📚 How to Substitute Correctly into the Quadratic Formula | 正确代入求根公式的方法
The quadratic formula is one of the most powerful tools in algebra, yet many students lose marks not because they don't know the formula, but because they substitute values incorrectly. This article will guide you through a reliable, step-by-step method to avoid those errors and solve any quadratic equation with confidence.
求根公式是代数中最强大的工具之一，然而许多学生失分并非因为不记得公式，而是因为代入数值时出错。本文将带你通过一套可靠、分步骤的方法，避免这些错误，自信地求解任何一元二次方程。
1. What Is the Quadratic Formula? | 什么是求根公式
The quadratic formula solves equations of the form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. The formula is:
求根公式适用于形如 ax² + bx + c = 0 的方程，其中 a、b、c 为实数，且 a ≠ 0。公式为：
x = (−b ± √b² − 4ac) / (2a)
Every symbol in this formula has a precise role. The key to correct substitution is recognising that a, b, and c are the numerical coefficients, including their signs, taken directly from the standard form of the equation.
公式中的每一个符号都有其精确的作用。正确代入的关键在于认识到 a、b、c 是方程标准形式中带符号的数值系数。
2. Step 1: Write the Equation in Standard Form | 步骤一：化为标准形式
Before substituting anything, make sure your equation is written as ax² + bx + c = 0. Move all terms to one side, leaving zero on the other side. For example, if you have 3x² + 5x = 2, rewrite it as 3x² + 5x − 2 = 0.
在代入任何数值之前，确保方程已写成 ax² + bx + c = 0 的形式。将所有项移到一边，另一边为零。例如，若方程为 3x² + 5x = 2，应改写为 3x² + 5x − 2 = 0。
If the equation is not in standard form, your a, b, and c will be wrong from the start. Always double-check that the order is: quadratic term, linear term, constant term, equal to zero.
如果方程不是标准形式，你的 a、b、c 从一开始就会出错。务必检查顺序是否为：二次项、一次项、常数项，等于零。
3. Step 2: Identify a, b, and c Carefully | 步骤二：仔细辨认 a、b、c
Once the equation is in standard form, identify the three coefficients:
当方程化为标准形式后，辨认三个系数：
a is the coefficient of the x² term.
a 是 x² 项的系数。
b is the coefficient of the x term.
b 是 x 项的系数。
c is the constant term (with no x).
c 是不含 x 的常数项。
For the equation 2x² − 7x + 3 = 0, we have a = 2, b = −7, c = 3. Notice that the minus sign belongs to b; it is not a separate symbol.
对于方程 2x² − 7x + 3 = 0，我们有 a = 2，b = −7，c = 3。注意负号属于 b，它不是单独存在的符号。
4. Step 3: Handle Negative Coefficients | 步骤三：处理负系数
Negative coefficients are the most common source of substitution errors. Write b as a single signed number. For example, in the equation x² + 4x − 5 = 0, b = 4, but c = −5. In the equation x² − 6x + 8 = 0, b = −6.
负系数是代入错误最常见的来源。将 b 写成一个带符号的数。例如，在方程 x² + 4x − 5 = 0 中，b = 4，但 c = −5。在方程 x² − 6x + 8 = 0 中，b = −6。
When substituting a negative b into the formula, remember that −b means the opposite of b. If b = −6, then −b = 6. This is often written as −(−6) = 6.
当将负的 b 代入公式时，记住 −b 表示 b 的相反数。若 b = −6，那么 −b = 6。这通常写作 −(−6) = 6。
Similarly, when c is negative, the term 4ac may become negative. For a = 2, c = −5, we have 4 × 2 × (−5) = −40. Then inside the square root, b² − 4ac means subtracting a negative, which becomes addition.
同样，当 c 为负数时，4ac 可能变为负数。对于 a = 2，c = −5，有 4 × 2 × (−5) = −40。那么在根号内，b² − 4ac 表示减去一个负数，实际上变为加法。
5. Step 4: Substitute into the Formula Exactly | 步骤四：精确代入公式
Write the quadratic formula first, then replace each letter with its numerical value in parentheses. For example, with a = 2, b = −7, c = 3, write:
先写出求根公式，然后用括号替换每个字母为数值。例如，a = 2，b = −7，c = 3，则写出：
x = (−(−7) ± √((−7)² − 4 × 2 × 3)) / (2 × 2)
Keep the parentheses in place until you are ready to simplify. This prevents sign errors and makes your working clear for the examiner. Never skip directly to mental arithmetic.
在化简之前保留括号。这能防止符号错误，也让阅卷老师看清你的步骤。切勿直接跳到心算。
6. Step 5: Simplify the Discriminant First | 步骤五：先化简判别式
The expression under the square root, Δ = b² − 4ac, is called the discriminant. Simplify it as a separate calculation before putting it into the square root. This reduces confusion and makes arithmetic easier.
根号内的表达式 Δ = b² − 4ac 称为判别式。先单独计算它的值，然后再放入根号。这样可以减少混淆，使计算更简单。
For the example above:
对于上面的例子：
Δ = (−7)² − 4 × 2 × 3 = 49 − 24 = 25
Since Δ = 25, the square root is √25 mes much simpler. Always calculate Δ first, especially when numbers are large or negative.
