Quadratic Equations and Functions | 二次方程与函数

📚 Quadratic Equations and Functions | 二次方程与函数

Quadratic equations and functions are a central topic in IGCSE Mathematics. They connect algebra, graphs, and problem-solving, appearing in both Paper 2 and Paper 4. Mastering this topic builds confidence for higher-level studies and helps you secure crucial marks.

二次方程与函数是IGCSE数学的核心内容。它们将代数、图像和问题解决联系起来，出现在Paper 2和Paper 4中。掌握这一主题能增强你继续深入学习的信心，并帮助你在考试中拿下关键分数。

1. Standard Form of a Quadratic Equation | 二次方程的标准形式

A quadratic equation in one variable can be written in the form ax² + bx + c = 0, where a, b, and c are real numbers, and a ≠ 0. The coefficient a determines the shape of the parabola, while b and c affect its position.

含一个未知数的二次方程可以写成 ax² + bx + c = 0 的形式，其中 a、b、c 是实数，且 a ≠ 0。系数 a 决定抛物线的形状，而 b 和 c 影响其位置。

For example, in the equation 3x² − 5x + 2 = 0, we have a = 3, b = −5, and c = 2.

例如，在方程 3x² − 5x + 2 = 0 中，a = 3，b = −5，c = 2。

If a = 0, the equation becomes linear, which is a different type of problem. Therefore, a quadratic equation must always include an x² term.

如果 a = 0，方程就变成线性方程，那是另一类问题。因此，二次方程必须始终包含 x² 项。

2. Solving by Factorisation | 用因式分解法求解

Factorisation is often the quickest method when the quadratic expression can be written as a product of two linear factors. For example, x² + 7x + 12 = 0 can be factored as (x + 3)(x + 4) = 0.

当二次表达式可以写成两个一次因式的乘积时，因式分解法通常是最快的方法。例如，x² + 7x + 12 = 0 可以分解为 (x + 3)(x + 4) = 0。

Using the zero product property, if (x + 3)(x + 4) = 0, then x + 3 = 0 or x + 4 = 0. Thus x = −3 or x = −4.

利用零积性质，如果 (x + 3)(x + 4) = 0，那么 x + 3 = 0 或 x + 4 = 0。因此 x = −3 或 x = −4。

To factorise ax² + bx + c, look for two numbers that multiply to give ac and add to give b. For x² + 7x + 12, the numbers 3 and 4 multiply to 12 and add to 7.

要分解 ax² + bx + c，需要找到两个数，它们相乘等于 ac，相加等于 b。对于 x² + 7x + 12，3 和 4 相乘为 12，相加为 7。

Check if the expression can be factored before trying other methods.
Be careful with signs when b or c is negative.
Always expand your factors to verify correctness.

在尝试其他方法之前，先检查表达式是否可分解。
当 b 或 c 为负数时，要特别注意符号。
始终展开因式验证正确性。

3. Solving by Completing the Square | 用配方法求解

Completing the square rewrites x² + bx as (x + b/2)² − (b/2)². This is useful when factorisation is not obvious or when the quadratic has irrational roots.

配方法将 x² + bx 改写为 (x + b/2)² − (b/2)²。当因式分解不明显或二次方程有无理根时，这种方法很有用。

For example, solve x² + 6x + 1 = 0. First, write x² + 6x as (x + 3)² − 9. Then the equation becomes (x + 3)² − 9 + 1 = 0, so (x + 3)² = 8.

例如，解 x² + 6x + 1 = 0。首先，将 x² + 6x 写成 (x + 3)² − 9。然后方程变为 (x + 3)² − 9 + 1 = 0，所以 (x + 3)² = 8。

Taking square roots gives x + 3 = ±√8, so x = −3 ± 2√2.

两边开平方得 x + 3 = ±√8，所以 x = −3 ± 2√2。

(x + b/2)² = x² + bx + (b/2)²

The general form for completing the square is x² + bx + c = (x + p)² + q, where p = b/2 and q = c − p².

配方法的一般形式为 x² + bx + c = (x + p)² + q，其中 p = b/2，q = c − p²。

4. The Quadratic Formula | 二次公式

The quadratic formula solves any quadratic equation ax² + bx + c = 0 directly. It is especially useful when factorisation fails or when the coefficients are difficult to handle.

二次公式可以直接求解任意二次方程 ax² + bx + c = 0。当因式分解失败或系数难以处理时，它尤其有用。

x = (−b ± √(b² − 4ac)) / (2a)

To use the formula, substitute the values of a, b, and c into the equation. Simplify carefully, especially the square root and the sign of b.

使用公式时，将 a、b、c 的值代入方程即可。要小心化简，特别是根号和 b 的符号。

For example, solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, c = −3. Therefore, x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = (2 ± √10) / 2.

例如，解 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。因此，x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = (2 ± √10) / 2。

Always check the discriminant first, because a negative value means no real roots.

先检查判别式，因为如果判别式为负，则没有实数根。

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

The discriminant, Δ = b² − 4ac, tells us how many real roots a quadratic equation has. It appears inside the square root of the quadratic formula.

