Quadratic Equations & Inequalities | 二次方程与不等式

📚 Quadratic Equations & Inequalities | 二次方程与不等式

Quadratic equations and inequalities are fundamental topics in the IGCSE Mathematics syllabus. They appear in many exam questions, from solving for roots to interpreting graphs and modelling real-life scenarios.

二次方程与不等式是 IGCSE 数学教学大纲中的基础专题。它们在许多考试题目中出现，从求根、解释图像到建模现实情景皆有涉及。

1. Standard Form and Basic Definitions | 标准形式与基本定义

A quadratic equation is any equation that can be expressed in the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of the variable x is 2.

二次方程是任何可以表示为 ax² + bx + c = 0 形式的方程，其中 a、b、c 是常数，且 a ≠ 0。变量 x 的最高次数为 2。

A quadratic expression has three terms: the quadratic term ax², the linear term bx and the constant term c. When all terms are on one side and the other side is zero, we call it the standard form.

二次表达式包含三项：二次项 ax²、一次项 bx 和常数项 c。当所有项移到一侧且另一侧为零时，我们称之为标准形式。

2. Solving by Factorisation | 因式分解法

Factorisation is often the fastest method when the equation has simple integer roots. We rewrite ax² + bx + c as (px + q)(rx + s), then set each bracket equal to zero.

当方程具有简单的整数根时，因式分解往往是最快的方法。我们将 ax² + bx + c 改写成 (px + q)(rx + s)，然后令每个括号等于零。

x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0

So x = 2 or x = 3. Always expand your brackets to check the factorisation is correct.

因此 x = 2 或 x = 3。务必展开括号验证分解是否正确。

3. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² + q. This is useful for solving equations where factorisation is difficult and for finding the turning point of a parabola.

配方法将二次式改写为 (x + p)² + q 的形式。这对于不易因式分解的方程求解以及寻找抛物线顶点非常有用。

x² + 6x + 2 = 0 → (x + 3)² − 7 = 0

Then (x + 3)² = 7, so x = −3 ± √7. Remember to halve the coefficient of x inside the bracket.

然后 (x + 3)² = 7，所以 x = −3 ± √7。注意括号内 x 的系数要除以 2。

4. The Quadratic Formula | 二次求根公式

For any quadratic equation ax² + bx + c = 0, the solutions can be found using the quadratic formula. This method always works, even when factorisation is impossible.

对于任意二次方程 ax² + bx + c = 0，可以使用二次求根公式求解。这种方法始终有效，即使无法因式分解。

x = (−b ± √(b² − 4ac)) / 2a

Substitute the values of a, b and c carefully. Simplify the square root as much as possible before dividing by 2a.

仔细代入 a、b、c 的值。先尽可能化简根号，再除以 2a。

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

The expression b² − 4ac inside the quadratic formula is called the discriminant. It tells us how many real roots a quadratic equation has.

二次求根公式中的 b² − 4ac 称为判别式。它告诉我们二次方程有多少个实数根。

If b² − 4ac > 0, there are two distinct real roots.

若 b² − 4ac > 0，则有两个不同的实数根。
If b² − 4ac = 0, there is exactly one repeated real root.

若 b² − 4ac = 0，则有一个重根（实数根）。
If b² − 4ac < 0, there are no real roots (two complex roots).

若 b² − 4ac < 0，则没有实数根（有两个复数根）。

For example, x² + 2x + 5 = 0 has discriminant 4 − 20 = −16, so it has no real solutions.

例如，x² + 2x + 5 = 0 的判别式为 4 − 20 = −16，因此没有实数解。

6. Graphs of Quadratic Functions | 二次函数的图像

The graph of y = ax² + bx + c is a parabola. When a > 0, the parabola opens upward and has a minimum point. When a < 0, it opens downward and has a maximum point.

y = ax² + bx + c 的图像是一条抛物线。当 a > 0 时，抛物线开口向上，有最小值点；当 a < 0 时，开口向下，有最大值点。

The x-intercepts of the graph correspond to the roots of the equation ax² + bx + c = 0. The y-intercept is the value of c.

抛物线与 x 轴的交点对应方程 ax² + bx + c = 0 的根。与 y 轴的交点则是 c 的值。

Vertex (turning point): x = −b / 2a

The vertical line through the vertex is the axis of symmetry.

经过顶点的竖直线是抛物线的对称轴。

7. Solving Quadratic Equations by Graphing | 利用图像求解二次方程

To solve ax² + bx + c = 0 graphically, plot the parabola y = ax² + bx + c and identify the x-intercepts. The x-values where the curve crosses the x-axis are the solutions.

要以图像方式求解 ax² + bx + c = 0，可绘制抛物线 y = ax² + bx + c 并找出与 x 轴的交点。曲线与 x 轴相交处的 x 值就是解。

If the parabola does not cross the x-axis, the equation has no real roots. If it just touches the axis, there is exactly one repeated root.

如果抛物线不与 x 轴相交，则方程没有实数根。如果它只与 x 轴相切，则恰有一个重根。

8. Solving Quadratic Inequalities | 解二次不等式

A quadratic inequality involves a quadratic expression with an inequality sign, such as x² − 5x + 6 > 0. Solving it means finding the range of x values that make the inequality true.

二次不等式涉及带有不等号的一次表达式，例如 x² − 5x + 6 > 0。求解它意味着找出使不等式成立的 x 的取值范围。

First solve the corresponding equation to find the critical points. Then test intervals on a number line or use the graph of the parabola to determine the sign.

首先解对应的方程找出临界点。然后使用数轴上的区间测试，或根据抛物线图像确定符号。

x² − 5x + 6 > 0 → x < 2 or x > 3

Remember to switch the inequality sign when multiplying or dividing by a negative number, but note that squaring both sides may require extra care.

记住在乘以或除以负数时要改变不等号方向，但注意对两边平方可能需要额外小心。

9. Applications in Word Problems | 应用题建模

Quadratic equations often model area, projectile motion, profit, or number relationships. A typical problem gives a condition, such as the sum of the squares of two consecutive integers equals 113.

二次方程常被用于建模面积、抛体运动、利润或数字关系。一个典型问题是：两个连续整数的平方和等于 113。

n² + (n + 1)² = 113 → 2n² + 2n + 1 = 113

Simplify to 2n² + 2n − 112 = 0, divide by 2 to get n² + n − 56 = 0, then factorise to find n = 7 or n = −8. Check which answer fits the context.

化简得 2n² + 2n − 112 = 0，除以 2 得 n² + n − 56 = 0，然后因式分解得 n = 7 或 n = −8。检查哪一个答案符合题意。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

A common mistake is forgetting to set the equation to zero before factorising or using the quadratic formula. Another is incorrectly simplifying √(b² − 4ac) when it is a perfect square.

一个常见错误是在因式分解或使用求根公式前忘记将方程化为零。另一个是在 b² − 4ac 是完全平方数时错误地化简根号。

Always check your answers by substituting them back into the original equation. In multiple-choice questions, avoid guessing; use the discriminant to quickly rule out options.

始终将答案代回原方程进行检验。在选择题中，不要凭猜测；使用判别式快速排除选项。

When drawing a quadratic graph, plot at least five points, including the vertex and the y-intercept. Label the axes and the intercepts clearly to earn full method marks.

绘制二次函数图像时，至少取五个点，包括顶点和 y 轴截距。清楚标注坐标轴和截距，以获得完整方法分。

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Quadratic Equations & Inequalities | 二次方程与不等式