因为 Δ = 25，所以根号下为 √25 = 5。此时公式变得简单多了。尤其是数字较大或为负数时，一定要先计算 Δ。
7. Step 6: Simplify the Numerator | 步骤六：化简分子
After simplifying the discriminant, work on the numerator. For our example, the numerator is −b ± √Δ = 7 ± 5. This gives two values: 12 and 2.
化简判别式后，处理分子。在我们的例子中，分子为 −b ± √Δ = 7 ± 5。这给出两个值：12 和 2。
The denominator is 2a = 4. Therefore the two solutions are x = 12/4 = 3 and x = 2/4 = 1/2. Always simplify fractions completely.
分母为 2a = 4。因此两个解为 x = 12/4 = 3 和 x = 2/4 = 1/2。务必把分数化到最简。
l. One uses the plus sign, the other uses the minus sign. Do not combine them incorrectly.
记住公式因 ± 符号产生两个答案。一个用加号，一个用减号。不要错误合并它们。
8. Common Mistakes and How to Avoid Them | 常见错误及避免方法
Here is a table of common substitution mistakes and their solutions.
下表列出了常见的代入错误及相应解决方法。
| Mistake | 错误 | Example | 示例 | Correct Approach | 正确做法 |
|---|---|---|
| Ignoring a negative b | For x² − 5x + 6 = 0, using 5 instead of −5 | Write b = −5, then −b = 5 |
| Missing the constant term sign | For 2x² + 3x − 4 = 0, using c = 4 | Use c = −4 |
| Forgetting to divide by 2a | x = (−b ± √Δ), without the denominator | Always keep the denominator (2a) |
| Simplifying √ before calculating Δ | Writing √49 − 24 as √49 − √24 | Calculate the whole discriminant first |
These errors are easy to make under time pressure. The remedy is to write every step down and to use parentheses when substituting negative numbers.
这些错误在时间压力下很容易出现。解决办法是把每一步写清楚，并在代入负数时加上括号。
9. Worked Example 1 | 例题一
Solve 2x² − 4x − 6 = 0 using the quadratic formula.
用求根公式解 2x² − 4x − 6 = 0。
Here a = 2, b = −4, c = −6. Substitute:
此处 a = 2，b = −4，c = −6。代入：
x = (−(−4) ± √((−4)² − 4 × 2 × (−6))) / (2 × 2)
Simplify the discriminant:
化简判别式：
Δ = 16 − (4 × 2 × (−6)) = 16 + 48 = 64
Then √Δ = 8. So:
于是 √Δ = 8。因此：
x = (4 ± 8) / 4
The two solutions are x = (4 + 8)/4 = 3 and x = (4 − 8)/4 = −1.
两个解为 x = (4 + 8)/4 = 3 和 x = (4 − 8)/4 = −1。
10. Worked Example 2: When b Is Negative and c Is Negative | 例题二：b 为负且 c 为负时
Solve 3x² − 5x − 2 = 0.
解方程 3x² − 5x − 2 = 0。
Here a = 3, b = −5, c = −2. Substitute carefully:
此处 a = 3，b = −5，c = −2。小心代入：
x = (−(−5) ± √((−5)² − 4 × 3 × (−2))) / (2 × 3)
Simplify the discriminant:
化简判别式：
Δ = 25 − (−24) = 25 + 24 = 49
So √Δ = 7. Thus:
所以 √Δ = 7。因此：
x = (5 ± 7) / 6
The solutions are x = (5 + 7)/6 = 2 and x = (5 − 7)/6 = −1/3.
解为 x = (5 + 7)/6 = 2 和 x = (5 − 7)/6 = −1/3。
11. Worked Example 3: A Fractional or Irrational Answer | 例题三：分数或无理数答案
Solve x² + 2x − 4 = 0.
解方程 x² + 2x − 4 = 0。
Here a = 1, b = 2, c = −4. Substitute:
此处 a = 1，b = 2，c = −4。代入：
x = (−2 ± √(2² − 4 × 1 × (−4))) / (2 × 1)
Simplify:
化简：
Δ = 4 + 16 = 20
Since √20 = 2√5, the solutions are:
因为 √20 = 2√5，解为：
x = (−2 ± 2√5) / 2 = −1 ± √5
Notice that you can divide both terms in the numerator by 2. The final answers are x = −1 + √5 and x = −1 − √5. Do not rush when simplifying radicals.
注意分子两项都可除以 2。最终答案为 x = −1 + √5 和 x = −1 − √5。化简根式时不要心急。
12. Final Tips | 最后提示
The quadratic formula is reliable only when you substitute correctly. Always follow the same routine: write in standard form, identify a, b, c with signs, substitute with parentheses, simplify the discriminant, then simplify the whole expression. Check your answers by plugging them back into the original equation.
求根公式只有在正确代入时才是可靠的。始终遵循同样的流程：化为标准形式，带符号辨认 a、b、c，用括号代入，化简判别式，再化简整个表达式。最后将答案代回原方程进行检验。
With practice, this method becomes automatic. The more carefully you write out your steps, the fewer careless errors you will make.
通过练习，这种方法会成为你的本能。步骤写得越清楚，粗心错误就越少。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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