IB Mathematics: Solving Quadratic Equations — Methods & Techniques | IB数学：一元二次方程求解方法与技巧

📚 IB Mathematics: Solving Quadratic Equations — Methods & Techniques | IB数学：一元二次方程求解方法与技巧

Quadratic equations are a fundamental topic in IB Mathematics, appearing in both Analysis & Approaches (AA) and Applications & Interpretation (AI). Mastering the various solving methods is essential not only for Paper 1 and Paper 2, but also for calculus, optimisation and modelling problems. This guide covers the essential techniques, mathematical reasoning and exam strategies for solving quadratic equations accurately and efficiently.

二次方程是 IB 数学的核心内容，在“分析与方法”（AA）和“应用与解释”（AI）两门课程中都会出现。掌握不同的求解方法，不仅对 Paper 1 和 Paper 2 至关重要，也是学习微积分、优化与建模问题的基础。本指南将系统梳理一元二次方程的求解技巧、数学原理与考试策略，帮助考生准确高效地解题。

1. Standard Form and Discriminant | 标准形式与判别式

The general form of a quadratic equation is ax² + bx + c = 0, where a ≠ 0. The coefficients a, b, c are real numbers; in IB questions they are often integers, fractions or parameters. The discriminant, Δ = b² − 4ac, is a powerful tool that tells us the nature of the roots without solving the equation fully.

一元二次方程的一般形式为 ax² + bx + c = 0，其中 a ≠ 0。系数 a、b、c 为实数；在 IB 题目中它们常为整数、分数或参数。判别式 Δ = b² − 4ac 是一个强有力的工具，无需完整解方程即可判断根的性质。

If Δ > 0, the equation has two distinct real roots.

若 Δ > 0，方程有两个不相等的实数根。
If Δ = 0, the equation has exactly one real root (a repeated root).

若 Δ = 0，方程有且仅有一个实数根（重根）。
If Δ < 0, the equation has no real roots; instead, it has two complex conjugate roots.

若 Δ < 0，方程没有实数根，而是有两个共轭复数根。

判别式 Δ	实数根个数	图像与 x 轴的交点
Δ > 0	两个不同实数根	抛物线与 x 轴相交于两点
Δ = 0	一个重根	抛物线与 x 轴相切于一点
Δ < 0	无实数根	抛物线与 x 轴不相交

In IB Paper 1, you must be able to compute the discriminant quickly and interpret it. In Paper 2, the discriminant is often used in combination with inequalities, for example when a quadratic function has no real roots for a certain range of a parameter.

在 IB Paper 1 中，你需要快速计算判别式并解释其含义。在 Paper 2 中，判别式经常与不等式结合使用，例如当二次函数在参数的某一取值范围内没有实数根时。

2. Factoring Method | 因式分解法

Factoring is the most direct method when the quadratic expression can be rewritten as a product of two linear expressions: (px + q)(rx + s) = 0. By the zero product property, if the product is zero, then at least one factor must be zero. Thus we set each factor equal to zero and solve for x.

因式分解是最直接的方法，适用于二次表达式可以改写为两个一次表达式乘积的情形：(px + q)(rx + s) = 0。根据零积性质，如果乘积为零，则至少有一个因式为零。因此，我们令每个因式分别等于零，再解出 x。

For a monic quadratic x² + bx + c, we look for two numbers whose sum is b and whose product is c. For a non-monic quadratic ax² + bx + c, we can use the “factor by grouping” method: find two numbers that multiply to ac and add to b, then split the middle term.

对于首项系数为 1 的二次式 x² + bx + c，我们寻找两个数，使其和为 b、积为 c。对于非首项系数为 1 的二次式 ax² + bx + c，我们可以使用“分组分解法”：寻找两个数，其积为 ac、和为 b，然后拆分中间项。

x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 or x = 3

Special products are very helpful in IB exams: the difference of squares a² − b² = (a − b)(a + b), and the perfect square trinomial a² ± 2ab + b² = (a ± b)². Recognising these patterns saves time and reduces errors.

特殊乘积形式在 IB 考试中非常有帮助：平方差公式 a² − b² = (a − b)(a + b)，以及完全平方公式 a² ± 2ab + b² = (a ± b)²。识别这些模式可以节省时间并减少错误。

3. Completing the Square | 配方法

Completing the square transforms a quadratic equation into a perfect square plus a constant. This method is especially important in IB because it leads directly to the vertex form of a quadratic function, and it is the basis for deriving the quadratic formula.

