📚 Mastering Quadratic Equations for IGCSE Mathematics | 掌握 IGCSE 数学二次方程
A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in algebra, graphs, problem-solving and even in coordinate geometry. This revision guide explains the standard form, four main solving methods, the discriminant, graph interpretation and exam-style applications.
二次方程是 IGCSE 数学中最重要的主题之一。它出现在代数、图像、应用题甚至坐标几何中。本复习指南将讲解标准形式、四种主要解法、判别式、图像解读以及考试型应用。
1. What Is a Quadratic Equation? | 什么是二次方程
a, b, c are constants, with a ≠ 0. If a were zero, the equation would become linear rather than quadratic.
二次方程是任何可以整理为 ax² + bx + c = 0 形式的方程，其中 x 是未知数，a、b、c 是常数，且 a ≠ 0。如果 a 等于零，方程就会变成一次方程而不是二次方程。
The word quadratic comes from the Latin word quadratus, meaning square, because the highest power of x is 2. This squared term gives the equation its curved graph and usually produces two possible solutions.
quadratic 一词来自拉丁文 quadratus，意为“平方”，因为 x 的最高次数是 2。这个平方项使方程图像成为曲线，并且通常会产生两个可能的解。
2. Standard Form and Coefficients | 标准形式与系数
Before solving, always rearrange the equation into standard form. Collect all terms on one side so that the other side is zero, then write them in descending powers of x.
求解前，一定要先把方程整理成标准形式。把所有项移到一边，使另一边为零，然后按 x 的降幂排列。
ax² + bx + c = 0, a ≠ 0
The coefficient a is attached to x², the coefficient b is attached to x, and c is the constant term. Identifying a, b and c correctly is essential when using the quadratic formula or the discriminant.
系数 a 是 x² 的系数，系数 b 是 x 的系数，c 是常数项。正确识别 a、b 和 c 对使用二次公式或判别式至关重要。
3. Solving by Factorisation | 因式分解法
Factorisation is usually the fastest method when the quadratic has simple integer roots. After writing the equation in standard form, look for two numbers that multiply to give ac and add to give b.
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。将方程写成标准形式后，寻找两个数，使它们相乘得到 ac，相加得到 b。
x² + 5x + 6 = 0 → (x + 2)(x + 3) = 0
= -2 and x = -3. Always check by substituting the values back into the original equation.
表达式因式分解后，令每个括号等于零。两个解为 x = -2 和 x = -3。始终将值代回原方程进行检验。
4. Solving by Completing the Square | 配方法
Completing the square is useful when factorisation is not obvious, and it also helps with finding the vertex of a quadratic graph. The aim is to rewrite the quadratic expression as a squared bracket plus or minus a constant.
当因式分解不明显时，配方法很有用，它也有助于求二次图像顶点。目的是把二次表达式改写为一个平方括号加上或减去一个常数。
x² + 6x + 5 = 0 → (x + 3)² - 4 = 0
x = -5. Completing the square is especially important when the coefficient of x² is not 1.
由此解 (x + 3)² = 4，所以 x + 3 = ±2。得到 x = -1 或 x = -5。当 x² 的系数不是 1 时，配方法尤其重要。
5. The Quadratic Formula | 二次公式
sed easily. It is derived from completing the square on the general form ax² + bx + c = 0.
二次公式适用于所有二次方程，包括那些不容易因式分解的方程。它是由对一般形式 ax² + bx + c = 0 配方推导出来的。
x = (-b ± √b² - 4ac) / 2a
eans there are two possible answers: one using the plus sign and one using the minus sign.
代入 a、b 和 c 的值时要仔细，注意负号。符号 ± 表示有两个可能的答案：一个使用加号，另一个使用减号。
6. The Discriminant | 判别式
The discriminant is the expression under the square root in the quadratic formula. It tells us the nature of the roots without solving the equation fully.
判别式是二次公式中平方根下的表达式。它可以在不完整求解方程的情况下告诉我们根的性质。
Δ = b² - 4ac
ated real root. If Δ < 0, it has no real roots, meaning the graph does not cross the x-axis.
