Solving Quadratic Equations Using the Quadratic Formula | 使用求根公式解二次方程

📚 Solving Quadratic Equations Using the Quadratic Formula | 使用求根公式解二次方程

The quadratic formula is one of the most powerful and reliable methods for solving quadratic equations. Unlike factorisation, which only works for special cases, the quadratic formula can solve any quadratic equation of the form ax² + bx + c = 0, provided that a ≠ 0. It is especially useful when the equation cannot be factorised easily or when the roots involve surds. This article explains how to use the formula correctly, understand its components, and avoid common mistakes in the Edexcel IGCSE examination.

二次求根公式是解二次方程最强大、最可靠的方法之一。与仅适用于特殊情况的因式分解不同，求根公式可以求解任何形如 ax² + bx + c = 0（其中 a ≠ 0）的二次方程。它特别适用于无法简便因式分解或根包含无理数的情况。本文讲解如何正确使用该公式、理解其各部分含义，并避免在 Edexcel IGCSE 考试中的常见错误。

1. What Is a Quadratic Equation? | 什么是二次方程？

A quadratic equation is a polynomial equation of degree 2. Its standard form is written as ax² + bx + c = 0, where a, b and c are constants, and a is not equal to zero. The values of x that satisfy this equation are called the roots or solutions. On a graph, these roots represent the points where the curve y = ax² + bx + c crosses the x-axis. A quadratic equation may have two distinct real roots, one repeated real root, or no real roots at all.

二次方程是最高次数为 2 的多项式方程。其标准形式写作 ax² + bx + c = 0，其中 a、b、c 是常数，且 a 不等于零。满足该方程的 x 值称为方程的根或解。在图象上，这些根对应曲线 y = ax² + bx + c 与 x 轴的交点。二次方程可能有两个不同的实数根、一个重根，或者没有实数根。

2. The Quadratic Formula | 二次求根公式

For the quadratic equation ax² + bx + c = 0, the solutions are given by the quadratic formula. This formula is derived from the method of completing the square and works for every quadratic equation. You must memorise it for your examination, as it is not provided in the Edexcel IGCSE formula booklet.

对于二次方程 ax² + bx + c = 0，其解由二次求根公式给出。该公式源于配方法，适用于所有二次方程。考试时必须牢记该公式，因为 Edexcel IGCSE 公式册中并不提供。

x = (−b ± √(b² − 4ac)) / (2a)

In this formula, the symbol ± means that you calculate two values: one by adding the square root, and one by subtracting it. The expression b² − 4ac is called the discriminant, which we will discuss later.

公式中的符号 ± 表示需要计算两个值：一个加上平方根，一个减去平方根。表达式 b² − 4ac 被称为判别式，我们稍后会详细讨论。

3. Understanding the Discriminant | 理解判别式

The discriminant, often denoted by Δ (uppercase delta), is the part under the square root: Δ = b² − 4ac. It determines the nature of the roots without solving the full equation. If Δ > 0, there are two distinct real roots. If Δ = 0, there is exactly one repeated real root. If Δ < 0, there are no real roots, because you cannot take the square root of a negative number in the set of real numbers.

判别式通常用大写希腊字母 Δ 表示，是根号下的部分：Δ = b² − 4ac。它决定了根的性质，无需解整个方程。若 Δ > 0，则有两个不同的实数根；若 Δ = 0，则有一个重复的实数根；若 Δ < 0，则没有实数根，因为在实数范围内不能对负数开平方。

Discriminant 判别式	Number and type of roots 根的个数与类型
Δ > 0	Two distinct real roots 两个不同实数根
Δ = 0	One repeated real root 一个重根
Δ < 0	No real roots 无实数根

4. Step-by-Step Procedure | 分步计算步骤

To solve a quadratic equation using the quadratic formula, follow these steps carefully. First, write the equation in standard form ax² + bx + c = 0. If the original equation is not in this form, rearrange it. Second, identify the values of a, b and c. Third, substitute these values into the formula. Fourth, calculate the discriminant b² − 4ac. Fifth, evaluate the square root if it is a perfect square, or leave it as a surd if not. Finally, simplify the expression to obtain the two solutions.

使用求根公式解二次方程时，请仔细遵循以下步骤。首先，将方程写成标准形式 ax² + bx + c = 0；若原方程不是该形式，需先整理。第二，确定 a、b、c 的值。第三，将这些值代入公式。第四，计算判别式 b² − 4ac。第五，若它是最简平方数则求平方根，否则保留为无理数。最后，化简表达式得到两个解。

Step 1: Ensure the equation is in the form ax² + bx + c = 0.
Step 2: List a, b, and c clearly, including their signs.
Step 3: Substitute into x = (−b ± √(b² − 4ac)) / (2a).
Step 4: Simplify under the square root first.
Step 5: Simplify the fraction for each of the two cases.

