Algebra 10 | 代数10

📚 Algebra 10 | 代数10

Welcome to the tenth article in our Edexcel IGCSE Mathematics Algebra revision series. This unit focuses on quadratic equations and their graphs – a central topic that bridges algebraic manipulation and graphical interpretation. You will learn to solve quadratics by factorising, completing the square, and using the quadratic formula, as well as how to sketch and interpret parabolas. Mastering these skills is essential for both the Foundation and Higher tier papers.

欢迎来到 Edexcel IGCSE 数学代数复习系列的第十篇文章。本单元聚焦于二次方程及其图像——这是一个连接代数运算与图像解读的核心主题。你将学习如何用因式分解、配方法和二次公式解二次方程，以及如何绘制和解读抛物线。掌握这些技能对基础卷和进阶卷都至关重要。

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

A quadratic equation is a polynomial equation of degree two. Its general form is ax² + bx + c = 0 graph of the corresponding function y = ax² + bx + c is a smooth curve called a parabola.

二次方程是一个二次多项式方程。其一般形式为 ax² + bx + c = 0，其中 a、b、c 是常数且 a ≠ 0。变量 x 的最高次数是 2，这意味着对应函数 y = ax² + bx + c 的图像是一条光滑曲线，称为抛物线。

In the Edexcel IGCSE specification, you will work with quadratics that have real solutions, although some equations may have no real roots. Recognising the standard form immediately is the first step to choosing a suitable solution method.

在 Edexcel IGCSE 考试大纲中，你将处理具有实数解的二次方程，尽管有些方程可能没有实根。立即识别出标准形式是选择合适解法的第一步。

2. Standard Form and Coefficients | 标准形式与系数

coefficient of x², b is the coefficient of x, and c is the constant term. For example, 3x² – 5x + 2 = 0 has a = 3, b = –5 and c = 2.

在解题之前，务必将方程整理成标准形式 ax² + bx + c = 0。其中，a 是 x² 的系数，b 是 x 的系数，c 是常数项。例如，3x² – 5x + 2 = 0 中 a = 3，b = –5，c = 2。

If the equation is not set to zero, move all terms to one side. An equation like 2x² = 8x – 6 should be rewritten as 2x² – proceeding. This discipline prevents sign errors when factorising or applying the formula.

若方程未设为零，请将所有项移到一边。像 2x² = 8x – 6 这样的方程应先改写为 2x² – 8x + 6 = 0 再继续求解。这一习惯能防止因式分解或代入公式时出现符号错误。

3. Solving by Factorising | 因式分解法解二次方程

When a quadratic can be expressed as a product of two linear brackets, we can use the null factor law. If (px + q)(rx + s) = 0 = 0 or rx + s = 0. This method is quick and reliable for equations that factorise neatly.

当二次式可表示为两个一次括号的乘积时，我们可以使用零因子法则。如果 (px + q)(rx + s) = 0，那么 px + q = 0 或 rx + s = 0。对于能整齐分解因式的方程，这种方法既快捷又可靠。

Example: Solve x² – 7x + 12 = 0. Look for two numbers that multiply to 12 and add to –7. They are –3 and –4. So (x – 3)(x – x = 4. Always check by expanding the brackets to ensure the original equation is obtained.

例如：解 x² – 7x + 12 = 0。寻找两个数，乘积为 12，和为 –7。这两个数是 –3 和 –4。因此 (x – 3)(x – 4) = 0，得出 x = 3 或 x = 4。务必重新展开括号进行验证，以确保得到原方程。

4. The Difference of Two Squares | 平方差公式

A special case of factorising is the difference of two squares: a² – b² = (a – b)(a + b). In quadratic equations like x² – 9 = 0, we identify a = x and b = 3, so it factorises to (x – 3)(x + 3) = 0, giving solutions x = 3 and x = –3. This appears frequently in IGCSE problems.