配方法将二次方程转化为“完全平方 + 常数项”的形式。这一方法在 IB 中尤为重要，因为它能直接导出二次函数的顶点式，也是推导求根公式的基础。

The general procedure for x² + bx + c = 0 is:

解 x² + bx + c = 0 的一般步骤如下：

Move the constant term to the other side: x² + bx = −c.

将常数项移到等号另一侧：x² + bx = −c。
Add (b/2)² to both sides: x² + bx + (b/2)² = −c + (b/2)².

两边同时加上 (b/2)²：x² + bx + (b/2)² = −c + (b/2)²。
Rewrite the left side as (x + b/2)² and solve using square roots.

将左侧改写为 (x + b/2)²，然后借助平方根求解。

For example, x² + 6x + 2 = 0:

例如，x² + 6x + 2 = 0：

x² + 6x = −2 → (x + 3)² = 7 → x = −3 ± √7

Notice that the vertex of the parabola y = (x + h)² + k is (−h, k). In the example above, y = (x + 3)² − 7 has vertex (−3, −7). This connection between algebra and graphing is frequently tested in IB questions.

注意，抛物线 y = (x + h)² + k 的顶点为 (−h, k)。在上面的例子中，y = (x + 3)² − 7 的顶点为 (−3, −7)。这种代数与图像之间的联系经常出现在 IB 考题中。

If the coefficient of x² is not 1, first factor it out from the first two terms. For example, for 2x² + 8x + 5, write 2(x² + 4x) + 5, then complete the square inside the brackets.

如果 x² 的系数不为 1，先从前两项中提取该系数。例如，对于 2x² + 8x + 5，可写成 2(x² + 4x) + 5，再在括号内配方。

4. Quadratic Formula | 求根公式

The quadratic formula is the most general method and works for every quadratic equation, including those that are hard to factor. It is derived by completing the square on the general form ax² + bx + c = 0.

求根公式是最通用的方法，适用于所有二次方程，包括那些难以因式分解的方程。它通过对一般形式 ax² + bx + c = 0 配方推导得出。

x = (−b ± √(b² − 4ac)) / (2a)

Substitute the values of a, b, c carefully. Notice that the expression under the square root is exactly the discriminant Δ. If Δ < 0, the square root of a negative number leads to complex roots; for example, x = (−b ± i√(−Δ)) / (2a).

代入 a、b、c 的值时要格外仔细。注意根号下的表达式正是判别式 Δ。若 Δ < 0，负数的平方根会产生复数根；例如 x = (−b ± i√(−Δ)) / (2a)。

In IB Paper 1, you are not allowed a calculator, so you must leave answers in exact form. This often means simplifying surds: for instance, √12 = 2√3. In Paper 2, you may use a graphing calculator to obtain decimal approximations, but showing the exact working remains valuable.

在 IB Paper 1 中不允许使用计算器，因此答案必须以精确形式保留。这通常意味着需要化简根式：例如 √12 = 2√3。在 Paper 2 中，你可以使用图形计算器获得小数近似值，但写出精确的解答过程仍然很有价值。

A useful tip: before applying the formula, always check whether factoring or completing the square is faster. The formula is the “safety net”, but it can be time-consuming when the coefficients are large.

一个实用的技巧：在使用公式之前，先判断因式分解或配方法是否更快。公式是“安全网”，但当系数较大时可能比较耗时。

5. Graphical Interpretation | 图像解释

The graph of a quadratic function y = ax² + bx + c is a parabola. The roots of the equation ax² + bx + c = 0 are the x-coordinates of the points where the parabola intersects the x-axis. The discriminant tells us whether these intersections exist and how many there are.

二次函数 y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的根就是抛物线与 x 轴交点的横坐标。判别式告诉我们这些交点是否存在以及有多少个。

The axis of symmetry is given by x = −b/(2a), and the vertex lies on this axis. Substituting this x-value into the function gives the y-coordinate of the vertex. If a > 0, the parabola opens upward and the vertex is a minimum; if a < 0, it opens downward and the vertex is a maximum.

