📚 Using the Quadratic Formula Correctly: Substituting a, b and c | 正确使用求根公式：代入 a、b 和 c 的值
In many Edexcel A-Level Mathematics questions, especially on quadratics, the mark scheme rewards a clear method step: attempt to use the correct formula with the values of a, b and c substituted. This article focuses on how to identify a, b and c, how to substitute them without sign errors, and how to interpret the discriminant.
在许多爱德思 A-Level 数学考题中，尤其是二次方程题目，评分标准会奖励一个清晰的方法步骤：尝试使用正确公式并代入 a、b 和 c 的值。本文重点讲解如何识别 a、b 和 c，如何在代入时避免符号错误，以及如何理解判别式。
1. Why the Quadratic Formula Matters | 为什么求根公式重要
The quadratic formula solves any equation of the form ax² + bx + c = 0. It is especially useful when factorisation is difficult or impossible. Examiners often award method marks for writing the formula and substituting the correct values.
求根公式可以求解任何形如 ax² + bx + c = 0 的方程。当因式分解较难或无法进行时，它特别有用。考官经常对写出公式并代入正确数值给予方法分。
x = (−b ± √(b² − 4ac)) / (2a)
Here a, b and c are the coefficients of x², x and the constant term. You must use the exact signs from the equation. This formula is not just a mechanical tool: it also links directly to the discriminant and the shape of the quadratic graph.
这里的 a、b 和 c 分别是 x²、x 的系数和常数项。你必须使用方程中原来的符号。这个公式不仅仅是一个机械工具：它还直接联系判别式和二次函数图像的性质。
2. The Standard Form ax² + bx + c = 0 | 一般式 ax² + bx + c = 0
Before using the formula, the equation must be rearranged into the standard form ax² + bx + c = 0. All terms must be on one side, with zero on the other side. If the equation is not equal to zero, the values of a, b and c will be wrong.
在使用公式之前，必须先将方程整理成一般式 ax² + bx + c = 0。所有项都要移到一边，另一边等于 0。如果方程不等于 0，a、b 和 c 的值就会出错。
For example, 3x² − 5x = 2 becomes 3x² − 5x − 2 = 0, so a = 3, b = −5 and c = −2. Do not take c as 2. This is one of the most common errors in exam solutions.
例如，3x² − 5x = 2 要写成 3x² − 5x − 2 = 0，所以 a = 3，b = −5，c = −2。不要把 c 当成 2。这是考试解答中最常见的错误之一。
3. Identifying a, b and c from a Given Equation | 从给定方程识别 a、b 和 c
If the equation is already in standard form, simply read off the coefficients. For 2x² + 7x − 15 = 0, a = 2, b = 7 and c = −15. Include the minus sign with c.
如果方程已经是一般式，直接读出系数即可。对于 2x² + 7x − 15 = 0，a = 2，b = 7，c = −15。c 要包含负号。
If a term is missing, its coefficient is 0. For x² − 4 = 0, a = 1, b = 0 and c = −4. For 3x² + 5x = 0, c = 0. The coefficient a is never zero in a quadratic equation, because otherwise the equation would be linear.
如果缺少某一项，其系数为 0。对于 x² − 4 = 0，a = 1，b = 0，c = −4。对于 3x² + 5x = 0，c = 0。在二次方程中，系数 a 永远不为 0，因为否则方程就变成一次方程。
Always write down a, b and c on a separate line before substituting. This helps you avoid copying the wrong sign and makes your working clear to the examiner.
在代入之前，始终单独一行写出 a、b 和 c 的值。这有助于避免抄错符号，也能让考官看清你的解题过程。
4. Substituting Values Step by Step | 逐步代入数值
Write the formula first, then replace a, b and c with the values you identified. Use brackets around negative numbers to prevent sign errors. This is especially important when b is negative, because −b will become positive.
先写出公式，然后用你识别出的 a、b 和 c 的值进行替换。负数要用括号括起来，防止符号错误。当 b 为负数时尤其重要，因为 −b 会变成正数。
Solve 2x² + 5x − 3 = 0: x = (−(5) ± √((5)² − 4 × 2 × (−3))) / (2 × 2)
Simplify the numerator and denominator separately. This gives x = (−5 ± √(25 + 24)) / 4 = (−5 ± √49) / 4. Then split the ± into two separate calculations only after the square root has been simplified.