如果 Δ > 0，方程有两个不同的实根。如果 Δ = 0，方程有一个重复的实根。如果 Δ < 0，方程没有实根，意味着图像不与 x 轴相交。
7. Graphical Interpretation | 图像解读
. When a > 0, the parabola opens upward and has a minimum point. When a < 0, it opens downward and has a maximum point.
y = ax² + bx + c 的图像是一条抛物线。当 a > 0 时，抛物线开口向上，有一个最低点。当 a < 0 时，抛物线开口向下，有一个最高点。
real roots of the quadratic equation ax² + bx + c = 0. The y-intercept is always at y = c.
图像与 x 轴的交点对应二次方程 ax² + bx + c = 0 的实根。y 轴截距始终在 y = c 处。
8. Problems Leading to Quadratic Equations | 应用题
Many IGCSE questions ask you to form a quadratic equation from a word problem. Common contexts include the area of a rectangle, the product of two consecutive numbers, or the height of a projectile.
许多 IGCSE 题目要求你从文字题中列出二次方程。常见情境包括矩形的面积、两个连续数的乘积或抛射物的高度。
(x + 3) = 40, which rearranges to x² + 3x - 40 = 0. Solve to find the possible dimensions.
例如，一个矩形的长为 x + 3，宽为 x，面积为 40，则 x(x + 3) = 40，整理为 x² + 3x - 40 = 0。解方程即可求出可能的尺寸。
9. Common Mistakes and Exam Tips | 常见错误与考试技巧
One frequent mistake is forgetting to set the equation to zero before factorising. Another is losing a negative sign when identifying b or c in the quadratic formula.
一个常见错误是在因式分解前忘记把方程设为零。另一个错误是在二次公式中识别 b 或 c 时丢掉负号。
Always show your method clearly, even if you use a calculator. In IGCSE exams, method marks are often awarded for correct substitution into the quadratic formula or correct factorisation.
即使使用计算器，也要清楚地写出步骤。在 IGCSE 考试中，正确代入二次公式或正确因式分解通常可以获得方法分。
10. Worked Example | 例题精讲
Solve the equation 2x² - 3x - 5 = 0 using the quadratic formula.
用二次公式解方程 2x² - 3x - 5 = 0。
d c = -5. Substituting into the formula gives x = (3 ± √((-3)² - 4 × 2 × (-5))) / (2 × 2).
这里 a = 2，b = -3，c = -5。代入公式得到 x = (3 ± √((-3)² - 4 × 2 × (-5))) / (2 × 2)。
x = (3 ± √9 + 40) / 4 = (3 ± √49) / 4 = (3 ± 7) / 4
olutions are x = 10/4 = 2.5 and x = -4/4 = -1. Therefore, the roots are x = 2.5 or x = -1.
两个解为 x = 10/4 = 2.5 和 x = -4/4 = -1。因此，根为 x = 2.5 或 x = -1。
11. Practice Checklist | 练习清单
Make sure you can rearrange any quadratic into standard form, factorise simple quadratics, complete the square, apply the quadratic formula accurately and interpret the discriminant.
确保你能够把任意二次方程整理成标准形式，对简单二次式进行因式分解，配方，准确应用二次公式并解释判别式。
- English: Solve x² - 7x + 12 = 0 by factorisation.
- 中文： 用因式分解法解 x² - 7x + 12 = 0。
- English: Use the quadratic formula to solve 3x² + 2x - 8 = 0.
- 中文： 用二次公式解 3x² + 2x - 8 = 0。
- English: Determine the nature of the roots of x² + 4x + 5 = 0.
- 中文： 判断 x² + 4x + 5 = 0 的根的性质。
12. Summary | 总结
Quadratic equations are solved by factorisation, completing the square or the quadratic formula. The discriminant determines how many real roots exist, and the graph is always a parabola.
二次方程可以通过因式分解、配方或二次公式来求解。判别式决定存在多少个实根，图像始终是一条抛物线。
Master this topic by practising a mix of pure algebra and word problems, and always check your solutions in the original equation.
通过练习纯代数题和文字题的混合题型来掌握这个主题，并且始终将解代入原方程进行检验。
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