Solving Quadratic Equations | 二次方程与函数

📚 Solving Quadratic Equations | 二次方程与函数

Quadratic equations are a cornerstone of IGCSE Mathematics. They appear in algebra, geometry, projectile motion, and many real-life contexts. Mastering the methods of solving and graphing quadratics is essential for achieving top marks in your exam.

二次方程是IGCSE数学的基石。它们出现在代数、几何、抛体运动以及许多现实生活情境中。掌握求解和作图的方法，是在考试中取得高分的关键。

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

A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2. Its standard form is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. If a = 0, the equation becomes linear, not quadratic.

二次方程是最高次数为2的多项式方程，即变量的最高幂是2。其标准形式为 ax² + bx + c = 0，其中a、b、c为常数，且a ≠ 0。如果a = 0，则方程变为一次方程，而非二次方程。

ax² + bx + c = 0

For example, 3x² – 2x + 5 = 0 is quadratic, whereas 2x + 1 = 0 is not. The solutions of a quadratic equation are called its roots, zeros, or x-intercepts. A quadratic equation usually has two roots, although they may be equal or non-real.

例如，3x² – 2x + 5 = 0 是二次方程，而 2x + 1 = 0 不是。二次方程的解称为根、零点或x截距。二次方程通常有两个根，可能相等，也可能不是实数根。

2. The Factoring Method | 因式分解法

Factoring is often the fastest method when the quadratic expression can be written as the product of two linear factors. The basic idea is to find two numbers that multiply to give ac and add to give b for the general form ax² + bx + c = 0. For the simple case where a = 1, we find two numbers that multiply to c and add to b.

当二次式可以写成两个一次因式的乘积时，因式分解通常是最快的方法。基本思想是：对于一般形式 ax² + bx + c = 0，找到两个数，它们的乘积等于 ac，和等于 b。对于a = 1的简单情况，只需找到两个数，乘积为c，和为b。

Let us solve x² + 5x + 6 = 0. We need two numbers whose product is 6 and sum is 5. They are 2 and 3. Therefore:

我们解 x² + 5x + 6 = 0。需要找到两个数，乘积为6，和为5。这两个数是2和3。因此：

x² + 5x + 6 = (x + 2)(x + 3) = 0

Setting each factor to zero gives x + 2 = 0 or x + 3 = 0, so x = -2 or x = -3. Always make sure the equation is set to zero before factoring.

令每个因式为零，得到 x + 2 = 0 或 x + 3 = 0，所以 x = -2 或 x = -3。分解前务必确保方程右边等于零。

3. Completing the Square | 配方法

Completing the square rewrites a quadratic expression in the form a(x – h)² + k. This is extremely useful for solving equations, finding the vertex, and simplifying integrals later in your studies. The method works by creating a perfect square trinomial.

配方法将二次式改写为 a(x – h)² + k 的形式。这在解方程、求顶点以及以后学习积分时非常有用。该方法通过构造完全平方三项式来实现。

For an equation x² + bx + c = 0, we take half of b, square it, then add and subtract that value. Consider x² + 6x + 2 = 0. Half of 6 is 3, and 3² = 9. So:

对于方程 x² + bx + c = 0，取b的一半，平方，然后加上再减去该值。考虑 x² + 6x + 2 = 0。6的一半是3，3² = 9。所以：

x² + 6x + 2 = (x + 3)² – 9 + 2 = (x + 3)² – 7 = 0

Then (x + 3)² = 7, so x = -3 ± √7. This method works even when the quadratic does not factor neatly.

于是 (x + 3)² = 7，因此 x = -3 ± √7。即使二次式不能简洁地分解，这种方法也有效。

4. The Quadratic Formula | 二次公式

The quadratic formula is a universal tool. It can solve any quadratic equation, including those that are difficult or impossible to factor. Given ax² + bx + c = 0, the solutions are found by:

二次公式是一个通用工具。它可以解任何二次方程，包括那些难以或无法分解的方程。对于 ax² + bx + c = 0，解为：

x = [-b ± √(b² – 4ac)] / (2a)

The symbol ± means there are two solutions: one with a plus sign and one with a minus sign. You should memorize this formula and practise substituting a, b and c carefully. For example, in 2x² – 4x – 3 = 0, a = 2, b = -4, c = -3.

