Algebra 10: Quadratic Equations & Functions | 代数10：二次方程与函数

📚 Algebra 10: Quadratic Equations & Functions | 代数10：二次方程与函数

This article provides a focused revision guide for Algebra 10 in the Edexcel IGCSE Mathematics syllabus. We will cover quadratic expressions, solving equations, graphing, inequalities, and common pitfalls, with every concept paired in English and Chinese.

本文为 Edexcel IGCSE 数学大纲中 Algebra 10 提供重点复习指南。我们将涵盖二次表达式、解方程、作图、不等式及常见错误，每个概念均以中英文配对呈现。

1. Quadratic Expressions | 二次表达式

A quadratic expression in one variable has the general form ax² + bx + c, where a, b and c are constants and a ≠ 0. The highest power of the variable is 2, which makes the graph a parabola.

一元二次表达式的一般形式为 ax² + bx + c，其中 a、b、c 为常数，且 a ≠ 0。变量的最高次数为 2，因此其图象呈抛物线。

The coefficient a controls the direction and width of the parabola.

系数 a 控制抛物线的开口方向和宽度。
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，开口向下。
c is the y-intercept, the point where the graph meets the y-axis.

c 是纵截距，即图象与 y 轴交点的纵坐标。

2. Factorising Quadratics | 因式分解二次式

Factorising is often the fastest way to solve a quadratic equation when the roots are rational. For x² + bx + c, look for two numbers whose sum is b and product is c.

当根为有理数时，因式分解通常是解二次方程最快的方法。对于 x² + bx + c，需要找到两个数，使其和为 b、积为 c。

x² + 5x + 6 = (x + 2)(x + 3)

For ax² + bx + c where a ≠ 1, use the method of finding factors of ac that add to b, then factor by grouping.

对于 a ≠ 1 的 ax² + bx + c，可先找到 ac 的因子，使其和为 b，再用分组分解法。

Always check by expanding the brackets again.

始终通过重新展开括号来验证。
Difference of two squares: x² – 9 = (x – 3)(x + 3).

平方差公式：x² – 9 = (x – 3)(x + 3)。
Perfect squares: x² + 6x + 9 = (x + 3)².

完全平方：x² + 6x + 9 = (x + 3)²。

3. Solving by Factorising | 用因式分解解方程

If a quadratic equation can be written as (x – p)(x – q) = 0, then either x = p or x = q. This is called the zero product property.

若二次方程可写成 (x – p)(x – q) = 0，则必有 x = p 或 x = q。这称为零乘积性质。

Solve x² – 3x – 10 = 0 ⇒ (x – 5)(x + 2) = 0 ⇒ x = 5 or x = -2

Make sure the equation is set to zero before factorising. Move all terms to one side first.

因式分解前必须确保方程右边为零。先将所有项移到等号一侧。

4. The Quadratic Formula | 二次公式

When factorising is difficult or impossible, use the quadratic formula. For ax² + bx + c = 0:

当因式分解困难或无法进行时，使用二次公式。对于 ax² + bx + c = 0：

x = (-b ± √(b² – 4ac)) / (2a)

Substitute the values of a, b and c carefully. This formula works for every quadratic equation.

小心代入 a、b、c 的值。该公式适用于所有二次方程。

First identify a, b, c before substituting.

代入前先明确 a、b、c。
Simplify the square root completely.

彻底化简根号内的数。
Give answers to an appropriate degree of accuracy if needed.

若需要，答案保留适当精度。

5. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x + h)² + k. This is useful for finding the vertex and solving equations.

配方法将二次式改写为 a(x + h)² + k 的形式。这有助于求顶点和解方程。

x² + 6x + 11 = (x + 3)² + 2

To complete the square for x² + bx: add (b/2)² and subtract it to keep the expression unchanged.

对 x² + bx 配方：加上 (b/2)²，再减去同一值以保持表达式不变。

x² + 8x = (x + 4)² – 16

When a ≠ 1, factor out a from the x² and x terms first.

当 a ≠ 1 时，先从 x² 项和 x 项提出 a。

6. The Discriminant | 判别式

The discriminant is Δ = b² – 4ac. It tells us the number of real roots of a quadratic equation without solving it.

判别式为 Δ = b² – 4ac。它不必解方程即可告诉我们二次方程实数根的个数。

Δ = b² – 4ac	Number of real roots	实数根个数
Δ > 0	Two distinct real roots	两个不等实根
Δ = 0	One repeated real root	一个重根
Δ < 0	No real roots	无实根

If the graph of y = ax² + bx + c does not cross the x-axis, then Δ < 0.

