Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are a fundamental part of the IGCSE Mathematics syllabus. They appear in almost every exam paper, either as direct questions or as tools for solving problems in other topics such as coordinate geometry, sequences, and practical word problems. Understanding how to solve them efficiently and accurately is essential for achieving a high grade.

二次方程是IGCSE数学教学大纲中的核心内容。它们几乎出现在每份试卷中，要么作为直接问题，要么作为解决坐标几何、数列和实际应用题等其它主题的工具。理解如何高效且准确地求解它们，是取得高分的关键。

1. Definition of Quadratic Equations | 二次方程的定义

A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2. The standard form is written as ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. If a = 0, the equation becomes linear and no longer quadratic.

二次方程是次数为2的多项式方程，即变量的最高次幂为2。标准形式写作 ax² + bx + c = 0，其中a、b、c为常数，且a≠0。如果a=0，方程变为线性方程，不再是二次方程。

For example, 2x² – 3x + 1 = 0 is a quadratic equation because it follows the standard form. The values of x that satisfy the equation are called its roots or solutions. In general, a quadratic equation can have at most two real roots, which may be equal or distinct.

例如，2x² – 3x + 1 = 0 是一个二次方程，因为它符合标准形式。满足该方程的x值称为方程的根或解。一般来说，一个二次方程最多有两个实根，它们可能相等或不同。

2. Solving by Factorisation | 因式分解法

Factorisation is the first method taught for solving quadratic equations. It involves writing the quadratic expression as a product of two linear factors. For example, x² – 5x + 6 = 0 can be factorised as (x – 2)(x – 3) = 0. Then, using the zero product property, we set each factor equal to zero: x – 2 = 0 or x – 3 = 0, giving x = 2 or x = 3.

因式分解法是学习求解二次方程的第一种方法。它将二次表达式写成两个线性因式的乘积。例如，x² – 5x + 6 = 0 可以因式分解为 (x – 2)(x – 3) = 0。然后利用零积性质，令每个因式等于零：x – 2 = 0 或 x – 3 = 0，得到 x = 2 或 x = 3。

To factorise successfully, you need to find two numbers that multiply to give the constant term c and add to give the coefficient b. In the above example, the numbers are 2 and 3 because 2 × 3 = 6 and 2 + 3 = 5. This method works best when the quadratic has simple integer roots.

要成功因式分解，需要找到两个数，它们的乘积等于常数项c，和等于系数b。在上例中，这两个数是2和3，因为2×3=6且2+3=5。当二次方程具有简单的整数根时，此方法最有效。

Always check if the equation is in standard form before factoring.
If the coefficient of x² is not 1, use the product-sum method or the “ac” method.
If the quadratic cannot be easily factorised, use another method.

在因式分解前，务必检查方程是否处于标准形式。
如果x²的系数不是1，使用乘积-和法或”ac”法。
如果二次式不容易分解，换用其它方法。

3. Solving by Completing the Square | 配方法

Completing the square is another algebraic method that transforms the quadratic into a perfect square plus a constant. The general approach is to rewrite ax² + bx + c = 0 as a(x + p)² + q = 0. For example, x² + 6x + 5 = 0 can be written as (x + 3)² – 4 = 0, because (x + 3)² = x² + 6x + 9, and then we subtract 4 to balance the original expression.

配方法是另一种代数方法，它将二次式转化为一个完全平方加一个常数。基本思路是将 ax² + bx + c = 0 改写为 a(x + p)² + q = 0。例如，x² + 6x + 5 = 0 可以写成 (x + 3)² – 4 = 0，因为 (x + 3)² = x² + 6x + 9，然后减去4以平衡原表达式。

The steps are: divide the coefficient of x by 2, square it, and add then subtract this value. This creates a perfect square trinomial. Finally, solve by taking the square root of both sides, remembering to include the ± sign.

步骤如下：将x的系数除以2，平方，然后加再减这个值。这样就构造出完全平方三项式。最后，对两边取平方根来求解，记得包含±号。

x² + 6x + 5 = 0 → (x + 3)² – 4 = 0 → (x + 3)² = 4 → x + 3 = ±2 → x = -1 or x = -5

The notation: From (x + 3)² = 4, we get x + 3 = ±√4 = ±2, so x = -1 or x = -5. This method is particularly useful when the quadratic cannot be factorised.

记法：由 (x + 3)² = 4，得到 x + 3 = ±√4 = ±2，因此 x = -1 或 x = -5。此方法在二次式无法因式分解时尤其有用。

4. The Quadratic Formula | 二次公式

The quadratic formula is a universal method that works for any quadratic equation. It is derived from the process of completing the square on the general form ax² + bx + c = 0. The formula provides the roots directly as x = (-b ± √(b² – 4ac)) / (2a).