因式分解的一种特殊情况是平方差公式：a² – b² = (a – b)(a + b)。对于像 x² – 9 = 0 这样的二次方程，我们确定 a = x，b = 3，因此它可分解为 (x – 3)(x + 3) = 0，得出解 x = 3 和 x = –3。这在 IGCSE 题目中经常出现。

Recognising this pattern can save time. Expressions such as 4x² – 25 are also differences of squares because 4x² = (2x)² and 25 = 5². Factorising gives (2x – 5)(2x + 5). Applied to an equation, you can solve directly.

识别这一模式能节省时间。像 4x² – 25 这样的式子也是平方差，因为 4x² = (2x)²，25 = 5²。分解后得到 (2x – 5)(2x + 5)。如果应用于方程，便可以直接求解。

5. Solving by Completing the Square | 配方法解二次方程

When factorising is not straightforward, completing the square transforms the quadratic into the form (x + p)² = q. Start with x² + bx, add (b/2)² to both sides to create a perfect square trinomial. This method also helps derive the quadratic formula and find the vertex of a parabola.

当因式分解并不简单时，配方法可将二次方程变为 (x + p)² = q 的形式。从 x² + bx 开始，两边同时加上 (b/2)² 来构造完全平方三项式。这种方法也有助于推导二次公式和求抛物线的顶点。

Example: Solve x² + 6x + 2 = 0. Move the constant: x² + 6x = – ² = 9 to both sides: x² + 6x + 9 = 7, so (x + 3)² = 7. Taking square roots gives x + 3 = ±√7, hence x = –3 ± √7. Answers are often left in surd form unless a decimal approximation is requested.

例如：解 x² + 6x + 2 = 0。将常数项移项：x² + 6x = –2。两边加上 (6/2)² = 9：x² + 6x + 9 = 7，因此 (x + 3)² = 7。开平方得 x + 3 = ±√7，故 x = –3 ± √7。答案通常保留为根式形式，除非题目要求小数近似值。

6. The Quadratic Formula | 二次公式

rived from completing the square on the general equation ax² + bx + c = 0. The formula is:

x = [ –b ± √b² – 4ac ] / (2a)

This formula works for any quadratic equation and is especially useful when coefficients are large or factorisation fails. Always substitute the values of a, b and c carefully, paying attention to negative signs.

二次公式由对一般方程 ax² + bx + c = 0 进行配方法推导得出。公式为：

x = [ –b ± √b² – 4ac ] / (2a)

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该公式适用于任何二次方程，在系数较大或因式分解失败时尤为有用。代入 a、b、c 的值时务必仔细，注意负号。

7. Discriminant and Number of Roots | 判别式与根的数量

The expression under the square root, D = b² – 4ac, is called the discriminant. It determines the nature of the roots without solving the equation:

根号下的表达式 D = b² – 4ac 称为判别式。它无需解方程即可判断根的性质：

• D > 0: two distinct real roots. | 两个不等实根。
• D = 0: exactly one real root (a repeated root). | 恰有一个实根（重根）。
• D < 0: no real roots (roots are complex). | 无实根（复数根）。

In IGCSE, you may be asked to find the value of k for which a quadratic has equal roots. Set the discriminant to zero and solve for k. This concept links algebra to graph sketching because it tells you whether the parabola touches, crosses or misses the x-axis.

在 IGCSE 考试中，你可能会被要求求出使二次方程有等根的 k 值。令判别式等于零，然后解出 k 即可。这一概念将代数与画图联系起来，因为它告诉你抛物线与 x 轴是相切、相交还是不相交。

8. Graphing Quadratic Functions | 二次函数图像

The graph of y = ax² + bx + c is a parabola. If a > 0, the parabola opens upwards (a 'U' shape); if a < 0, it opens downwards (an '∩' e the y-intercept (where x = 0), the roots (where y = 0), and the turning point or vertex.

y = ax² + bx + c 的图像是一条抛物线。若 a > 0，抛物线开口向上（呈 'U' 形）；若 a < 0，开口向下（呈 '∩' 形）。关键特征包括 y 轴截距（x = 0 处）、根（y = 0 处）以及转折点或顶点。

The vertex can be found by completing the square or by using the formula x = – the curve accurately. Edexcel IGCSE often asks for a sketch showing these points labelled.