若 y = ax² + bx + c 的图象与 x 轴无交点，则 Δ < 0。

7. Graphs of Quadratic Functions | 二次函数图象

The graph of y = ax² + bx + c is a parabola. Its shape and position depend on a, b, c and the discriminant.

y = ax² + bx + c 的图象是抛物线。其形状和位置取决于 a、b、c 与判别式。

The roots are the x-coordinates where the graph crosses the x-axis.

根是图象与 x 轴交点的横坐标。
The y-intercept is c.

纵截距是 c。
The line of symmetry passes through the vertex.

对称轴经过顶点。

To sketch a quadratic graph, find the roots, y-intercept and vertex, then draw a smooth curve.

画二次函数草图时，先求出根、纵截距和顶点，再画平滑曲线。

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

The vertex is the turning point of the parabola. Its x-coordinate can be found by x = -b / (2a), or directly from completed square form a(x – h)² + k, where (h, k) is the vertex.

顶点是抛物线的转向点。其横坐标可用 x = -b / (2a) 求得，或直接从配方法形式 a(x – h)² + k 中获得，其中 (h, k) 为顶点。

y = 2(x – 3)² + 5 ⇒ vertex = (3, 5), axis: x = 3

If a > 0, the vertex is a minimum point; if a < 0, it is a maximum point.

若 a > 0，顶点为最小值点；若 a < 0，顶点为最大值点。

The axis of symmetry is the vertical line x = h, which divides the parabola into two mirror images.

对称轴是竖直线 x = h，它将抛物线分为左右对称的两部分。

9. Quadratic Inequalities | 二次不等式

To solve a quadratic inequality, first rearrange it to one side and factorise. Then use a sign diagram or sketch the graph to determine the solution set.

解二次不等式时，先将所有项移到一侧并因式分解，然后利用符号图或函数图象来确定解集。

Solve x² – x – 6 < 0 ⇒ (x - 3)(x + 2) < 0 ⇒ -2 < x < 3

For a quadratic with a > 0, the expression is negative between the roots and positive outside the roots.

对于 a > 0 的二次式，其值在两根之间为负，在两根之外为正。

Pay attention to strict (<) and non-strict (≤) inequality signs.

注意严格不等式 (<) 与非严格不等式 (≤) 的区别。
If the inequality is ≤ 0, include the roots as closed circles.

若不等式为 ≤ 0，则根处用实心点表示包含。

10. Simultaneous Equations: Linear and Quadratic | 联立方程：线性与二次

Solve a linear equation and a quadratic equation together by substitution. Rearrange the linear equation to make one variable the subject, then substitute into the quadratic.

用代入法联立解线性方程与二次方程。先将线性方程改写为用一个变量表示另一个变量，再代入二次方程。

y = x + 1 and y = x² – 3x + 2 ⇒ x + 1 = x² – 3x + 2 ⇒ x² – 4x + 1 = 0

The resulting quadratic may have two, one, or zero solutions, corresponding to two intersection points, one tangent point, or no intersection.

所得二次方程可能有两个、一个或零个解，分别对应两个交点、一个切点或无交点。

11. Word Problems with Quadratics | 二次方程应用题

Many real-world problems involve areas, projectiles, and revenue. Set up a quadratic equation from the given information, then solve it and check that the answer makes sense in context.

许多实际问题涉及面积、抛体运动和收入。根据已知信息建立二次方程，然后求解并检查答案是否符合实际情境。

For example, a rectangle has length (x + 4) cm and width (x – 1) cm, with area 30 cm². Then (x + 4)(x – 1) = 30.

例如，一个长方形长为 (x + 4) cm，宽为 (x – 1) cm，面积为 30 cm²。则 (x + 4)(x – 1) = 30。

x² + 3x – 34 = 0

Reject negative solutions when they represent lengths or distances.

当负数解表示长度或距离时，应舍去。

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

Here are the most frequent errors students make and how to avoid them in the Edexcel IGCSE exam.

以下是学生最常犯的错误及在 Edexcel IGCSE 考试中如何避免它们。

Forgetting to set the equation to zero before factorising.

因式分解前忘记将方程化为零。
Losing the negative sign when substituting into the quadratic formula.

代入二次公式时遗漏负号。
Mistaking the sign of the vertex in completed square form: y = a(x – h)² + k has vertex (h, k), not (-h, k).

混淆配方法中顶点的符号：y = a(x – h)² + k 的顶点是 (h, k)，而不是 (-h, k)。
When solving inequalities, forgetting to reverse the inequality sign after multiplying by a negative.

解不等式时，乘以负数后忘记改变不等号方向。

Always show your working clearly, especially when using the quadratic formula. In Edexcel IGCSE, method marks are awarded even if the final answer is wrong.

务必清晰写出解题步骤，特别是使用二次公式时。在 Edexcel IGCSE 中，即使最终答案错误，也会给过程分。

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Algebra 10: Quadratic Equations & Functions | 代数10：二次方程与函数