Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations appear throughout the Edexcel IGCSE Mathematics syllabus, from factorising to graphing. Mastering them is essential for higher-level algebra and problem solving.

二次方程贯穿爱德思IGCSE数学大纲，从因式分解到图像问题都非常重要。掌握二次方程对高阶代数和实际问题求解至关重要。

1. The Standard Form | 标准形式

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

二次方程是指可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数，且 a ≠ 0。

The highest power of x is 2, so the graph is a parabola.

x 的最高次数为 2，因此图像是一条抛物线。
If a > 0, the parabola opens upwards; if a < 0, it opens downwards.

若 a > 0，抛物线开口向上；若 a < 0，抛物线开口向下。

2. Solving by Factorisation | 因式分解法

Factorisation is the quickest method when the quadratic can be written as a product of two linear factors.

当二次式可以写成两个一次因式的乘积时，因式分解是最快速的方法。

For example, solve x² − 5x + 6 = 0. Look for two numbers that multiply to 6 and add to −5. These are −2 and −3.

例如，解 x² − 5x + 6 = 0。寻找两个数，它们相乘得 6，相加得 −5。这两个数是 −2 和 −3。

(x − 2)(x − 3) = 0

Then set each bracket equal to zero: x = 2 or x = 3.

然后令每个括号等于零：x = 2 或 x = 3。

3. Special Factorisation Patterns | 特殊因式分解模式

Remember these common patterns:

记住这些常见模式：

Difference of two squares: x² − a² = (x − a)(x + a).

平方差：x² − a² = (x − a)(x + a)。
Perfect square: x² ± 2ax + a² = (x ± a)².

完全平方：x² ± 2ax + a² = (x ± a)²。

For instance, x² − 9 = 0 gives (x − 3)(x + 3) = 0, so x = ±3.

例如，x² − 9 = 0 可得 (x − 3)(x + 3) = 0，所以 x = ±3。

4. Completing the Square | 配方法

Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This is useful for finding turning points and solving equations.

配方法将 ax² + bx + c 改写为 a(x + p)² + q 的形式。这在求顶点坐标和解方程时非常有用。

For x² + 6x − 7 = 0, take half of 6, square it to get 9, then write:

对于 x² + 6x − 7 = 0，取 6 的一半，平方得 9，然后写成：

x² + 6x + 9 − 9 − 7 = (x + 3)² − 16 = 0

So (x + 3)² = 16, giving x + 3 = ±4, hence x = 1 or x = −7.

于是 (x + 3)² = 16，得到 x + 3 = ±4，故 x = 1 或 x = −7。

5. The Quadratic Formula | 二次公式

For any quadratic ax² + bx + c = 0, the solutions are given by the quadratic formula:

对于任意二次方程 ax² + bx + c = 0，解由二次公式给出：

x = (−b ± √(b² − 4ac)) / (2a)

This formula works for all quadratics, including those that do not factorise easily.

这个公式适用于所有二次方程，包括那些不易因式分解的方程。

For 2x² + 3x − 2 = 0, a = 2, b = 3, c = −2. Substitute:

对于 2x² + 3x − 2 = 0，a = 2，b = 3，c = −2。代入：

x = (−3 ± √(9 + 16)) / 4 = (−3 ± 5) / 4

Therefore x = ½ or x = −2.

因此 x = ½ 或 x = −2。

6. The Discriminant | 判别式

The expression b² − 4ac is called the discriminant. It tells us the number of real roots.

表达式 b² − 4ac 称为判别式。它告诉我们实数根的个数。

Discriminant Δ = b² − 4ac	Number of real roots	实数根个数
Δ > 0	Two distinct roots	两个不等的实数根
Δ = 0	One repeated root	一个重根
Δ < 0	No real roots	没有实数根

If Δ is a perfect square, the quadratic factors over integers.