顶点可通过配方法或使用公式 x = –b/(2a) 求得。一旦有了顶点和根，你就可以准确地描绘曲线。Edexcel IGCSE 经常要求画出标注这些点的草图。

9. Finding Roots from Graphs | 从图像求根

read off the solutions or use the graph to solve a related equation like x² + bx + c = k.

方程 ax² + bx + c = 0 的根是图像与 x 轴交点的横坐标。有时 IGCSE 题目会提供图像，要求你直接读出解，或者利用图像求解相关方程，如 x² + bx + c = k。

intersection points. This graphical method reinforces the connection between algebraic and visual representations.

要利用 y = x² + bx + c 的图像解 x² + bx + c = k，可画出水平线 y = k，并找出交点的横坐标。这种图像法加强了代数表述与视觉呈现之间的联系。

10. Sketching Parabolas | 绘制抛物线草图

A sketch does not need to be plotted point-by-point, but it must show the correct shape, the y-intercept, and any roots or turning points you have calculated. Always label these coordinates. If the parabola does not cross the x-axis (discriminant < 0), indicate that it lies entirely above or below the axis.

草图无需逐点描画，但必须显示正确的形状、y 轴截距以及所有计算出的根或转折点。务必标注这些坐标。如果抛物线不与 x 轴相交（判别式 < 0），则需表明它完全位于 x 轴上方或下方。

Example: Sketch y = x² – 4x + 3. Roots are x = 1 and 3. y-intercept is (0, 3). Vertex: x = –(–4)/(2×1) = 2, y = 2² – 8 + 3 = –1, so vertex is (2, –1). The parabola is U-shaped and passes through these points.

例如：绘制 y = x² – 4x + 3 的草图。根为 x = 1 和 3。y 截距为 (0, 3)。顶点：x = –(–4)/(2×1) = 2，y = 2² – 8 + 3 = –1，因此顶点为 (2, –1)。抛物线为 U 形并经过这些点。

11. Applications and Word Problems | 实际应用与文字题

Quadratic equations often appear in problems involving area, projectile motion, and number puzzles. Translate the written scenario into an algebraic equation, then solve using an appropriate method. Always check that your answer makes sense in the context – negative lengths or times are usually rejected.

二次方程常出现在涉及面积、抛体运动和数字谜题的问题中。将文字描述的场景转化为代数方程，然后选择合适的方法求解。务必要检查答案在情境中是否合理——长度或时间为负通常需舍去。

For example: 'The product of two consecutive positive integers is 56. Find the integers.' Let the first integer be n, then the next is n+1. Equation: n(n+1) = 56 → n² + n – 56 = 0. Factorising gives (n – 7)(n + 8) = 0, so n = 7 (since positive). The integers are 7 and 8.

例如：“两个连续正整数的乘积为 56，求这两个数。”设第一个整数为 n，则下一个为 n+1。方程：n(n+1) = 56 → n² + n – 56 = 0。因式分解得 (n – 7)(n + 8) = 0，由于为正数，取 n = 7。这两个数是 7 和 8。

12. Common Mistakes to Avoid | 常见错误及避免方法

. Another frequent error is mishandling the negative sign when using the quadratic formula – ically. Also, after finding roots, do not forget to present them clearly, often as x = ...

许多学生因在因式分解前忘记将方程设为零而失分。另一个常见错误是在使用二次公式时处理负号不当——务必用括号括起 -b 并系统地代入数值。此外，求出根之后，别忘了清晰地写出答案，通常为 x = ...

When sketching, a sketch that shows the wrong curvature (upward instead of downward) will be penalised. Always check the sign of a first. Lastly, in word problems, read the question again to ensure your solution fits all conditions, not just the equation.

画草图时，若抛物线凹向画反（应向上却画成向下）会被扣分。务必先检查 a 的符号。最后，在文字应用题中，再次审题以确保你的解满足所有条件，而非仅仅满足所列方程。

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