📚 Exercise 2H.1 | 练习 2H.1
Exercise 2H.1 in the IB Mathematics course is primarily designed to test your understanding of the discriminant of a quadratic equation and its role in determining the nature of roots. Whether you are working through the Analysis & Approaches or Applications & Interpretation syllabus, a confident grasp of this topic is essential for success on both Paper 1 and Paper 2 exams.
IB 数学课程中的练习 2H.1 主要考察你对二次方程判别式的理解，以及它如何决定根的性质。无论你学习的是分析与方法还是应用与解释课程，扎实掌握这一专题对 Paper 1 和 Paper 2 的考试都至关重要。
1. Standard Form of a Quadratic Equation | 二次方程的标准形式
use if a = 0 the equation becomes linear, losing the x² term that defines it as quadratic.
二次方程必须写成标准形式 ax² + bx + c = 0，其中 a、b、c 为实数常数且 a ≠ 0。a ≠ 0 这个条件至关重要，因为如果 a = 0，方程退化为一次方程，失去使它成为二次的 x² 项。
nalysis. For example, 3x² = 5x - 2 becomes 3x² - 5x + 2 = 0, with a = 3, b = -5 and c = 2.
在练习 2H.1 中，你经常需要先将方程整理成这种形式，再做进一步分析。例如，3x² = 5x - 2 变成 3x² - 5x + 2 = 0，其中 a = 3，b = -5，c = 2。
2. Solving Quadratic Equations | 求解二次方程
The most general method for solving ax² + bx + c = 0 is the quadratic formula: x = [ -b ± √b² - 4ac ] / 2a. The expression under the square root, b² - 4ac, is the discriminant and is denoted by the Greek letter delta, Δ.
求解 ax² + bx + c = 0 最通用的方法是二次公式：x = [ -b ± √b² - 4ac ] / 2a。根号下的表达式 b² - 4ac 就是判别式，用希腊字母 Δ 表示。
Factorization and completing the square are also valid techniques, but the quadratic formula works even when the roots are irrational or complex. Many questions in Exercise 2H.1 ask you to first calculate Δ and then decide on the best solving strategy.
因式分解和配方法也是有效技巧，但即使根是无理数或复数，二次公式仍然有效。练习 2H.1 中的许多题目要求你先计算 Δ，再决定最佳求解策略。
3. Introducing the Discriminant | 判别式介绍
4ac. Its value alone tells you the nature of the roots without requiring you to solve the equation fully. This is one of the most powerful shortcuts in algebra and a favourite topic for IB examiners.
判别式定义为 Δ = b² - 4ac。仅仅通过它的值你就能判断根的性质，而不需要完全解出方程。这是代数中最强大的捷径之一，也是 IB 考官偏爱的考点。
In Exercise 2H.1 you will frequently encounter questions like 'Find the value of k for which the equation has two equal roots' . These are entirely discriminant-based, asking you to set Δ = 0 and solve for the unknown parameter.
在练习 2H.1 中，你会频繁遇到像“求使方程有两个相等根的 k 值”这样的问题。它们完全依赖于判别式，需要你令 Δ = 0 并解出未知参数。
4. Nature of Roots: Real and Distinct | 根的性质：两个不等实根
e both real numbers. The graph of the quadratic function y = ax² + bx + c will cross the x‑axis at two different points.
如果 Δ > 0，二次方程有两个不等的实根。它们可能是有理数也可能是无理数，但都是实数。二次函数 y = ax² + bx + c 的图像会与 x 轴交于两个不同的点。
ance, if a = 1, b = 4 and c = k, the condition Δ > 0 becomes 16 - 4k > 0 → k < 4.
练习 2H.1 经常要求你根据计算出的 Δ 对根的性质作出判断，或者找出使 Δ 保持为正的参数范围。例如，若 a = 1, b = 4, c = k，条件 Δ > 0 变为 16 - 4k > 0 → k < 4。
5. Nature of Roots: Real and Equal | 根的性质：两个相等实根
e real root (also called a repeated or double root). Algebraically the two roots coincide, and the parabola touches the x‑axis at exactly one point — the vertex lies on the x‑axis.
当 Δ = 0 时，方程恰有一个实根（也称为重根或二重根）。代数上两个根重合，抛物线恰与 x 轴相切于一点——顶点落在 x 轴上。
Δ = 0 yields 4m² - 4(m+2) = 0, which simplifies to m² - m - 2 = 0, giving m = -1 or m = 2.
练习 2H.1 常通过这个条件来求未知系数。例如，给定 x² + 2mx + (m+2) = 0，令 Δ = 0 得到 4m² - 4(m+2) = 0，化简得 m² - m - 2 = 0，解得 m = -1 或 m = 2。
6. Nature of Roots: No Real Roots | 根的性质：没有实根
If Δ < 0, the quadratic equation has no real roots. Instead, it has a pair of complex conjugate roots. In the context of the IB course, this generally means the graph does not intersect the x‑axis at all, and the corresponding quadratic function is either always positive or always negative.
如果 Δ < 0，二次方程没有实根，而是有一对共轭复根。在 IB 课程的语境下，这通常意味着图像完全不与 x 轴相交，相应的二次函数要么恒正要么恒负。
This concept is heavily used in quadratic inequality problems, which also feature in Exercise 2H.1. A condition such as Δ < 0 combined with a > 0 guarantees that ax² + bx + c > 0 for all real x.