二次公式是一种通用方法，适用于任何二次方程。它来源于对一般形式 ax² + bx + c = 0 进行配方推导。该公式直接给出根：x = (-b ± √(b² – 4ac)) / (2a)。

x = (-b ± √(b² – 4ac)) / (2a)

To use this formula, identify the values of a, b, and c from the equation, substitute them into the formula, and simplify. For example, in the equation 2x² + 3x – 2 = 0, we have a = 2, b = 3, and c = -2. The formula gives x = (-3 ± √(9 – 4×2×(-2))) / (4) = (-3 ± √25) / 4 = (-3 ± 5) / 4, so x = 2/4 = 0.5 or x = -8/4 = -2.

使用此公式时，从方程中识别a、b、c的值，将其代入公式并简化。例如，在方程 2x² + 3x – 2 = 0 中，a = 2，b = 3，c = -2。公式给出 x = (-3 ± √(9 – 4×2×(-2))) / (4) = (-3 ± √25) / 4 = (-3 ± 5) / 4，因此 x = 2/4 = 0.5 或 x = -8/4 = -2。

The quadratic formula is reliable but requires careful calculation. Always simplify the discriminant (b² – 4ac) first before taking the square root, and simplify the final fraction if possible.

二次公式可靠，但需要仔细计算。在取平方根之前，务必先简化判别式 (b² – 4ac)，并在可能的情况下简化最终分数。

5. The Discriminant | 判别式

The discriminant is the expression b² – 4ac inside the square root of the quadratic formula. It tells us the nature and number of the roots without fully solving the equation. The value of the discriminant determines whether the equation has two distinct real roots, one repeated real root, or no real roots.

判别式是二次公式中平方根内的表达式 b² – 4ac。它无需完全解方程就能告诉我们根的性质和数量。判别式的值决定方程有两个不同的实根、一个重实根，还是没有实根。

Discriminant (b² – 4ac)	Nature of Roots
b² – 4ac > 0	Two distinct real roots
b² – 4ac = 0	One repeated real root
b² – 4ac < 0	No real roots

判别式 (b² – 4ac)	根的性质
b² – 4ac > 0	两个不同的实根
b² – 4ac = 0	一个重实根
b² – 4ac < 0	没有实根

If the discriminant is a perfect square and non-negative, the roots are rational; otherwise, they are irrational. If the discriminant is negative, the equation has complex roots, which are not covered in IGCSE Mathematics. This property is often tested in coordinate geometry problems involving intersections of lines and curves.

如果判别式是非负完全平方数，则根为有理数；否则为无理数。如果判别式为负，方程有复根，但IGCSE数学不涉及。这个性质常在线与曲线交点问题中考查。

6. Graphical Interpretation | 图形理解

The graph of a quadratic function y = ax² + bx + c is a parabola. The roots of the equation ax² + bx + c = 0 correspond to the x-intercepts of the parabola, where the curve crosses (or touches) the x-axis. The x-coordinate of the vertex is given by h = -b/(2a), and the y-coordinate is found by substituting h into the function.

二次函数 y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的根对应于抛物线的x截距，即曲线与x轴相交（或相切）的点。顶点的x坐标由 h = -b/(2a) 给出，y坐标通过将h代入函数求得。

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, which is the vertex. If it is negative, the parabola does not touch the x-axis at all. The coefficient a determines the direction of the parabola: a > 0 opens upwards, and a < 0 opens downwards.

如果判别式为正，抛物线与x轴相交于两个点。如果为零，抛物线与x轴相切于一个点，即顶点。如果为负，抛物线完全不与x轴相交。系数a决定抛物线的开口方向：a > 0 时开口向上，a < 0 时开口向下。

Example: For y = x² – 4x + 3, the roots are x = 1 and x = 3, and the vertex is at (2, -1). The graph helps visualise the solutions and is frequently used in exam questions about drawing or interpreting graphs.

例如：对于 y = x² – 4x + 3，根为 x = 1 和 x = 3，顶点在 (2, -1)。图形有助于直观理解解，并且经常出现在绘制或解释图形的考题中。

7. 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 of the roots is α + β = -b/a, and the product of the roots is αβ = c/a. These relations are very useful in constructing quadratic equations from given roots without solving them.

对于根为 α 和 β 的二次方程 ax² + bx + c = 0，根的和为 α + β = -b/a，根的积为 αβ = c/a。这些关系在已知根构造二次方程时非常有用，无需解方程。

α + β = -b/a, αβ = c/a

For example, if the roots are 2 and 3, then the sum is 5 and the product is 6. Therefore, the quadratic equation is x² – 5x + 6 = 0. This is because the general form is x² – (sum)x + (product) = 0 when a = 1.

例如，如果根为2和3，则和为5，积为6。因此，二次方程为 x² – 5x + 6 = 0。这是因为当a = 1时，一般形式为 x² – (和)x + (积) = 0。

These relationships also help in checking the correctness of your solutions. After solving, plug the roots back into the sum and product formulas to verify that they match the original coefficients.

