Quadratic Equations | 二次方程

📚 Quadratic Equations | 二次方程

Quadratic equations form a cornerstone of algebra in IB Mathematics. They appear in topics ranging from functions and graphs to complex numbers and optimisation. A deep understanding of quadratic equations is essential for success in both Analysis and Approaches (AA) and Applications and Interpretation (AI) courses. In this article, we review the key concepts, solution methods, and common pitfalls to help you master quadratics.

二次方程式是IB数学代数部分的基石。它们出现在函数、图形、复数以及优化等多个主题中。深刻理解二次方程对于在分析与方法和应用与解释两门课程中取得成功至关重要。本文回顾关键概念、解题方法和常见错误，助你掌握二次方程。

---

1. Standard Form of a Quadratic Equation | 二次方程的标准形式

A quadratic equation in one variable is any equation that can be written in the form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. The condition a ≠ 0 is crucial because if a = 0 the equation becomes linear. The expression ax² + bx + c is called a quadratic expression, and its highest power of x is 2.

一元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 是实数且 a ≠ 0。a ≠ 0 的条件至关重要，因为如果 a = 0，方程就变成了一次方程。表达式 ax² + bx + c 称为二次表达式，x 的最高次幂为 2。

In IB exams, you may need to identify the coefficients a, b and c even when the equation is not initially in standard form. For example, rearranging 5x – 3x² = 2 gives –3x² + 5x – 2 = 0, so a = –3, b = 5, c = –2. Recognising the coefficients correctly is essential for applying the quadratic formula or discriminant.

在 IB 考试中，即使方程最初不是标准形式，你可能也需要识别系数 a、b 和 c。例如，将 5x – 3x² = 2 整理后得到 –3x² + 5x – 2 = 0，因此 a = –3, b = 5, c = –2。正确识别系数对于应用求根公式或判别式至关重要。

---

2. Solving by Factorisation | 因式分解法

Factorisation relies on the zero-product property: if pq = 0, then p = 0 or q = 0. After writing the quadratic in standard form, we factor it into two linear factors. For example, x² – 5x + 6 = 0 factors as (x – 2)(x – 3) = 0, yielding solutions x = 2 or x = 3. This method is efficient when the quadratic has simple integer roots.

因式分解依赖于零积性质：若 pq = 0，则 p = 0 或 q = 0。将二次方程写成标准形式后，我们将其分解为两个一次因式。例如，x² – 5x + 6 = 0 分解为 (x – 2)(x – 3) = 0，得到解 x = 2 或 x = 3。当二次方程具有简单整数根时，该方法十分高效。

Not all quadratics factor nicely with integer coefficients. When a ≠ 1, techniques such as splitting the middle term or trial and error are often used. For 2x² + 7x + 3, we look for two numbers whose product is 2×3 = 6 and sum is 7; these are 6 and 1. Thus 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3). Always check your factorisation by expanding. If a quadratic does not factor over the integers, the discriminant can tell you whether it is worth attempting more complicated factorisation.

并非所有二次方程都能用整数系数轻松分解。当 a ≠ 1 时，常使用拆分中项或试探法。对于 2x² + 7x + 3，我们寻找两个数，其积为 2×3 = 6 且和为 7；这两个数是 6 和 1。因此 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。务必通过展开来验证因式分解。如果二次方程不能在整数范围内分解，判別式可以告诉你是否值得尝试更复杂的因式分解。

---

3. Completing the Square | 配方法

Completing the square transforms a quadratic expression into a perfect square plus or minus a constant. Starting from ax² + bx + c, we first factor a out of the x² and x terms, then add and subtract (b/(2a))² inside the brackets. The resulting vertex form a(x – h)² + k reveals the coordinates of the vertex (h, k) and is particularly useful for sketching graphs and solving equations when factorisation is not straightforward.

配方法将二次表达式转化为一个完全平方加上或减去一个常数。从 ax² + bx + c 开始，我们先将 a 从 x² 和 x 项中提出，然后在括号内加上再减去 (b/(2a))²。得到的顶点式 a(x – h)² + k 显示了顶点坐标 (h, k)，在不易因式分解时对画图和求解方程尤其有用。

For example, solve x² + 6x + 5 = 0. Rewrite as (x² + 6x + 9) – 9 + 5 = 0 ⇒ (x + 3)² – 4 = 0 ⇒ (x + 3)² = 4 ⇒ x + 3 = ±2 ⇒ x = –1 or x = –5. When a ≠ 1, say 2x² + 8x + 3 = 0, first factor out 2: 2[x² + 4x] + 3 = 0. Complete the square inside: 2[(x + 2)² – 4] + 3 = 0 ⇒ 2(x + 2)² – 8 + 3 = 0 ⇒ 2(x + 2)² = 5 ⇒ (x + 2)² = 5/2 ⇒ x = –2 ± √(5/2). Completing the square is also the method used to derive the quadratic formula.