步骤 1：确保方程为 ax² + bx + c = 0 形式。
步骤 2：清楚列出 a、b、c，包括它们的正负号。
步骤 3：代入 x = (−b ± √(b² − 4ac)) / (2a)。
步骤 4：先化简根号内的部分。
步骤 5：分别化简两个分数解。

5. Worked Example 1: Two Distinct Rational Roots | 例题 1：两个不同的有理根

Solve the equation x² − 5x + 6 = 0 using the quadratic formula. First, identify a = 1, b = −5, c = 6. Substitute into the formula:

使用求根公式解方程 x² − 5x + 6 = 0。首先确定 a = 1，b = −5，c = 6。代入公式得：

x = (−(−5) ± √((−5)² − 4 × 1 × 6)) / (2 × 1)

Simplify: x = (5 ± √(25 − 24)) / 2 = (5 ± √1) / 2 = (5 ± 1) / 2. Thus x = (5 + 1) / 2 = 3, or x = (5 − 1) / 2 = 2. So the roots are x = 3 and x = 2. This equation could also be solved by factorisation, and the quadratic formula confirms the same answers.

化简得 x = (5 ± √(25 − 24)) / 2 = (5 ± √1) / 2 = (5 ± 1) / 2。因此 x = (5 + 1) / 2 = 3，或 x = (5 − 1) / 2 = 2。所以根为 x = 3 和 x = 2。该方程也可以用因式分解求解，求根公式给出了同样的答案。

6. Worked Example 2: Irrational Roots | 例题 2：无理根

Solve 2x² + 4x − 3 = 0 using the quadratic formula. Here a = 2, b = 4, c = −3. Substitute:

用求根公式解 2x² + 4x − 3 = 0。这里 a = 2，b = 4，c = −3。代入得：

x = (−4 ± √(4² − 4 × 2 × (−3))) / (2 × 2)

Simplify the discriminant: 16 − (−24) = 16 + 24 = 40. Therefore x = (−4 ± √40) / 4. Since √40 = √(4 × 10) = 2√10, the solutions become x = (−4 ± 2√10) / 4. Divide each term by 2: x = (−2 ± √10) / 2. The two roots are x = (−2 + √10) / 2 and x = (−2 − √10) / 2.

化简判别式：16 − (−24) = 16 + 24 = 40。因此 x = (−4 ± √40) / 4。因为 √40 = √(4 × 10) = 2√10，所以解变为 x = (−4 ± 2√10) / 4。每项除以 2 得 x = (−2 ± √10) / 2。两个根为 x = (−2 + √10) / 2 和 x = (−2 − √10) / 2。

7. Working with Fractions and Negative Coefficients | 处理分数与负系数

Sometimes the equation contains fractions or negative coefficients. When a is negative, it is often easier to multiply the whole equation by −1 to make a positive, but this is not compulsory. If the equation involves fractions, multiply every term by the common denominator to obtain integer coefficients before applying the formula. For example, consider (1/2)x² − x − 1 = 0. Multiply all terms by 2 to get x² − 2x − 2 = 0, then use a = 1, b = −2, c = −2.

有时方程中含有分数或负系数。当 a 为负数时，通常可以将整个方程乘以 −1 使 a 变为正数，但这并非必须。若方程含有分数，先将每一项乘以公分母以得到整数系数，再使用公式。例如，考虑 (1/2)x² − x − 1 = 0。将所有项乘以 2 得 x² − 2x − 2 = 0，然后使用 a = 1，b = −2，c = −2。

Applying the formula gives x = (2 ± √(4 + 8)) / 2 = (2 ± √12) / 2 = (2 ± 2√3) / 2 = 1 ± √3. Notice how clearing fractions first simplifies the entire calculation and reduces the risk of arithmetic errors.

代入公式得 x = (2 ± √(4 + 8)) / 2 = (2 ± √12) / 2 = (2 ± 2√3) / 2 = 1 ± √3。注意先消去分数会使整个计算更简单，并降低算术错误的风险。

8. Common Mistakes to Avoid | 常见错误及避免方法

Many students make avoidable mistakes when using the quadratic formula. The most common error is forgetting to include the sign of b when substituting. For example, if b = −5, then −b equals 5, not −5. Another frequent mistake is incorrectly calculating b² when b is negative; square the value including the sign: (−5)² = 25. Also, students often forget to divide both the numerator and denominator by 2a, or they cancel incorrectly when the numerator has two terms and the denominator is a single term.

许多学生在使用求根公式时会犯一些可以避免的错误。最常见的错误是代入时忘记 b 的符号。例如，若 b = −5，则 −b 等于 5，而不是 −5。另一个常见错误是当 b 为负数时错误地计算 b²；平方时应包含符号：(−5)² = 25。此外，学生常常忘记将分子和分母同时除以 2a，或当分子有两项而分母是一项时错误地约分。

Always write the equation in standard form first.
Double-check the signs of a, b and c.
Simplify √(b² − 4ac) completely before the final answer.
Do not cancel a term in the numerator with part of the denominator unless it is a factor of the entire numerator.

