Mastering Quadratic Equations for IGCSE: Factorising, Completing the Square, the Quadratic Formula and Graphs | 掌握 IGCSE 二次方程：因式分解、配方法、求根公式与图像

📚 Mastering Quadratic Equations for IGCSE: Factorising, Completing the Square, the Quadratic Formula and Graphs | 掌握 IGCSE 二次方程：因式分解、配方法、求根公式与图像

A quadratic equation is one of the most important topics in the IGCSE Mathematics syllabus. It appears in algebra, graphs and problem-solving questions, so a confident command of solving methods and graphical interpretation is essential for a high grade.

二次方程是 IGCSE 数学大纲中最重要的主题之一。它出现在代数、图像和应用题中，因此熟练掌握求解方法和图像解读是获得高分的关键。

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1. Recognising a Quadratic Equation | 识别二次方程

nd c are constants and a ≠ 0. If a = 0, the equation becomes linear rather than quadratic.

二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数且 a ≠ 0。如果 a = 0，方程就变成一次方程而不是二次方程。

The term ax² is called the quadratic term, bx is the linear term and c is the constant term. In IGCSE questions, you may need to rearrange an equation into this standard form before solving it.

ax² 项称为二次项，bx 项称为一次项，c 项称为常数项。在 IGCSE 题目中，你可能需要先把方程整理成这种标准形式，再求解。

it can be rewritten as x² + 3x - 4 = 0. Similarly, 2x(x - 5) = 0 expands to 2x² - 10x = 0.

例如，x² + 3x = 4 是二次方程，因为它可以改写为 x² + 3x - 4 = 0。同样地，2x(x - 5) = 0 展开后得到 2x² - 10x = 0。

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2. Standard Form and Key Terms | 标准形式与关键术语

nnot be zero, but b or c can be zero, giving equations such as x² - 9 = 0 or 2x² + 3x = 0.

标准形式为 ax² + bx + c = 0。系数 a 不能为零，但 b 或 c 可以为零，例如得到 x² - 9 = 0 或 2x² + 3x = 0 这样的方程。

Roots or solutions are the values of x that make the equation true. A quadratic equation can have two real roots, one repeated real root or no real roots, depending on the discriminant.

根或解是使方程成立的 x 值。二次方程可以有两个实根、一个重根，或没有实根，这取决于判别式。

Standard form: ax² + bx + c = 0, a ≠ 0

You should always write the equation in descending powers of x before identifying a, b and c. This is especially important when using the quadratic formula.

在确定 a、b、c 之前，你应该始终按 x 的降幂排列方程。使用求根公式时这一点尤为重要。

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3. Solving by Factorisation | 因式分解法

s simple integer roots. For example, x² + 5x + 6 = 0 can be written as (x + 2)(x + 3) = 0.

当二次方程具有简单的整数根时，因式分解通常是最快的方法。例如，x² + 5x + 6 = 0 可以写成 (x + 2)(x + 3) = 0。

at least one factor must be zero. So set x + 2 = 0 or x + 3 = 0, giving x = -2 or x = -3.

使用零乘积性质：如果两个因式的乘积为零，那么至少有一个因式为零。因此令 x + 2 = 0 或 x + 3 = 0，得到 x = -2 或 x = -3。

(x + 2)(x + 3) = 0 ⇒ x = -2 or x = -3

3 = 0, split the middle term or use trial and error to factorise into (2x + 1)(x + 3) = 0.

对于首项系数不等于 1 的方程，例如 2x² + 7x + 3 = 0，可以拆分中项或用试错法因式分解为 (2x + 1)(x + 3) = 0。

3 = 0 gives x = -3. Always check by expanding the factors back to the original expression.

然后令每个因式等于零：2x + 1 = 0 得到 x = -1/2，x + 3 = 0 得到 x = -3。始终通过将因式展开回原表达式来检查。

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4. Solving by Completing the Square | 配方法

his method is useful when factorising is difficult and is also needed to find the turning point of a graph.

配方法将二次方程转化为 (x + p)² + q = 0 的形式。当因式分解困难时，这种方法很有用，而且求图像顶点时也需要它。

as (x + 3)² - 9. Then the equation becomes (x + 3)² - 7 = 0, so (x + 3)² = 7 and x = -3 ± √7.

对于 x² + 6x + 2 = 0，先将 x² + 6x 写成 (x + 3)² - 9。于是方程变为 (x + 3)² - 7 = 0，因此 (x + 3)² = 7，x = -3 ± √7。

x² + bx = (x + b/2)² - (b/2)²

rst. For example, 2x² + 8x + 3 = 0 becomes x² + 4x + 3/2 = 0 before completing the square.

当 x² 的系数不为 1 时，先将整个方程除以该系数。例如，2x² + 8x + 3 = 0 在配方之前先变为 x² + 4x + 3/2 = 0。

Completing the square also gives exact answers in surd form, which is often required in non-calculator IGCSE papers.

配方法还能给出保留根号形式的精确答案，这在 IGCSE 非计算器试卷中经常要求。

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5. The Quadratic Formula | 求根公式

The quadratic formula solves any quadratic equation by substituting a, b and c directly. It is given in the IGCSE formula sheet, but you must know how to apply it accurately.

求根公式通过直接代入 a、b、c 来解任何二次方程。IGCSE 公式表会给出该公式，但你必须知道如何准确应用。

x = (-b ± √b² - 4ac) ÷ 2a

For 3x² - 5x - 2 = 0, a = 3, b = -5, c = -2. Substitution gives x = (5 ± √25 + 24) ÷ 6 = (5 ± 7) ÷ 6, so x = 2 or x = -1/3.

对于 3x² - 5x - 2 = 0，a = 3，b = -5，c = -2。代入得到 x = (5 ± √25 + 24) ÷ 6 = (5 ± 7) ÷ 6，所以 x = 2 或 x = -1/3。

ve b values; -(-5) = 5, not -5. This is one of the most common sign errors in IGCSE exams.

始终完全化简根号和分数。注意 b 为负数时的符号；-(-5) = 5，而不是 -5。这是 IGCSE 考试中最常见的符号错误之一。

The quadratic formula works even when the roots are irrational or when factorisation is not possible, so it is a reliable final method for any quadratic.

即使根是无理数或无法因式分解，求根公式仍然有效，因此它是解任何二次方程的可靠最终方法。

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

The discriminant is the expression b² - 4ac inside the square root of the quadratic formula. It tells you the nature of the roots without solving the equation.

判别式是求根公式中根号内的表达式 b² - 4ac。它能在不求解方程的情况下判断根的性质。

Δ = b² - 4ac

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