如果 Δ 是完全平方数，则二次式可以在整数范围内因式分解。

7. Solving Quadratics by Graphical Method | 图像法解二次方程

The roots of ax² + bx + c = 0 are the x-coordinates where the curve y = ax² + bx + c crosses the x-axis.

ax² + bx + c = 0 的根就是曲线 y = ax² + bx + c 与 x 轴交点的横坐标。

To solve graphically, plot the parabola and read the x-intercepts. This method is useful when exact solutions are not required.

要通过图像求解，先绘制抛物线，然后读出 x 轴交点。当不需要精确解时，这种方法很有用。

Remember that the vertex lies on the axis of symmetry x = −b/(2a).

记住顶点位于对称轴 x = −b/(2a) 上。

8. Word Problems | 应用题

Many IGCSE problems involve forming a quadratic equation from a real-life situation. For example, the area of a rectangle is 40 cm², and its length is 3 cm longer than its width.

许多IGCSE题目需要根据实际情境建立二次方程。例如，一个矩形的面积是 40 cm²，长比宽多 3 cm。

Let width = x, then length = x + 3. The area is x(x + 3) = 40, so x² + 3x − 40 = 0.

设宽为 x，则长为 x + 3。面积为 x(x + 3) = 40，所以 x² + 3x − 40 = 0。

Factorise: (x + 8)(x − 5) = 0, so x = 5 (since width cannot be negative).

因式分解：(x + 8)(x − 5) = 0，所以 x = 5（因为宽不能为负）。

9. Solving Quadratic Inequalities | 解二次不等式

Once roots are found, quadratic inequalities can be solved by considering the sign of the parabola.

求出根之后，可以通过抛物线的符号来解二次不等式。

For x² − x − 6 < 0, factorise to (x − 3)(x + 2) < 0. The critical points are x = −2 and x = 3.

对于 x² − x − 6 < 0，因式分解为 (x − 3)(x + 2) < 0。临界点为 x = −2 和 x = 3。

Since the parabola opens upwards, the inequality is negative between the roots, so −2 < x < 3.

因为抛物线开口向上，不等式在两根之间取负值，所以 −2 < x < 3。

10. Sum and Product of Roots | 根的和与积

For a quadratic ax² + bx + c = 0, if the roots are α and β, then:

对于二次方程 ax² + bx + c = 0，若根为 α 和 β，则：

α + β = −b/a, αβ = c/a

This is useful when checking answers or constructing equations with given roots.

这在检查答案或根据已知根构造方程时非常有用。

For example, roots 2 and −5 give sum −3 and product −10, so the equation is x² + 3x − 10 = 0.

例如，根 2 和 −5 的和为 −3，积为 −10，所以方程为 x² + 3x − 10 = 0。

11. Common Mistakes to Avoid | 常见错误提醒

Do not forget to make the equation equal to zero before factorising or using the formula.

在因式分解或使用公式前，不要忘记将方程化为等于零的形式。
When taking the square root of both sides, remember to include both positive and negative roots.

两边开平方时，记得同时取正根和负根。
In the quadratic formula, be careful with signs when substituting negative coefficients.

在二次公式中，代入负系数时要特别注意符号。
Always check your solutions by substituting them back into the original equation.

始终将解代回原方程进行检查。

12. Exam Tips | 考试技巧

In the Edexcel IGCSE exam, show all working clearly. If a question says ‘give your answer correct to 2 decimal places’, use the quadratic formula or completing the square, not factorisation.

在爱德思IGCSE考试中，要清晰展示全部步骤。如果题目要求”将答案精确到小数点后两位”，应使用二次公式或配方法，而不是因式分解。

For calculator papers, you can check your factorised answer by expanding it mentally. Always present the roots in the required form, whether exact or as decimals.

在允许使用计算器的试卷中，可以用心算展开因式来检验答案。无论要求精确值还是小数，务必按要求形式写出根。

Practising all four methods—factorisation, completing the square, formula and graphing—will give you flexibility and speed.

练习四种方法——因式分解、配方法、公式法和图像法——会让你更加灵活和快速。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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Solving Quadratic Equations | 解二次方程