这些关系也有助于检查解的正确性。解完后，将根代入和与积的公式，验证它们是否与原系数一致。

8. Applications in Real Life | 实际应用

Quadratic equations are widely used in real-world contexts, including physics, engineering, and economics. For instance, the motion of an object under gravity can be modelled by a quadratic equation, where the height h(t) = -gt²/2 + v₀t + h₀. Solving this equation helps find the time when the object hits the ground.

二次方程广泛应用于物理、工程和经济等现实场景。例如，物体在重力作用下的运动可以用二次方程建模，高度 h(t) = -gt²/2 + v₀t + h₀。求解该方程可以帮助找到物体落地的时间。

In business, profit or revenue functions are often quadratic. Finding the break-even points requires solving a quadratic equation. In geometry, the area of a rectangle with given perimeter and area can lead to a quadratic equation for the dimensions.

在商业中，利润或收益函数通常为二次函数。寻找盈亏平衡点需要求解二次方程。在几何学中，已知周长和面积的矩形边长问题也会归结为二次方程。

Example: A rectangular garden has a length 3 meters greater than its width, and the area is 40 m². Let the width be x. Then x(x + 3) = 40, leading to x² + 3x – 40 = 0. Solving gives x = 5 or x = -8. Since width cannot be negative, the width is 5 m, and the length is 8 m.

例如：一个矩形花园的长度比宽度大3米，面积为40平方米。设宽度为x，则 x(x + 3) = 40，得出 x² + 3x – 40 = 0。求解得 x = 5 或 x = -8。宽度不能为负，所以宽度为5米，长度为8米。

9. Common Mistakes and Tips | 常见错误与提示

One common mistake is forgetting to set the equation to zero before solving. For example, if the equation is x² = 4x, some students divide by x, losing the root x = 0. The correct approach is to rewrite it as x² – 4x = 0, factorise to get x(x – 4) = 0, and then solve to find x = 0 or x = 4.

一个常见错误是求解前忘记将方程移项归零。例如，如果方程是 x² = 4x，有些学生两边除以x，从而丢失了根 x = 0。正确做法是改写为 x² – 4x = 0，因式分解得到 x(x – 4) = 0，然后求解得到 x = 0 或 x = 4。

Another error is mismanaging the negative signs in the quadratic formula. Always enclose negative values in brackets when substituting into the formula. For example, if c = -2, then b² – 4ac becomes b² – 4a(-2) = b² + 8a.

另一个错误是在使用二次公式时处理负号不当。代入公式时，请始终将负值用括号括起来。例如，如果 c = -2，则 b² – 4ac 变为 b² – 4a(-2) = b² + 8a。

Always check if a = 0; if so, it is not a quadratic.
Set the equation in standard form before applying any method.
Simplify surds where possible, e.g., √8 = 2√2.
Double-check your solutions by substituting them back into the original equation.

始终检查a是否为0；如果为0，则不是二次方程。
在使用任何方法前，将方程化为标准形式。
尽量简化根式，例如 √8 = 2√2。
通过将解代回原方程来检查解的正确性。

10. Practice Questions | 练习问题

To master solving quadratic equations, consistent practice is essential. Below are a few practice questions covering different methods. Try to solve them using the appropriate technique.

为掌握解二次方程，坚持练习至关重要。以下是一些涵盖不同方法的练习问题。尝试使用合适的技巧求解。

1. Solve x² – 7x + 12 = 0 by factorisation.

2. Solve x² + 8x + 15 = 0 by completing the square.

3. Solve 2x² – 5x – 3 = 0 using the quadratic formula.

4. Find the discriminant of 3x² + 6x + 2 = 0 and state the nature of roots.

5. If the sum and product of roots are 4 and 3 respectively, write down the quadratic equation.

1. 用因式分解法解 x² – 7x + 12 = 0。

2. 用配方法解 x² + 8x + 15 = 0。

3. 用二次公式解 2x² – 5x – 3 = 0。

4. 求 3x² + 6x + 2 = 0 的判别式，并说明根的性质。

5. 如果根的和与积分别为4和3，写出二次方程。

Answers: 1. x = 3 or x = 4. 2. x = -3 or x = -5. 3. x = 3 or x = -0.5. 4. Discriminant = 36 – 24 = 12 > 0, two distinct real roots. 5. x² – 4x + 3 = 0.

答案：1. x = 3 或 x = 4。2. x = -3 或 x = -5。3. x = 3 或 x = -0.5。4. 判别式 = 36 – 24 = 12 > 0，两个不同的实根。5. x² – 4x + 3 = 0。

In conclusion, solving quadratic equations is a vital skill for IGCSE Mathematics. Master the three main methods—factorisation, completing the square, and the quadratic formula—and understand the discriminant to analyse solutions. With practice, these techniques become second nature, allowing you to tackle any quadratic problem confidently.

总之，解二次方程是IGCSE数学中的关键技能。掌握三种主要方法——因式分解法、配方法和二次公式——并理解判别式以分析解。通过练习，这些技巧会成为本能，让你自信地处理任何二次方程问题。

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Solving Quadratic Equations | 解二次方程