Algebra in English · 用英语学代数
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| coefficient/ˌkəʊɪˈfɪʃənt/ | 系数 | xì shù |
| constant/ˈkɒnstənt/ | 常数 | cháng shù |
| term/tɜːm/ | 项 | xiàng |
| inequality/ɪniːˈkwɒlɪti/ | 不等式 | bù děng shì |
| equation/ɪˈkweɪʒn/ | 方程 | fāng chéng |
| identity/aɪˈdentɪti/ | 恒等式 | héng děng shì |
| expand/ekˈspænd/ | 展开 | zhǎn kāi |
| factorise/ˈfæktəraɪz/ | 因式分解 | yīn shì fēn jiě |
| discriminant/dɪˈskrɪmɪnənt/ | 判别式 | pàn bié shì |
| simultaneous equations/ˌsɪməlˈteɪnɪəs ɪˈkweɪʒnz/ | 联立方程 | lián lì fāng chéng |
Represent an unknown relationship
- In $5x^2-3x+7$, 5 is the coefficient 系数 of $x^2$, 7 is a constant 常数, and each separated piece is a term 项.
- A variable can represent many possible values, not just a number waiting to be found. Translate the stated relationship before selecting an operation.
In the expression 5x² − 3x + 7, what is 5 called? · 在表达式 5x² − 3x + 7 中,5 叫什么?
A coefficient multiplies a term with a letter in it. The constant here is 7, which has no letter. · 系数是乘在带字母的项前面的数。这里的常数是 7,它不含字母。
Equality and inequality have different solutions
- An equation 方程 states equality; an identity 恒等式 holds for every allowed value in its domain. An inequality 不等式 uses a comparison and may have a range of solutions.
- Add or subtract the same amount on both sides. Multiplying or dividing an inequality by a negative number reverses its sign; dividing by a possibly zero expression needs separate care.
(x + 1)² = x² + 2x + 1 is an identity, not an equation to solve. · (x + 1)² = x² + 2x + 1 是恒等式,而不是待解的方程。
It is true for every value of x, so there is nothing to solve — it is a fact about the algebra. · 它对每一个 x 都成立,因此无需求解——它是关于代数本身的一个事实。
Expand and factorise
- To expand 展开, distribute multiplication across every bracket term. To factorise 因式分解, express a sum as a product.
- A zero product has at least one zero factor. From $(x-4)(x+3)=0$, obtain $x=4$ or $x=-3$ and check both in the original equation.
Expand (x + 3)(x − 5) and give the coefficient of x. · 展开 (x + 3)(x − 5),写出 x 的系数。
The x terms are −5x and +3x, giving −2x. The full expansion is x² − 2x − 15. · x 的项是 −5x 与 +3x,合并得 −2x。完整展开式为 x² − 2x − 15。
Choose a method and check every condition
- For $ax^2+bx+c=0$ with $a\ne0$, use factorisation or $x=(-b\pm\sqrt{b^2-4ac})/(2a)$. The discriminant 判别式 is $b^2-4ac$: positive, zero and negative give two, one and no distinct real roots respectively.
- Simultaneous equations 联立方程 require a solution satisfying all equations. Elimination or substitution can reveal one solution, no solution or infinitely many; check the original pair.
One solution of x² − 2x − 15 = 0 is negative. What is it? · x² − 2x − 15 = 0 有一个负数解,它是多少?
Factorising gives (x − 5)(x + 3) = 0, so x = 5 or x = −3. · 因式分解得 (x − 5)(x + 3) = 0,所以 x = 5 或 x = −3。
Balance two ticket purchases. $3a+2s=46$ and $2a+4s=44$. Doubling the first equation gives $6a+4s=92$. Subtraction gives $4a=48$, so $a=12$ and $s=5$. Both original totals are satisfied.
Solve −2x > 6. · 解不等式 −2x > 6。
Dividing both sides by −2 reverses the sign. Test x = −4: −2(−4) = 8 > 6 ✓. · 两边同除以 −2 时不等号要反向。代入 x = −4 检验:−2(−4) = 8 > 6 ✓。
A quadratic has discriminant b² − 4ac = 0. How many real solutions does it have? · 某个二次方程的判别式 b² − 4ac = 0,它有几个实数解?
The ± in the formula adds and subtracts zero, so both roots are the same value. · 公式中的 ± 加减的都是零,因此两个根是同一个值。
Interpret roots after solving. A rectangle with width x and length $x+3$ can give roots 5 and negative 8. Only positive dimensions fit the physical problem. Keep both algebraic roots before explaining the selection; practise this on sheet 1.2.
For $4-2x\leq10$, subtract 4 to get $-2x\leq6$, then divide by negative 2 to get $x\geq-3$. The boundary is included. A single trial value can expose a mistake but does not prove an entire solution set.