Advanced Math
| English | 中文 | Pinyin |
|---|---|---|
| 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.
Read the relationship
- Factor a quadratic when possible. The zero-product rule applies to a product equal to zero.
- Solve nonlinear equations and systems.
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.
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.
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.
Which answer fits this case?
Transform equivalent polynomial and rational expressions
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.
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.