这一概念在二次不等式的题目中被大量使用，练习 2H.1 中也会出现。像 Δ < 0 且 a > 0 的条件保证了 ax² + bx + c > 0 对所有实数 x 成立。
7. Graphical Interpretation | 图像解释
he sign of a determines whether the parabola opens upwards (a > 0) or downwards (a < 0), while Δ tells you how many x‑intercepts there are.
将判别式与抛物线联系起来是一项关键技能。对于 y = ax² + bx + c，a 的符号决定抛物线开口向上（a > 0）还是向下（a < 0），而 Δ 告诉你与 x 轴有几个交点。
Exercise 2H.1 might present a sketch of a parabola and ask you to deduce the sign of Δ, or give an inequality describing the position of a graph relative to the x‑axis. Being able to move fluently between algebra and geometry will save time in the exam.
练习 2H.1 可能会给出一个抛物线草图，让你推断 Δ 的符号，或者给出一个描述图像相对于 x 轴位置的不等式。能够在代数与几何之间自如转换会为考试节省时间。
8. Discriminant and Coefficient Problems | 判别式与系数问题
A classic style of question in Exercise 2H.1 is: ' set of values of p for which the equation x² + (p+1)x + 4 = 0 has two distinct real roots.' by writing Δ > 0, substitute a = 1, b = p+1, c = 4 and solve the resulting inequality.
练习 2H.1 中一类经典的题型是：“求使方程 x² + (p+1)x + 4 = 0 有两个不等实根的 p 的取值范围。”你从写出 Δ > 0 开始，代入 a = 1, b = p+1, c = 4 并解所得的不等式。
Often the inequality will be quadratic itself, requiring you to use the discriminant a second time or factorisation to determine the final interval. Carefully check the direction of the inequality sign and whether the endpoint(s) should be included.
通常这个不等式本身也是二次的，需要你再次使用判别式或因式分解来确定最终区间。要仔细检查不等号的方向以及端点是否应包含在内。
9. Tangency Condition | 相切条件
sk for the value of m that makes the line tangent to the curve. Setting the two equations equal yields a quadratic whose discriminant must be zero.
Δ = 0 的条件也代表一条直线与二次曲线相切。练习 2H.1 可能会给出直线 y = mx + k 和二次曲线 y = x² + bx + c，并求使直线与曲线相切的 m 值。令两个方程相等得到一个二次式，其判别式必须为零。
implify the combined equation carefully and double-check the algebra before setting Δ = 0.
这种方法在解析几何与二次函数混合题中极为常见。一定要仔细化简合并后的方程，并在令 Δ = 0 之前双重检查代数运算。
10. Quadratic Inequalities and the Discriminant | 二次不等式与判别式
Exercise 2H.1 often extends the discriminant idea to inequalities such as 'Find the range of k for which kx² + 3x + 5 > 0 for all real x'. For the quadratic to be always positive, we need a > 0 and Δ < 0 simultaneously.
练习 2H.1 经常将判别式的思想拓展到不等式，如“求使得对所有实数 x 都有 kx² + 3x + 5 > 0 的 k 的范围”。要使二次式恒正，我们需要同时满足 a > 0 和 Δ < 0。
on is no longer quadratic. After finding the intersection of the conditions, write the final answer using set notation or interval notation as required by the IB markscheme.
务必单独处理 x² 的系数，因为如果 a = 0，表达式就不再是二次的。求出条件的交集后，按照 IB 评分标准的要求使用集合符号或区间记号书写最终答案。
11. Sum and Product of Roots | 根的和与积
formulas let you construct or manipulate quadratic equations without explicitly finding the roots.
虽然练习 2H.1 侧重于判别式，但许多题目也会联系到根的和与积：根的和 = -b/a，根的积 = c/a。这些公式让你无需显式求出根就能构造或操作二次方程。
For instance, you might be asked to find a quadratic whose roots are twice the roots of a given equation. Using the sum and product relationships with the original equation, combined with a change of variable, is a neat discriminant-free method that often appears alongside Exercise 2H.1 style questions.
例如，你可能被要求求一个二次方程，其根是给定方程根的两倍。利用原方程的根的和与积关系，再结合换元，是一种巧妙的避开判别式的方法，常与练习 2H.1 类题目一同出现。
12. Common Pitfalls and Exam Tips | 常见错误与考试技巧
A frequent mistake in Exercise 2H.1 is forgetting that a must be non‑ k whether a = 0 leads to a degenerate case that must be excluded or considered separately.
练习 2H.1 中一个常见错误是忘记 a 必须非零。当参数出现在 x² 的系数中时，始终要检查 a = 0 是否导致退化情形，它必须被排除或单独考虑。
if b = -4, b² = 16, not -16. Finally, present your final answer in the exact form required — leave roots as surds or in terms of π, and never approximate unless instructed.
另一个陷阱是在计算 Δ 时弄错 b 的符号。系数平方时要仔细；例如 b = -4 时 b² = 16，而不是 -16。最后，按要求的精确形式呈现最终答案——根保留为根式或用 π 表示，除非明确要求，否则不要取近似值。
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