Algebra in English
| English | Chinese | Pinyin |
|---|---|---|
| coefficient | 系数 | xì shù |
| constant | 常数 | cháng shù |
| term | 项 | xiàng |
| equation | 方程 | fāng chéng |
| identity | 恒等式 | héng děng shì |
| inequality | 不等式 | bù děng shì |
| expand | 展开 | zhǎn kāi |
| factorise | 因式分解 | yīn shì fēn jiě |
| discriminant | 判别式 | pàn bié shì |
| simultaneous equations | 联立方程 | lián lì fāng chéng |
A letter is a number you do not know yet
- $5x^2 - 3x + 7$ looks like a code. It is a sentence about a number.
- $5$ is the coefficient 系数 of $x^2$. $7$ is a constant 常数. Each piece is a term 项.
- Naming the parts is what lets a teacher, or a question, tell you which part to change.
In the expression 5x² − 3x + 7, what is 5 called?
A coefficient multiplies a term with a letter in it. The constant here is 7, which has no letter.
Equation, identity, inequality
- An equation 方程 says two expressions are equal, and you solve it for a value.
- An identity 恒等式 is true for every value: $(x+1)^2 = x^2 + 2x + 1$ always.
- An inequality 不等式 compares with $<$, $\le$, $>$, $\ge$, and its answer is a range.
(x + 1)² = x² + 2x + 1 is an identity, not an equation to solve.
It is true for every value of x, so there is nothing to solve — it is a fact about the algebra.
Expand and factorise
- To expand 展开 is to remove brackets: $(x+3)(x-5) = x^2 - 2x - 15$.
- To factorise 因式分解 is to put them back: $x^2 - 2x - 15 = (x-5)(x+3)$.
- They are the same fact read in two directions, and every quadratic method depends on it.
Expand (x + 3)(x − 5) and give the coefficient of x.
The x terms are −5x and +3x, giving −2x. The full expansion is x² − 2x − 15.
Solve $x^2 - 2x - 15 = 0$.
Factorise: $(x-5)(x+3) = 0$.
A product is zero only when one factor is zero, so $x = 5$ or $x = -3$.
Check the first: $25 - 10 - 15 = 0$ ✓. Checking costs ten seconds and catches a sign error, which is the error you are most likely to make.
One solution of x² − 2x − 15 = 0 is negative. What is it?
Factorising gives (x − 5)(x + 3) = 0, so x = 5 or x = −3.
When factorising fails
- Use the formula $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
- $b^2 - 4ac$ is the discriminant 判别式. Positive gives two solutions, zero gives one, negative gives none that are real.
- Simultaneous equations 联立方程 are two equations in two unknowns: eliminate one letter, solve, then substitute back.
Solve −2x > 6.
Dividing both sides by −2 reverses the sign. Test x = −4: −2(−4) = 8 > 6 ✓.
A quadratic has discriminant b² − 4ac = 0. How many real solutions does it have?
The ± in the formula adds and subtracts zero, so both roots are the same value.
An inequality reverses when you multiply or divide by a negative number. From $-2x > 6$ it follows that $x < -3$, not $x > -3$. Test one value from your answer in the original inequality — it takes five seconds and catches this every time.
Expanding and factorising are the same fact read in two directions. Every quadratic method — factorising, the formula, completing the square — is a way of getting from one direction to the other when the obvious route is blocked.