Algebra basics and manipulation · 代数基础与变形
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| algebra/ˈældʒɪbrə/ | 代数 | dài shù |
| variable/ˈveərɪəbl/ | 变量 | biàn liàng |
| substitute/ˈsʌbstɪtjuːt/ | 代入 | dài rù |
| like terms/laɪk tɜːmz/ | 同类项 | tóng lèi xiàng |
| expanding/ekˈspændɪŋ/ | 展开 | zhǎn kāi |
| factorising/ˈfæktəraɪzɪŋ/ | 因式分解 | yīn shì fēn jiě |
| difference of two squares/ˈdɪfrəns ɒv tuː skweəz/ | 平方差 | píng fāng chà |
The language of the unknown
- Al-Khwarizmi, a 9th-century Persian mathematician, wrote the first book on "al-jabr" — restoring and balancing. That's where "algebra 代数" comes from.
- He used words instead of letters. We use letters because they're faster — and because a single letter can stand for any number.
Algebra manipulation route · 代数变形路径
Follow expression work from collecting terms to solving. · 跟随表达式运算流程，从合并同类项到求解。
Variables 变量 and substitution
- A variable is a letter that stands for a number. Its value can change.
- To substitute 代入 means to replace a letter with a specific number.
- Example: $3x^2 - 2y$ at $x = 4$, $y = 5$: $\;3(4)^2 - 2(5) = 48 - 10 = 38$.
Square first, then multiply. $3x^2$ at $x = 4$ is $3 \times 16 = 48$, not $(3 \times 4)^2 = 144$. Indices apply only to the letter immediately before them.
Collecting like terms: add terms with the same letter
Find the value of 3x² − 2y when x = 4 and y = 5. · 求当 3x² − 2y 在 x = 4 且 y = 5.
3 × 4² − 2 × 5 = 3 × 16 − 10 = 48 − 10 = 38.
Collecting like terms 同类项
- Like terms have the same letter(s) raised to the same power(s).
- $2a^2 + 5a^2 = 7a^2$ (same letter, same power — collect them).
- $3x + 2y$ cannot be simplified — different letters.
Simplify 2a² + 5a² + 3a − a. What is the coefficient of a²? · 化简 2a² + 5a² + 3a − a。a² 的系数是多少？
2a² + 5a² = 7a²; 3a − a = 2a. The coefficient of a² is 7. · 2a² + 5a² = 7a²; 3a − a = 2a。a² 的系数是 7。
Expanding 展开 brackets
- Single bracket: $3(2x + 5) = 6x + 15$.
- Double brackets (FOIL): $(2x + 1)(x - 4) = 2x^2 - 8x + x - 4 = 2x^2 - 7x - 4$.
Quick check. $(x + 3)(x + 5) = x^2 + 5x + 3x + 15 = x^2 + 8x + 15$. The constant term is $3 \times 5 = 15$.
Expand (2x + 1)(x − 4) = 2x² + bx + c. What is the constant term c? · 展开 (2x + 1)(x − 4) = 2x² + bx + c。常数项 c 是多少？
(2x+1)(x−4) = 2x² − 8x + x − 4 = 2x² − 7x − 4, so c = −4. · (2x+1)(x−4) = 2x² − 8x + x − 4 = 2x² − 7x − 4，因此 c = −4。
(x + 3)(x + 5) = x² + 8x + 15.
x² + 5x + 3x + 15 = x² + 8x + 15. ✓
Factorising 因式分解
- Factorising is the reverse of expanding — write as a product.
- Common factor: $9x^2 + 15xy = 3x(3x + 5y)$.
- Extended quadratic factorisation: $x^2 + 5x + 6 = (x + 2)(x + 3)$.
- Extended difference of two squares 平方差: $9x^2 - 16 = (3x + 4)(3x - 4)$.
Factorising a difference of two squares ($a^2 - b^2$) is like comparing two square areas.
Factorise the difference of two squares 9x² − 16. · 因式分解两平方差 9x² − 16。
a² − b² = (a+b)(a−b), with a = 3x, b = 4. · a² − b² = (a+b)(a−b)，其中 a = 3x, b = 4。
Match each expression to its factorisation. · 将每个表达式与其因式分解形式匹配。
x²+7x+10: factors of 10 that add to 7 are 2 and 5. x²−9 = x²−3². 6x+9: HCF is 3. x²+4x+4 is a perfect square. · x²+7x+10：积为 10 且和为 7 的因子是 2 和 5。x²−9 = x²−3²。6x+9：最大公约数是 3。x²+4x+4 是完全平方式。
You've got it
- substitute numbers for letters; collect like terms
- expand = multiply out brackets; factorise = write as a product
- (Extended) $a^2 - b^2 = (a+b)(a-b)$ — the difference of two squares