📚 IGCSE Maths: Solving Quadratic Equations | IGCSE 数学：解二次方程
Quadratic equations are a central topic in IGCSE Mathematics, appearing in algebra, graphs, and real-world modelling. A quadratic equation involves an x² term and can usually be solved by factorising, using the quadratic formula, completing the square, or reading roots from a graph. Mastering this skill is essential for both Core and Extended tier candidates.
二次方程是 IGCSE 数学的核心内容，出现在代数、图像和实际建模中。二次方程含有 x² 项，通常可以用因式分解法、求根公式、配方法或通过图像读取根来求解。掌握这一技能对核心层和扩展层考生都至关重要。
1. Recognising a Quadratic Equation | 认识二次方程
ations always contain an x² term, which may be the only variable term, as in 3x² − 12 = 0.
二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b 和 c 是常数，且 a ≠ 0。如果 a = 0，方程就变成一次方程，而不是二次方程。二次方程总是含有 x² 项，有时它可能是唯一的变量项，例如 3x² − 12 = 0。
tic. Identifying the type of equation is the first step before choosing a solution method.
二次方程中 x 的最高次数是 2。像 x³ + 2x − 1 = 0 这样的方程是三次方程，像 1/x + x = 5 这样的方程是有理方程，都不是二次方程。识别方程类型是选择解题方法前的第一步。
2. The Standard Form ax² + bx + c = 0 | 标准形式 ax² + bx + c = 0
side is set to zero. For example, 5x − 3 = 2x² becomes 2x² − 5x + 3 = 0 after rearranging.
在求解二次方程之前，通常最好将它整理成标准形式：ax² + bx + c = 0。将所有项移到一边，右边设为零。例如，5x − 3 = 2x² 整理后变成 2x² − 5x + 3 = 0。
In standard form, the coefficient a is the number in front of x², b is the number in front of x, and c is the constant term. These values are needed for the quadratic formula and the discriminant. If an equation has brackets or fractions, expand or clear them first.
在标准形式中，系数 a 是 x² 前面的数，b 是 x 前面的数，c 是常数项。求根公式和判别式需要用到这些值。如果方程含有括号或分数，先展开或去分母。
ax² + bx + c = 0, a ≠ 0
3. Solving by Factorising: The Zero Product Property | 因式分解法：零乘积性质
erty: if the product of two factors equals zero, then at least one factor must equal zero.
当二次表达式可以因式分解成两个一次括号时，因式分解法通常是最快的方法。一旦方程变成 (x + p)(x + q) = 0 的形式，我们就使用零乘积性质：如果两个因式的乘积等于零，那么至少有一个因式必须等于零。
= 0, giving x = −2, and x + 3 = 0, giving x = −3. The two solutions are x = −2 and x = −3.
例如，要解 x² + 5x + 6 = 0，将左边因式分解为 (x + 2)(x + 3)。然后令 x + 2 = 0，得到 x = −2；令 x + 3 = 0，得到 x = −3。两个解是 x = −2 和 x = −3。
Always check that the original equation is equal to zero before applying the zero product property. If the right-hand side is not zero, factorising alone cannot directly give the roots.
在应用零乘积性质之前，一定要确认原方程等于零。如果右边不是零，仅靠因式分解不能直接给出根。
4. Factorising Simple Trinomials | 因式分解简单三项式
A quadratic trinomial of the form x² + bx + c can often be factorised by finding two numbers that multiply to give c and add to give b. For x² + 7x + 10, the numbers 2 and 5 multiply to 10 and add to 7, so the factorised form is (x + 2)(x + 5).
形如 x² + bx + c 的二次三项式通常可以通过寻找两个数来因式分解，这两个数相乘得 c，相加得 b。对于 x² + 7x + 10，数 2 和 5 相乘得 10，相加得 7，因此因式分解形式为 (x + 2)(x + 5)。
When the constant term is negative, the two numbers must have opposite signs. For x² − 2x − 8, the numbers −4 and 2 multiply to −8 and add to −2, so the factorised form is (x − 4)(x + 2).