符号±表示有两个解：一个加号，一个减号。你应该记牢这个公式，并练习仔细代入a、b、c。例如在 2x² – 4x – 3 = 0 中，a = 2，b = -4，c = -3。

Substituting gives x = [4 ± √(16 + 24)] / 4 = [4 ± √40] / 4, which simplifies to 1 ± √10/2. Always check your signs and the value of the discriminant.

代入得 x = [4 ± √(16 + 24)] / 4 = [4 ± √40] / 4，化简为 1 ± √10/2。务必检查符号和判别式的值。

5. The Discriminant and the Nature of Roots | 判别式与根的性质

The discriminant, denoted by Δ, is the part of the quadratic formula under the square root: Δ = b² – 4ac. It tells us how many and what kind of roots a quadratic equation has, without solving it fully.

判别式，记作Δ，是二次公式中根号下的部分：Δ = b² – 4ac。它告诉我们二次方程有多少根以及根的性质，而无需完全求解。

Discriminant	Nature of Roots	判别式	根的性质
Δ > 0	Two distinct real roots	Δ > 0	两个不等实根
Δ = 0	One repeated real root	Δ = 0	一个重根（相等实根）
Δ < 0	No real roots (two complex roots)	Δ < 0	无实根（两个复数根）

For example, the equation x² – 4x + 4 = 0 has Δ = 16 – 16 = 0, so it has one repeated root x = 2. If Δ is a perfect square, the roots are rational, which also means the quadratic factorises.

例如，方程 x² – 4x + 4 = 0 的 Δ = 16 – 16 = 0，因此有一个重根 x = 2。如果Δ是完全平方数，则根为有理数，也意味着二次式能因式分解。

6. Graphing Quadratic Functions | 二次函数图像

A quadratic function f(x) = ax² + bx + c has a U-shaped graph called a parabola. If a > 0, the parabola opens upward like a smiling face; if a < 0, it opens downward like a sad face. The graph is symmetric, and its maximum or minimum point is called the vertex.

二次函数 f(x) = ax² + bx + c 的图像是一条U形曲线，称为抛物线。若a > 0，抛物线开口向上，如笑脸；若a < 0，开口向下，如哭脸。图像是对称的，其最高点或最低点称为顶点。

The x-intercepts correspond to the real roots of the equation f(x) = 0. The y-intercept is found by setting x = 0, giving (0, c). To sketch a parabola accurately, you should locate the vertex, the y-intercept, and the x-intercepts, then plot additional points if needed.

x截距对应方程 f(x) = 0 的实根。y截距通过令x = 0得到，为 (0, c)。为了准确画出抛物线草图，你需要确定顶点、y截距和x截距，必要时再描几个辅助点。

y = ax² + bx + c

7. Vertex and Axis of Symmetry | 顶点与对称轴

The vertex is the turning point of the parabola. Its x-coordinate is always given by x = -b/(2a). The vertical line passing through this x-coordinate is called the axis of symmetry, and it divides the parabola into two mirror images.

顶点是抛物线的转折点。其x坐标始终为 x = -b/(2a)。经过该x坐标的垂直线称为对称轴，它将抛物线分为两半镜像。

To find the y-coordinate of the vertex, substitute x = -b/(2a) back into the original function. For example, for f(x) = x² – 4x + 5, we have a = 1 and b = -4, so x = 4/(2×1) = 2. Then f(2) = 4 – 8 + 5 = 1. The vertex is (2, 1).

要求顶点的y坐标，将 x = -b/(2a) 代回原函数。例如，对于 f(x) = x² – 4x + 5，a = 1，b = -4，所以 x = 4/(2×1) = 2。然后 f(2) = 4 – 8 + 5 = 1。顶点为 (2, 1)。

If the quadratic is written in vertex form y = a(x – h)² + k, then the vertex is directly (h, k) and the axis of symmetry is x = h. This form is obtained by completing the square.