对称轴为 x = −b/(2a)，顶点位于该对称轴上。将这个 x 值代入函数即可得到顶点的 y 坐标。若 a > 0，抛物线开口向上，顶点为最小值；若 a < 0，抛物线开口向下，顶点为最大值。

y = x² − 2x − 3 → roots at x = −1 and x = 3; axis x = 1; vertex (1, −4)

In IB, you are often asked to sketch a parabola given its equation. Key features to label are the roots, the y-intercept, the axis of symmetry and the vertex. On Paper 2, you can use the GDC to verify your sketch, but Paper 1 requires you to calculate these features by hand.

在 IB 中，你经常需要根据方程画出抛物线草图。需要标注的关键特征包括根、y 截距、对称轴和顶点。在 Paper 2 中，你可以用图形计算器验证草图，但 Paper 1 要求你手算这些特征。

6. Sum and Product of Roots | 根与系数的关系

For a quadratic equation ax² + bx + c = 0 with roots α and β, Vieta’s formulas give a direct relationship between the roots and the coefficients:

对于根为 α 和 β 的二次方程 ax² + bx + c = 0，韦达定理给出了根与系数之间的直接关系：

α + β = −b/a and αβ = c/a

If a = 1, the equation can be written as x² − (sum of roots)x + (product of roots) = 0. This is extremely useful for constructing quadratic equations from given roots, a common IB question type.

如果 a = 1，方程可写为 x² − (两根之和)x + (两根之积) = 0。这在根据已知根构造二次方程时非常有用，是 IB 常见题型。

Another common application is finding symmetric expressions in α and β without solving the equation. For example:

另一个常见应用是在不解方程的情况下求 α 和 β 的对称式。例如：

α² + β² = (α + β)² − 2αβ

(α − β)² = (α + β)² − 4αβ

These identities appear frequently in IB Paper 1 and can save a great deal of time if you remember them. Always state which root is α and which is β when you use them, unless the expression is symmetric.

这些恒等式在 IB Paper 1 中经常出现，记住它们可以节省大量时间。使用这些公式时，除非表达式是对称的，否则要说明哪个根是 α、哪个根是 β。

7. Word Problems and Modelling | 应用题与建模

Quadratic equations are used to model a wide range of real-world situations: projectile motion, area and perimeter problems, revenue and profit functions, and population growth. In IB Mathematics, these appear in both Paper 2 and the Internal Assessment contexts.

二次方程被广泛用于模拟现实世界中的各种情况：抛体运动、面积与周长问题、收入与利润函数、人口增长等。在 IB 数学中，这些问题出现在 Paper 2 以及内部评估的背景下。

For example, a rectangle has a length that is 3 meters longer than its width, and its area is 10 m². If the width is x, then x(x + 3) = 10, so x² + 3x − 10 = 0, giving x = 2 or x = −5. Since length cannot be negative, we discard x = −5.

例如，一个矩形的长比宽长 3 米，面积为 10 平方米。设宽为 x，则 x(x + 3) = 10，即 x² + 3x − 10 = 0，解得 x = 2 或 x = −5。由于长度不能为负数，我们舍去 x = −5。

In projectile motion problems, the height of an object is often given by h(t) = −gt² + v₀t + h₀. Solving h(t) = 0 gives the times when the object is on the ground. The vertex gives the maximum height and the time at which it occurs.

在抛体运动问题中，物体的高度常表示为 h(t) = −gt² + v₀t + h₀。求解 h(t) = 0 可得到物体落地的时间。顶点则给出最大高度及其发生的时间。

When solving word problems, always define your variable clearly, translate the words into an equation with appropriate units, solve algebraically, then interpret the roots in the context of the problem. Reject any root that is not physically meaningful.

在解答应用题时，务必先明确变量，将文字翻译为带合适单位的方程，再进行代数求解，最后结合问题背景解释根的含义。舍去任何在物理意义上不成立的根。

8. Common Mistakes and Exam Tips | 常见错误与考试技巧

Many IB students lose marks on quadratic equations due to avoidable errors. Here are the most common pitfalls and how to avoid them.

许多 IB 学生在二次方程上失分，往往是因为一些可以避免的错误。以下是最常见的陷阱及其避免方法。

Forgetting to write the equation in standard form before factoring. Always rearrange to ax² + bx + c = 0 first.

因式分解前忘记把方程写成标准形式。务必先整理为 ax² + bx + c = 0。
Losing the negative root when solving x² = k. Remember that x = ±√k, not just x = √k.

解 x² = k 时丢失负根。记住 x = ±√k，而不只是 x = √k。
Incorrectly simplifying surds. For example, √8 = 2√2, not 4√2.

