inside the square root, Δ = b² – 4ac, is called the discriminant. It tells us how many real roots the quadratic equation has.

根号内的表达式 Δ = b² – 4ac 称为判别式。它告诉我们二次方程有多少个实数根。

Discriminant 判别式	Number of real roots 实数根个数
Δ > 0	Two different real roots 两个不同的实数根
Δ = 0	One repeated real root 一个重根
Δ < 0	No real roots 没有实数根

Example: For x² + 4x + 9 = 0, Δ = 16 – 36 = -20 < 0, so there are no real roots.

例如：对于 x² + 4x + 9 = 0，Δ = 16 – 36 = -20 < 0，所以没有实数根。

6. Completing the Square | 配方法

Completing the square rewrites a quadratic expression in the form (x + p)² + q. This is very useful for solving equations and finding turning points.

配方法将二次表达式改写为 (x + p)² + q 的形式。这对于解方程和求顶点坐标非常有用。

Example: Solve x² + 6x + 4 = 0 by completing the square.

例如：用配方法解 x² + 6x + 4 = 0。

Step 1: Move the constant to the other side: x² + 6x = -4.

第一步：把常数移到另一边： x² + 6x = -4。

Step 2: Add (6/2)² = 9 to both sides: x² + 6x + 9 = 5.

第二步：两边都加上 (6/2)² = 9： x² + 6x + 9 = 5。

Step 3: Write the left side as a square: (x + 3)² = 5.

第三步：将左边写成完全平方： (x + 3)² = 5。

Step 4: Take the square root: x + 3 = ±√5, so x = -3 ± √5.

第四步：两边开平方： x + 3 = ±√5，所以 x = -3 ± √5。

7. Solving Word Problems | 应用题

Quadratic equations often arise in geometry and number problems. Follow a clear strategy: define the unknown, form an equation, solve it and check the answer makes sense.

二次方程常出现在几何和数字问题中。解题策略是：设未知数、列方程、解方程，并检查答案是否合理。

Example: A rectangle has length 5 cm greater than its width. Its area is 36 cm². Find the width.

例如：一个长方形的长比宽大5 cm，面积为36 cm²。求宽。

Let the width be x cm. Then the length is x + 5 cm. The area equation is x(x + 5) = 36.

设宽为 x cm，则长为 x + 5 cm。面积方程为 x(x + 5) = 36。

Expanding gives x² + 5x – 36 = 0. Factorising: (x + 9)(x – 4) = 0. So x = -9 or x = 4. Since length cannot be negative, the width is 4 cm.

展开得 x² + 5x – 36 = 0。因式分解： (x + 9)(x – 4) = 0。所以 x = -9 或 x = 4。因为长度不能为负，所以宽为4 cm。

8. Graphs and Roots | 图像与根

The graph of a quadratic function y = ax² + bx + c is a parabola. The x-intercepts of the graph are the roots of the equation ax² + bx + c = 0.

二次函数 y = ax² + bx + c 的图像是一条抛物线。图像与x轴的交点就是方程 ax² + bx + c = 0 的根。

If the discriminant is positive, the parabola crosses the x-axis at two points. If it is zero, the parabola touches the x-axis at one point. If it is negative, the parabola does not touch the x-axis.

若判别式为正，抛物线与x轴交于两点；若为零，抛物线与x轴相切于一点；若为负，抛物线不与x轴相交。

The vertex (turning point) of the parabola has x-coordinate x = -b/(2a). Substituting this back gives the y-coordinate.

抛物线的顶点（转折点）的x坐标为 x = -b/(2a)。将该值代回可得到y坐标。

Example: For y = x² – 4x + 3, the roots are x = 1 and x = 3. The x-coordinate of the vertex is x = 2, so the vertex is at (2, -1).

例如：对于 y = x² – 4x + 3，方程的根为 x = 1 和 x = 3。顶点的x坐标为 x = 2，因此顶点坐标为 (2, -1)。

9. Sum and Product of Roots | 根的和与积

For a quadratic equation ax² + bx + c = 0 with roots α and β, the sum and product of the roots have simple formulas.

对于二次方程 ax² + bx + c = 0，若其两根为 α 和 β，则根的和与积有简单的公式。

α + β = -b/a, αβ = c/a

Example: For x² – 5x + 6 = 0, the roots are 2 and 3. Their sum is 5 and product is 6. This matches -b/a = 5 and c/a = 6.

例如：对于 x² – 5x + 6 = 0，两根为2和3，它们的和为5，积为6，与 -b/a = 5 和 c/a = 6 相符。

This relationship can also be used to construct a quadratic equation with given roots.

这一关系也可用于构造具有给定根的二次方程。

10. Common Mistakes | 易错警示

Here are some typical errors students make and how to avoid them:

下面是学生常犯的一些错误以及如何避免它们：

Forgetting to write the equation in the form ax² + bx + c = 0 before factorising.

在因式分解前忘记将方程写成 ax² + bx + c = 0 的形式。
Losing the ± sign when taking the square root.

开平方时遗漏正负号 ±。
Substituting a coefficient with the wrong sign into the quadratic formula.

将系数代入求根公式时弄错符号。
Concluding that x = -2 is invalid in a word problem without checking the context.

在应用题中不结合实际情况检查，直接断定 x = -2 是不合理的。

Always check your solutions in the original equation and read each question carefully.

务必将解代回原方程检验，并仔细审题。

11. Practice Questions | 练习

Try the following questions to test your understanding.

请尝试以下问题来检验你的理解。

1. Solve x² – 7x + 10 = 0. Answer: x = 2 or x = 5.

1. 解方程 x² – 7x + 10 = 0。答案： x = 2 或 x = 5。
2. Solve 3x² + 5x – 2 = 0 using the quadratic formula. Answer: x = 1/3 or x = -2.

2. 用求根公式解方程 3x² + 5x – 2 = 0。答案： x = 1/3 或 x = -2。
3. Find the discriminant of 2x² – 4x + 5 = 0 and state the number of real roots. Answer: Δ = -24, no real roots.

3. 求 2x² – 4x + 5 = 0 的判别式，并判断实数根的个数。答案： Δ = -24，没有实数根。

If you made mistakes, review the relevant section and try again. Practice is the key to success in quadratic equations.

如果做错了，请回顾相应章节并重新尝试。练习是掌握二次方程的关键。

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Solving Quadratic Equations | 解二次方程