当常数项为负时，两个数必须异号。对于 x² − 2x − 8，数 −4 和 2 相乘得 −8，相加得 −2，因此因式分解形式为 (x − 4)(x + 2)。
If the coefficient of x² is not 1, such as 2x² + 5x + 3, use trial and error or grouping. The factorised form is (2x + 3)(x + 1), which expands back to the original expression.
如果 x² 的系数不是 1，比如 2x² + 5x + 3，可以使用试错法或分组法。因式分解形式为 (2x + 3)(x + 1)，展开后得到原表达式。
5. Special Cases: Difference of Squares | 特殊情况：平方差
16 can be written as (x + 4)(x − 4), so the solutions to x² − 16 = 0 are x = −4 and x = 4.
一个特殊的因式分解模式是平方差：a² − b² = (a + b)(a − b)。这种形式在 IGCSE 考题中经常出现。例如，x² − 16 可以写成 (x + 4)(x − 4)，所以 x² − 16 = 0 的解是 x = −4 和 x = 4。
= (3x)² and 25 = 5². The factorised form is (3x + 5)(3x − 5), giving x = −5/3 and x = 5/3.
平方差也可能带有系数或多个项。对于 9x² − 25 = 0，识别出 9x² = (3x)² 和 25 = 5²。因式分解形式为 (3x + 5)(3x − 5)，得到 x = −5/3 和 x = 5/3。
Remember that a sum of squares, such as x² + 9, cannot be factorised using real numbers. Only a difference of squares fits this pattern.
请记住，平方和（比如 x² + 9）不能用实数因式分解。只有平方差符合这个模式。
6. The Quadratic Formula | 求根公式
or finding both roots. For the standard form ax² + bx + c = 0, the solutions are given by:
当二次方程不容易因式分解时，求根公式提供了一种可靠的求两个根的方法。对于标准形式 ax² + bx + c = 0，解为：
x = (-b ± √b² − 4ac) / 2a
, substitute a = 2, b = 3, c = −2 into the formula and simplify to get x = 1/2 and x = −2.
符号 ± 表示通常有两个解：一个用加号，一个用减号。对于 2x² + 3x − 2 = 0，将 a = 2、b = 3、c = −2 代入公式并化简，得到 x = 1/2 和 x = −2。
The quadratic formula is especially useful when coefficients are not integers or when the roots are surds. It is given in the IGCSE formula sheet, but you should memorise it to save time.
当系数不是整数或根是无理数时，求根公式特别有用。IGCSE 公式表中给出了该公式，但你仍应记住它以节省时间。
7. The Discriminant: How Many Roots? | 判别式：有多少个根？
It tells us the nature and number of real solutions without actually solving the equation.
判别式是求根公式中平方根号下的部分：Δ = b² − 4ac。它告诉我们在不实际解方程的情况下，实数解的性质和数量。
ne repeated real root. If Δ < 0, the equation 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数根。如果 Δ = 0，方程有一个重复的实数根。如果 Δ < 0，方程没有实数根，这意味着图像不与 x 轴相交。
−4, so there are no real solutions. This is quick to check and is a common exam question.
例如，x² + 4x + 5 = 0 的判别式 Δ = 4² − 4 × 1 × 5 = −4，因此没有实数解。这一点检查起来很快，也是常见的考题。
8. Completing the Square | 配方法
Completing the square rewrites a quadratic expression in the form (x + p)² + q. This is useful for finding the vertex of a parabola and for solving equations that do not factorise neatly. For x² + 6x + 5, halve the coefficient 6 to get 3, square it to get 9, and rewrite:
配方法将二次表达式改写成 (x + p)² + q 的形式。这对于求抛物线的顶点和求解不容易因式分解的方程很有用。对于 x² + 6x + 5，将系数 6 除以 2 得到 3，平方得 9，然后改写：
x² + 6x + 5 = (x + 3)² − 4
0 can then be written as (x + 3)² − 4 = 0. Solving gives x + 3 = ±2, so x = −1 and x = −5.
方程 x² + 6x + 5 = 0 然后可以写成 (x + 3)² − 4 = 0。求解得到 x + 3 = ±2，所以 x = −1 和 x = −5。
When the coefficient of x² is not 1, factorise it out first. Completing the square is also the method used to derive the quadratic formula.
