Advanced Math · 高等数学
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| root/ruːt/ | 根 | gēn |
| domain/dəˈmeɪn/ | 定义域 | dìng yì yù |
A decision before an answer
- A quadratic can describe a ball’s height, but only the times before it lands belong to the physical model.
- Your goal: Transform equivalent polynomial and rational expressions.
答案前的判断
- 二次函数可描述球体高度，但仅落地前的时段属于该物理模型的适用范围。
- 你的目标：变换等价的整式与分式表达式。
Read the relationship
- Factor a quadratic when possible. The zero-product rule applies to a product equal to zero.
- Solve nonlinear equations and systems.
阅读关系
- 尽可能对二次三项式进行因式分解。积为零法则适用于乘积等于零的情形。
- 求解非线性方程及方程组。
The minimum of y=(x−4)²+2 is:
A square is nonnegative and is zero at x=4.
Use the defining rule
- Completing the square reveals a vertex: x²−6x+5=(x−3)²−4.
- Interpret quadratic and exponential functions.
运用定义规则
- 配方法可揭示顶点形式：x²−6x+5=(x−3)²−4。
- 解读二次函数和指数函数。
A quantity 100 grows 10% twice. Final value?
100×1.1²=121; growth compounds.
Check the conditions
- For rational expressions, record excluded denominator values before cancelling factors.
- Interpret quadratic and exponential functions.
Solve x²−5x+6=0. Factor: (x−2)(x−3)=0. Thus x=2 or 3. If instead (x²−5x+6)/(x−2)=0, x=2 is excluded; only 3 remains.
检查条件
- 对于有理表达式，在约去公因式前记录使分母为零的排除值。
- 解读二次函数和指数函数。
求解 x²−5x+6=0。因式分解：(x−2)(x−3)=0。因此 x=2 或 3。若改为 (x²−5x+6)/(x−2)=0，则 x=2 为排除值；仅余 3。
The roots of x²−9=0 are 3 and ____.
The factorisation is (x−3)(x+3)=0.
Apply the task format
- An exponential model multiplies by a fixed factor per interval. A 20% increase means multiply by 1.2, not add 0.2.
- Interpret quadratic and exponential functions.
Squaring an equation can add extraneous roots. Substitute into the original equation, not just the transformed one.
应用题目格式
- 指数模型按固定因子乘以每个区间。增长 20% 意味着乘以 1.2，而非加上 0.2。
- 解读二次函数和指数函数。
对方程两边平方可能引入增根。应代入原方程检验，而不仅仅是代入变换后的方程。
Which answer fits this case? · 哪个答案符合此案例？
Transform equivalent polynomial and rational expressions · 变换等价的多项式与有理表达式
Cancelling (x−2) makes x=2 allowed in the original rational expression.
Its original denominator remains zero at 2.
Keep the distinctions
- root 根 — An input for which a function equals zero.
- domain 定义域 — The allowed input values.
- Transform equivalent polynomial and rational expressions.
- Solve nonlinear equations and systems.
- Interpret quadratic and exponential functions.
保持区别
- root 根 — 使函数值为零的输入值。
- domain 定义域 — 允许的输入值集合。
- 转换等价的多项式和有理表达式。
- 求解非线性方程及方程组。
- 解读二次函数和指数函数。
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.