例如，解方程 x² + 6x + 5 = 0。改写为 (x² + 6x + 9) – 9 + 5 = 0 ⇒ (x + 3)² – 4 = 0 ⇒ (x + 3)² = 4 ⇒ x + 3 = ±2 ⇒ x = –1 或 x = –5。当 a ≠ 1 时，如 2x² + 8x + 3 = 0，先提出因数 2：2[x² + 4x] + 3 = 0。在括号内配方：2[(x + 2)² – 4] + 3 = 0 ⇒ 2(x + 2)² – 8 + 3 = 0 ⇒ 2(x + 2)² = 5 ⇒ (x + 2)² = 5/2 ⇒ x = –2 ± √(5/2)。配方法也是推导求根公式的方法。

---

4. The Quadratic Formula | 求根公式

The quadratic formula is a universal tool for solving any quadratic equation. It states that the solutions of ax² + bx + c = 0 are given by:

x = [ –b ± √(b² – 4ac) ] / (2a)

求根公式是解任何二次方程的通用工具。它给出二次方程 ax² + bx + c = 0 的解为：

x = [ –b ± √(b² – 4ac) ] / (2a)

This formula is derived by completing the square on the general quadratic. You must memorise it and be able to apply it accurately. In IB exams, showing the clear substitution of a, b and c into the formula often earns method marks, even if an arithmetic slip occurs later. For instance, solve 3x² – 4x – 2 = 0: a = 3, b = –4, c = –2. Then x = [4 ± √(16 + 24)] / 6 = [4 ± √40] / 6 = [4 ± 2√10] / 6 = [2 ± √10] / 3.

该公式通过对一般二次式进行配方法推导而来。你必须熟记并能够准确应用。在 IB 考试中，清晰展示将 a、b 和 c 代入公式的过程通常能获得方法分，即使后续出现计算失误。例如，解 3x² – 4x – 2 = 0：a = 3, b = –4, c = –2。那么 x = [4 ± √(16 + 24)] / 6 = [4 ± √40] / 6 = [4 ± 2√10] / 6 = [2 ± √10] / 3。

---

5. Discriminant and Nature of Roots | 判别式与根的性质

The discriminant Δ is the expression under the square root in the quadratic formula: Δ = b² – 4ac. It determines the nature of the roots without solving the equation. If Δ > 0, the equation has two distinct real roots. If Δ = 0, there is exactly one real root (a repeated root). If Δ < 0, the equation has two complex conjugate roots and no real solutions.

判别式 Δ 是求根公式中平方根下的表达式：Δ = b² – 4ac。它无需解方程即可确定根的性质。若 Δ > 0，方程有两个不相等的实数根。若 Δ = 0，恰好有一个实数根（重根）。若 Δ < 0，方程有两个共轭复数根，没有实数解。

In typical IB questions, you may be asked to find the values of a parameter k for which a quadratic has a given number of real roots. For example, for x² + kx + 9 = 0 to have two distinct real roots, we require Δ = k² – 36 > 0 ⇒ k < –6 or k > 6. For a repeated root, k² – 36 = 0 ⇒ k = ±6. The discriminant also helps when sketching graphs: it tells you whether the parabola cuts the x‑axis, touches it, or lies completely above or below it.

在典型的 IB 问题中，你可能会被要求找出参数 k 的值，使得二次方程具有指定个数的实数根。例如，要使 x² + kx + 9 = 0 有两个不相等的实数根，需要 Δ = k² – 36 > 0 ⇒ k < –6 或 k > 6。若要有重根，则 k² – 36 = 0 ⇒ k = ±6。判别式在画图时也很有帮助：它能告诉你抛物线与 x 轴相交、相切，还是完全位于 x 轴上方或下方。

---

6. Sum and Product of Roots | 根的和与积

For a quadratic equation ax² + bx + c = 0 with roots α and β, the sum and product of the roots are given by Vieta’s formulas:

α + β = –b/a,

Published by TutorHao | IB Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）