判别式 Δ = b² − 4ac 告诉我们二次方程有多少个实数根。它出现在二次公式的根号内。

Δ > 0	Two distinct real roots
Δ = 0	One repeated real root
Δ < 0	No real roots

Δ > 0	两个不同实数根
Δ = 0	一个重复实数根
Δ < 0	没有实数根

If Δ is a perfect square, the roots are rational; otherwise, they are irrational. This information helps you decide which solving method to use.

如果 Δ 是完全平方数，根是有理数；否则是无理数。这个信息能帮助你决定使用哪种求解方法。

6. 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 roots is α + β = −b/a, and the product is αβ = c/a.

对于根为 α 和 β 的二次方程 ax² + bx + c = 0，根之和为 α + β = −b/a，根之积为 αβ = c/a。

α + β = −b/a, αβ = c/a

This relationship is useful for forming quadratic equations from given roots without solving the original equation.

这个关系在已知根的情况下构造二次方程时非常有用，无需解原方程。

For instance, if a quadratic equation has roots 2 and 5, then the sum is 7 and the product is 10. Therefore, the equation is x² − 7x + 10 = 0.

例如，如果某二次方程的根是 2 和 5，那么和为 7，积为 10。因此，方程为 x² − 7x + 10 = 0。

7. Quadratic Functions and Their Graphs | 二次函数与图像

The graph of a quadratic function y = ax² + bx + c is a parabola. If a > 0, it opens upwards and has a minimum point; if a < 0, it opens downwards and has a maximum point.

二次函数 y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，抛物线开口向上，有最小值点；如果 a < 0，抛物线开口向下，有最大值点。

The vertex of the parabola is the turning point. Its x-coordinate is given by x = −b/(2a). The y-coordinate is found by substituting this x-value into the function.

抛物线的顶点是转向点。其 x 坐标由 x = −b/(2a) 给出。y 坐标通过将该 x 值代入函数求得。

The axis of symmetry is the vertical line through the vertex, with equation x = −b/(2a).

对称轴是通过顶点的竖直线，方程为 x = −b/(2a)。

8. Sketching Quadratic Graphs | 绘制二次函数图像

To sketch a quadratic graph accurately, you need to find three key features: the roots, the vertex, and the y-intercept. These points define the overall shape.

要准确绘制二次函数图像，需要找到三个关键特征：根、顶点和 y 截距。这些点决定整体形状。

Find the roots by solving y = 0, which gives the x-intercepts.
Find the y-intercept by setting x = 0.
Find the vertex using x = −b/(2a).

通过解 y = 0 求根，即可得到 x 截距。
通过令 x = 0 求 y 截距。
使用 x = −b/(2a) 求顶点。

Plot these points on a coordinate grid, then draw a smooth curve through them. Make sure the curve shows the correct minimum or maximum point.

在坐标网格上标出这些点，然后画一条平滑曲线穿过它们。确保曲线显示出正确的最小值点或最大值点。

9. Solving Quadratic Inequalities | 二次不等式的求解

Quadratic inequalities involve expressions such as x² − x − 6 > 0. To solve them, first find the roots of the corresponding equation, then test intervals on a number line.

二次不等式涉及如 x² − x − 6 > 0 的表达式。要解这类不等式，首先求对应方程的根，然后在数轴上测试区间。

For x² − x − 6 > 0, the roots are x = 3 and x = −2. The quadratic is positive when x < −2 or x > 3, because the graph opens upwards.

对于 x² − x − 6 > 0，根为 x = 3 和 x = −2。由于图像开口向上，该二次式在 x < −2 或 x > 3 时为正。

Therefore, the solution set is x ≤ −2 or x ≥ 3, depending on whether the inequality is strict.

因此，解集为 x ≤ −2 或 x ≥ 3，具体取决于不等式是否为严格不等式。

10. Applications of Quadratics | 二次方程的应用

Quadratic equations model many real-life situations, such as projectile motion, area optimisation, and profit calculations. In these problems, we often need to interpret the meaningful root.

二次方程可以模拟许多现实情境，如抛体运动、面积优化和利润计算。在这些问题中，我们通常需要解释有意义的根。

For example, a rectangle has area 24 m² and its length is 2 m longer than its width. If the width is x, then x(x + 2) = 24, which gives x² + 2x − 24 = 0.

例如，一个矩形面积为 24 m²，其长比宽长 2 m。如果宽为 x，则 x(x + 2) = 24，即 x² + 2x − 24 = 0。

Solving this gives x = 4 or x = −6. Since width cannot be negative, the width is 4 m and the length is 6 m.

求解得 x = 4 或 x = −6。由于宽不能为负，所以宽为 4 m，长为 6 m。

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

Many students lose marks by making simple errors. Always rewrite the equation in the form ax² + bx + c = 0 before solving.

许多学生因为简单错误而失分。求解前，一定要将方程改写为 ax² + bx + c = 0 的形式。

Do not forget that a negative discriminant means no real roots.
Check your factoring by expanding.
When using the quadratic formula, be careful with negative signs.
In word problems, reject negative or impossible answers.

不要忘记判别式为负意味着没有实数根。
通过展开来检查你的因式分解。
使用二次公式时，注意负号的处理。
在应用题中，舍弃负数或不可能的解。

Practice drawing parabolas and solving equations until the methods become automatic. This will save you time in the exam.

多练习绘制抛物线和求解方程，直到这些方法变得熟练。这将帮助你在考试中节省时间。

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Quadratic Equations and Functions | 二次方程与函数