Algebra basics and manipulation
Introduced| English | Meaning |
|---|---|
| algebra/ˈældʒɪbrə/ | Mathematics that uses symbols to represent quantities and express relationships between them. |
| variable/ˈveərɪəbl/ | A quantity or named value that can change or take different values. |
| substitute/ˈsʌbstɪtjuːt/ | To replace a variable or expression with a specified value or another expression. |
| like terms/laɪk tɜːmz/ | Terms with exactly the same variable parts, which can be collected: 3x + 4x = 7x. |
| expanding/ekˈspændɪŋ/ | expanding |
| factorising/ˈfæktəraɪzɪŋ/ | The reverse of expanding — write as a product. |
| difference of two squares/ˈdɪfrəns ɒv tuː skweəz/ | The pattern a² − b² = (a − b)(a + b). |
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
Follow expression work from collecting terms to solving.
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.
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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?
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2a² + 5a² = 7a²; 3a − a = 2a. The coefficient of a² is 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?
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(2x+1)(x−4) = 2x² − 8x + x − 4 = 2x² − 7x − 4, so c = −4.
(x + 3)(x + 5) = x² + 8x + 15.
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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.
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a² − b² = (a+b)(a−b), with a = 3x, b = 4.
Match each expression to its factorisation.
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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.
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
Write from a value of 14
- Find 2 x squared minus y when x is 3 and y is 4.
- Square before multiplying.
Find the value of 2 x squared minus y when x equals 3 and y equals 4.
18 minus 4 is 14.
What is the value?
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2 times 9 minus 4 is 14.