若二次式写成顶点式 y = a(x – h)² + k，则顶点直接为 (h, k)，对称轴为 x = h。这种形式可通过配方法得到。

8. Solving Quadratic Inequalities | 二次不等式

Quadratic inequalities involve expressions like ax² + bx + c > 0, < 0, ≥ 0, or ≤ 0. To solve them, we first find the critical values where the quadratic equals zero, then test intervals on a number line.

二次不等式涉及形如 ax² + bx + c > 0、< 0、≥ 0 或 ≤ 0 的式子。求解时，先找到二次式等于零的临界值，然后在数轴上测试各区间。

Let us solve x² – x – 2 < 0. Factor to get (x - 2)(x + 1) < 0. The roots are -1 and 2. These split the number line into three intervals: x < -1, -1 < x < 2, and x > 2. Test each interval by picking a sample point.

我们来解 x² – x – 2 < 0。分解得 (x - 2)(x + 1) < 0。根为 -1 和 2。它们将数轴分为三个区间：x < -1、-1 < x < 2、x > 2。选取样本点测试每个区间。

For x = 0, the product (0 – 2)(0 + 1) = -2 < 0, so the inequality is true in -1 < x < 2. Thus the solution is -1 < x < 2. Sketching the parabola can help you confirm the sign pattern quickly.

对于 x = 0，乘积 (0 – 2)(0 + 1) = -2 < 0，因此不等式在 -1 < x < 2 内成立。所以解集为 -1 < x < 2。画出抛物线草图可以帮助你快速确认符号规律。

9. Applications in Word Problems | 应用题

Quadratic equations often arise in real-world problems involving areas, projectile motion, and business profit. The key is to translate the given information into a quadratic equation, solve it, and then check which solutions make sense in the context.

二次方程常出现在涉及面积、抛体运动和商业利润的现实问题中。关键是将已知信息转化为二次方程，求解，然后检查哪些解在情境中合理。

For example, a rectangle has length (x + 5) cm and width (x – 2) cm, with area 40 cm². The equation is (x + 5)(x – 2) = 40. Expanding gives x² + 3x – 10 = 40, so x² + 3x – 50 = 0.

例如，矩形的长为 (x + 5) 厘米，宽为 (x – 2) 厘米，面积为 40 平方厘米。方程为 (x + 5)(x – 2) = 40。展开得 x² + 3x – 10 = 40，即 x² + 3x – 50 = 0。

Using the quadratic formula, the roots are not nice numbers, but only the positive root gives a valid width. This shows why it is important to interpret results and reject nonsensical negative lengths.

使用二次公式，根不是整数，但只有正根给出有效宽度。这说明为什么我们必须解释结果，并舍去无意义的负长度。

10. Common Mistakes and Tips | 常见错误与技巧

One common mistake is forgetting to rearrange the equation into the standard form ax² + bx + c = 0 before factoring or applying the formula. Always move all terms to one side first. Another mistake is dropping the ± sign when taking square roots.

一个常见错误是在分解或应用公式之前，忘记将方程整理成标准形式 ax² + bx + c = 0。务必先将所有项移到一边。另一个错误是在开平方时漏掉 ± 号。

Always check if a quadratic can be factorised before using the formula.
先检查二次方程能否因式分解，再使用公式。
When using the quadratic formula, write down a, b, c with their signs.
使用二次公式时，写上a、b、c及它们的符号。
Remember that a quadratic equation can have zero, one, or two real roots.
记住二次方程可以没有、有一个或两个实根。
Practice sketching parabolas to build visual understanding.
多练习画抛物线，建立直观理解。
In word problems, always reject roots that do not fit the physical context.
在应用题中，始终舍去不符合实际情境的根。

By mastering these techniques and avoiding these mistakes, you will be well prepared for any quadratic question in your IGCSE exam. Keep practising with a variety of examples to build speed and confidence.

通过掌握这些技巧并避免这些错误，你将能够从容应对IGCSE考试中的任何二次方程问题。坚持练习各种题型，以提升速度和信心。

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Solving Quadratic Equations | 二次方程与函数