始终先将方程写成标准形式。
仔细检查 a、b、c 的符号。
在得出最终答案前，将 √(b² − 4ac) 化到最简。
除非分母是分子整项的公因数，否则不要在分子中的某一项与分母之间任意约分。

9. When to Use the Formula Instead of Factorisation | 何时使用公式而不是因式分解

In IGCSE examinations, you should choose the most efficient method. Factorisation is quick when roots are integers or simple fractions, but it is not always obvious. The quadratic formula is a universal method: it works even when the equation has irrational or complex roots. In questions that specifically ask for solutions “given to 2 decimal places” or “in surd form”, the formula is usually the required approach. If a question directs you to use the quadratic formula, you must show your substitution clearly.

在 IGCSE 考试中，应选择最有效的方法。当根为整数或简单分数时，因式分解较快，但并不总是明显。求根公式是通用方法：即使方程有无理根或复数根也能使用。在题目明确要求“精确到两位小数”或“以无理数形式”给出解时，通常需要采用公式法。如果题目指定使用求根公式，你必须清楚展示代入过程。

For example, if you encounter an equation like x² + 3x − 7 = 0, factorisation is not straightforward, so the quadratic formula is the best choice. Its discriminant is 9 + 28 = 37, giving roots x = (−3 ± √37) / 2.

例如，若遇到方程 x² + 3x − 7 = 0，因式分解并不直接，因此求根公式是最佳选择。其判别式为 9 + 28 = 37，得到根 x = (−3 ± √37) / 2。

10. Practice Questions | 练习题目

Try the following questions to test your understanding. For each, use the quadratic formula and give your answer in the requested form.

尝试以下题目来检验你的理解。对每一题使用求根公式，并按题目要求给出答案。

Solve x² + 7x + 10 = 0. (Give integer answers)
Solve 3x² − 6x + 2 = 0. (Give answers in surd form)
Solve 5x² + 2x − 4 = 0. (Give answers to 2 decimal places)
Find the discriminant of 2x² − 3x + 5 = 0 and state the number of real roots.

解 x² + 7x + 10 = 0。（给出整数答案）
解 3x² − 6x + 2 = 0。（给出无理数形式答案）
解 5x² + 2x − 4 = 0。（答案保留两位小数）
求 2x² − 3x + 5 = 0 的判别式，并指出实数根的个数。

Answers: 1. x = −2 or x = −5. 2. x = (6 ± √(36 − 24)) / 6 = (6 ± √12) / 6 = (3 ± √3) / 3. 3. Using a calculator, x ≈ 0.68 or x ≈ −1.08. 4. Discriminant = 9 − 40 = −31, so no real roots.

答案：1. x = −2 或 x = −5。2. x = (6 ± √(36 − 24)) / 6 = (6 ± √12) / 6 = (3 ± √3) / 3。3. 使用计算器得 x ≈ 0.68 或 x ≈ −1.08。4. 判别式 = 9 − 40 = −31，因此没有实数根。

11. Exam Tips for Edexcel IGCSE | Edexcel IGCSE 考试技巧

In the examination, always align your working clearly. Write the equation in standard form first, then list a = …, b = …, c = … before substitution. This helps you and the examiner to follow your method. If the question asks for exact values, keep surds in your final answer; do not round unless instructed. Check whether the discriminant is a perfect square: if so, the roots are rational and factorisation might have been possible. Also, remember that if the discriminant is negative, you should state that there are no real solutions rather than trying to calculate a square root of a negative number.

考试中，务必书写清晰。先将方程写成标准形式，然后列出 a = …，b = …，c = …，再代入。这有助于你和阅卷人理解你的方法。若题目要求精确值，最终答案应保留无理数形式；除非题目要求，不要四舍五入。注意判别式是否为完全平方数：若是，则根为有理数，或许原先可以用因式分解。另外，若判别式为负，应直接说明没有实数解，而不要尝试对负数开平方。

12. Summary | 总结

The quadratic formula x = (−b ± √(b² − 4ac)) / (2a) is an essential tool for solving any quadratic equation in the form ax² + bx + c = 0. It is particularly valuable when factorisation is difficult or impossible. The discriminant b² − 4ac reveals the nature of the roots: positive gives two distinct real roots, zero gives one repeated root, and negative gives no real roots. By practising substitution and careful simplification, you can solve quadratic equations accurately and quickly in your IGCSE examination.

求根公式 x = (−b ± √(b² − 4ac)) / (2a) 是解形如 ax² + bx + c = 0 的任意二次方程的重要工具。当因式分解困难或不可能时，它尤为有价值。判别式 b² − 4ac 揭示根的性质：正数对应两个不同实数根，零对应一个重根，负数对应没有实数根。通过练习代入和仔细化简，你可以在 IGCSE 考试中准确快速地解出二次方程。

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Solving Quadratic Equations Using the Quadratic Formula | 使用求根公式解二次方程