根式化简错误。例如 √8 = 2√2，而不是 4√2。
Dividing both sides by a variable term that could equal zero. This can destroy a solution.

两边同时除以可能为零的变量项。这可能导致丢失解。
Substituting a, b, c incorrectly into the quadratic formula. Double-check the signs.

将 a、b、c 代入求根公式时出错。请再检查一遍符号。

Exam tips: always verify your roots by substituting them back into the original equation. In Paper 2, use the GDC’s polynomial solver or graphing feature to check exact answers. In Paper 1, show your working line by line; partial marks are awarded for correct intermediate steps.

考试技巧：务必将根代回原方程检验。在 Paper 2 中，使用图形计算器的多项式求解或绘图功能来核对精确答案。在 Paper 1 中，逐步书写过程；正确的中间步骤可以获得步骤分。

9. Practice Questions | 练习与自测

The best way to master quadratic equations is through consistent practice. Try the following questions that closely follow the IB style.

掌握二次方程最好的方式就是持续练习。请尝试以下贴近 IB 风格的题目。

Solve 3x² − 10x + 3 = 0 by factoring.

用因式分解法解 3x² − 10x + 3 = 0。
Write y = 2x² + 12x + 7 in vertex form and state the coordinates of the vertex.

将 y = 2x² + 12x + 7 写成顶点式，并写出顶点坐标。
For what values of k does the equation x² + kx + 9 = 0 have exactly one real root?

当 k 取何值时，方程 x² + kx + 9 = 0 恰好有一个实数根？
If α and β are the roots of 2x² − 6x + 1 = 0, find α² + β².

若 α 和 β 是 2x² − 6x + 1 = 0 的两个根，求 α² + β²。

Answers:

参考答案：

3x² − 10x + 3 = (3x − 1)(x − 3) = 0, so x = 1/3 or x = 3.

3x² − 10x + 3 = (3x − 1)(x − 3) = 0，所以 x = 1/3 或 x = 3。
y = 2(x + 3)² − 11, vertex at (−3, −11).

y = 2(x + 3)² − 11，顶点为 (−3, −11)。
Set Δ = 0: k² − 36 = 0, so k = ±6.

令 Δ = 0：k² − 36 = 0，所以 k = ±6。
α + β = 3, αβ = 1/2, so α² + β² = 3² − 2(1/2) = 8.

α + β = 3，αβ = 1/2，所以 α² + β² = 3² − 2(1/2) = 8。

If you found these questions challenging, review the corresponding sections above before moving on. If they were straightforward, try writing your own word problem that leads to a quadratic equation and solve it.

如果你觉得这些题有难度，请先复习上面对应的章节再继续。如果觉得轻松，不妨自己编写一个能归结为二次方程的应用题并求解。

10. Summary and Final Advice | 总结与建议

Quadratic equations are one of the most tested topics in IB Mathematics. You should be comfortable moving between the standard form, factored form and vertex form, and choosing the most efficient method for each equation.

二次方程是 IB 数学中考查频率最高的主题之一。你应该熟练掌握标准式、因式分解式与顶点式之间的转换，并能针对不同的方程选择最高效的方法。

A quick decision guide: if the expression factors easily, use factoring; if the coefficient of x is even, completing the square may be clean; if the equation is messy or unfactorable, use the quadratic formula. Always check the discriminant first when asked about the nature of the roots.

快速决策指南：如果表达式容易分解，使用因式分解；如果一次项系数为偶数，配方可能更简洁；如果方程复杂或无法分解，使用求根公式。当题目问及根的性质时，首先检查判别式。

Remember that many IB calculus problems, such as finding maximum or minimum points, require you to solve a quadratic equation at some stage. A solid command of this topic will pay dividends throughout the whole syllabus.

请记住，许多 IB 微积分问题（如求最大值或最小值）在某个阶段都需要你解二次方程。扎实掌握这一主题将贯穿整个课程体系，让你持续受益。

Finally, practise with past papers, time yourself, and aim not just for the correct answer but for a clear, well-structured solution. Accuracy and speed will improve together with deliberate practice.

最后，请使用历年真题进行练习，并为自己计时。不仅要追求正确答案，还要追求清晰、结构完整的解答过程。在刻意练习中，准确度和速度会同步提升。

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IB Mathematics: Solving Quadratic Equations — Methods & Techniques | IB数学：一元二次方程求解方法与技巧