分别化简分子和分母。得到 x = (−5 ± √(25 + 24)) / 4 = (−5 ± √49) / 4。只有在平方根化简之后，再把 ± 拆成两个独立的计算。
5. The Discriminant: b² − 4ac | 判别式 b² − 4ac
The expression under the square root, b² − 4ac, is called the discriminant. It tells you how many real roots the equation has before you finish the calculation. This can save time and helps you check whether your final answers are sensible.
平方根下的表达式 b² − 4ac 称为判别式。它可以在你完成计算之前告诉你方程有多少个实根。这可以节省时间，也有助于检查最终答案是否合理。
- If b² − 4ac > 0: two distinct real roots. | 如果 b² − 4ac > 0：两个不同的实根。
- If b² − 4ac = 0: one repeated real root. | 如果 b² − 4ac = 0：一个重根。
- If b² − 4ac < 0: no real roots, but two complex roots if you study further mathematics. | 如果 b² − 4ac < 0：没有实根，但如果学习进阶数学，则有两个复数根。
In A-Level Mathematics, you are mainly expected to comment on the number of real roots. The discriminant is also useful when a question asks you to find the range of values of k for which a quadratic has real roots.
在 A-Level 数学中，主要要求你判断实根的个数。当题目要求你找出使二次方程有实根的 k 的取值范围时，判别式也非常有用。
6. Worked Example: Two Real Roots | 例题：两个实根
Solve x² − 5x + 6 = 0. Here a = 1, b = −5 and c = 6. Substituting gives x = (5 ± √(25 − 24)) / 2 = (5 ± 1) / 2.
求解 x² − 5x + 6 = 0。这里 a = 1，b = −5，c = 6。代入得到 x = (5 ± √(25 − 24)) / 2 = (5 ± 1) / 2。
So x = 3 or x = 2. The discriminant is 1, which is positive, confirming two real roots. You can also check by factorising: (x − 2)(x − 3) = 0 gives the same answers.
所以 x = 3 或 x = 2。判别式为 1，是正数，确认有两个实根。你也可以通过因式分解来验证：(x − 2)(x − 3) = 0 得到相同的答案。
7. Worked Example: Repeated Root | 例题：重根
Solve 4x² − 12x + 9 = 0. Here a = 4, b = −12 and c = 9. The discriminant is (−12)² − 4 × 4 × 9 = 144 − 144 = 0.
求解 4x² − 12x + 9 = 0。这里 a = 4，b = −12，c = 9。判别式为 (−12)² − 4 × 4 × 9 = 144 − 144 = 0。
Therefore x = (12 ± √0) / 8 = 12 / 8 = 3/2. There is only one solution, called a repeated root. The graph of y = 4x² − 12x + 9 touches the x-axis at x = 3/2 but does not cross it.
因此 x = (12 ± √0) / 8 = 12 / 8 = 3/2。只有一个解，称为重根。y = 4x² − 12x + 9 的图像在 x = 3/2 处与 x 轴相切但不穿过。
8. Worked Example: Negative and Fractional Coefficients | 例题：负数与分数系数
Solve 2x² − 3x − 5 = 0. Here a = 2, b = −3 and c = −5. Substitute: x = (3 ± √(9 + 40)) / 4 = (3 ± √49) / 4.
求解 2x² − 3x − 5 = 0。这里 a = 2，b = −3，c = −5。代入：x = (3 ± √(9 + 40)) / 4 = (3 ± √49) / 4。
This gives x = (3 + 7) / 4 = 10/4 = 5/2 or x = (3 − 7) / 4 = −4/4 = −1. Notice that c = −5 made the term under the square root become 9 + 40, not 9 − 40.
得到 x = (3 + 7) / 4 = 10/4 = 5/2 或 x = (3 − 7) / 4 = −4/4 = −1。注意 c = −5 使得平方根下的项变成 9 + 40，而不是 9 − 40。
If the equation had fractional coefficients, such as x² − (1/2)x − 1/2 = 0, you could multiply through by 2 to get 2x² − x − 1 = 0, but the formula also works directly. For a direct substitution, a = 1, b = −1/2 and c = −1/2.
如果方程有分数系数，例如 x² − (1/2)x − 1/2 = 0，你可以两边乘以 2 得到 2x² − x − 1 = 0，但公式也可以直接使用。直接代入时，a = 1，b = −1/2，c = −1/2。
9. Common Mistakes to Avoid | 常见错误及避免方法
A very common error is forgetting to rearrange the equation to equal 0 before identifying c. For example, x² + 3x = 10 gives c = −
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