当 x² 的系数不是 1 时，先把它提出来。配方法也是推导求根公式所用的方法。
9. Graphical Interpretation: Parabolas | 图像解释：抛物线
abola. If the graph crosses the x-axis at x = p and x = q, then p and q are the solutions.
y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的根就是这条抛物线的 x 轴截距。如果图像在 x = p 和 x = q 处与 x 轴相交，那么 p 和 q 就是解。
If the parabola only touches the x-axis at one point, the quadratic has one repeated root. If it never touches the x-axis, the discriminant is negative and there are no real roots.
如果抛物线只在一点与 x 轴相切，那么二次方程有一个重复的根。如果它从不与 x 轴相交，则判别式为负，没有实数根。
In IGCSE graph questions, you may need to estimate roots from a given diagram or sketch a parabola to solve an equation. Understanding the link between algebra and graphs improves accuracy.
在 IGCSE 图像题中，你可能需要根据给定的图估算根，或者画抛物线来解方程。理解代数与图像之间的联系可以提高准确性。
10. Word Problems and Applications | 文字题与应用
Quadratic equations often appear in word problems involving area, projectile motion, product of consecutive numbers, or profit. The key is to define a variable, build an equation, simplify to standard form, and then solve using an appropriate method.
二次方程常出现在涉及面积、抛体运动、连续数乘积或利润的文字题中。关键是定义一个变量，建立方程，化简为标准形式，然后用适当的方法求解。
es to x² + 3x − 40 = 0. Factorising gives (x + 8)(x − 5) = 0, so x = 5 is the valid width.
示例：一个矩形的长比宽多 3 cm，面积为 40 cm²。设宽为 x，那么长为 x + 3。方程为 x(x + 3) = 40，化简为 x² + 3x − 40 = 0。因式分解得 (x + 8)(x − 5) = 0，因此 x = 5 是有效的宽。
Always discard solutions that do not make sense in the real-world context, such as negative lengths or times. Clearly state the final answer with correct units.
一定要舍去在现实情境中没有意义的解，比如负的长度或时间。最后要用正确的单位清楚地写出答案。
11. Common Mistakes to Avoid | 常见错误避免
rise as (x + 4)(x − 1) = 0. Dividing both sides by x too early can also cause a lost root.
一个常见错误是在因式分解前忘记将方程设为零。如果你有 x² + 3x = 4，必须先写成 x² + 3x − 4 = 0，然后因式分解为 (x + 4)(x − 1) = 0。过早两边除以 x 也可能导致丢根。
hen substituting. For x² − 5x + 6 = 0, a = 1, b = −5, c = 6, so the formula gives x = (5 ± √25 − 24) / 2.
另一个错误是在求根公式中处理负号不当。仔细写出 a、b 和 c，代入时使用括号。对于 x² − 5x + 6 = 0，a = 1、b = −5、c = 6，因此公式给出 x = (5 ± √25 − 24) / 2。
Finally, always check your solutions by substituting them back into the original equation. This catches sign errors and factorising mistakes quickly.
最后，一定要把解代回原方程进行检验。这能快速发现符号错误和因式分解错误。
12. Exam Tips and Practice Strategy | 考试技巧与练习策略
In the IGCSE exam, show all steps clearly: write the standard form, state the method, show factorising or formula substitution, and list both roots. Method marks are often awarded even if a small arithmetic error occurs later.
在 IGCSE 考试中，要清楚地写出所有步骤：写出标准形式，说明方法，展示因式分解或公式代入，并列出两个根。即使后面出现小的算术错误，方法分通常也能拿到。
Practise a mix of factorising, formula, completing the square, and graph-based questions. Begin with straightforward factorising, then move to questions with non-unit leading coefficients, surds, and word problems.
练习时要混合因式分解、公式、配方法和基于图像的题目。从简单的因式分解开始，然后过渡到首项系数不为 1、含有无理数根以及文字题的问题。
Using the quadratic formula as a check is smart. After factorising, quickly substitute the roots into the original equation. This builds confidence and improves accuracy under time pressure.
用求根公式进行检验是聪明的做法。因式分解后，快速将根代入原方程。这可以建立信心，并在时间压力下提高准确性。
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