Appearance
2.1 — What Algebra Actually Is
A shopkeeper says: "I sold some shirts at ₹450 each, took ₹200 off the total as a festival discount, and collected ₹3,400." How many shirts?
You could guess. Seven shirts is ₹3,150 minus 200, ₹2,950 — too low. Eight is ₹3,600 minus 200, ₹3,400 — got it. Guessing worked because the numbers were small.
Algebra is what you do instead of guessing. You give the unknown a name, write down what you know about it as a sentence, and then rearrange that sentence until the unknown is alone on one side. The answer falls out. Nothing is guessed, and the method works whether the answer is 8 or 8,000,000.
450x - 200 = 3400
That is the shopkeeper's sentence in symbols. The rest of this chapter is about what such sentences mean and how to rearrange them safely.
1. Where the word comes from
Around 820 CE in Baghdad, al-Khwārizmī — the same man whose name became "algorithm" in Chapter 1.1 — wrote al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala, "The Compendious Book on Calculation by Restoration and Balancing".
Al-jabr, restoration, is the operation of moving a subtracted term to the other side: turning x - 5 = 3 into x = 3 + 5. Al-muqābala, balancing, is cancelling equal amounts from both sides. Those two moves are still the whole toolkit, and the first of them gave Europe the word algebra.
The word also travelled into Spanish as algebrista, meaning a bone-setter — someone who restores broken bones — and barbers in medieval Spain who set bones advertised themselves that way. Restoration is restoration.
2. A letter is not a mystery, it is a placeholder
The single obstacle that stops people learning algebra is the feeling that letters are strange. They are not. A letter is a box waiting to be filled, and it does three quite different jobs depending on the sentence it sits in.
An unknown. In 450x - 200 = 3400, the letter x stands for one specific number that we do not yet know. The job is to find it. There is one right answer.
A variable. In A = \pi r^2, the letter r is not one hidden number — it is any radius you like, and A changes as r changes. Nothing is being solved; a relationship is being described.
A parameter or a constant. In y = mx + c, the letters m and c hold fixed but unspecified values that define which particular line you mean, while x and y move along it.
The same symbol can play different roles in different sentences, and knowing which role it plays is most of the skill. When a problem confuses you, ask first: is this letter one number I am hunting, or a slot that anything can go into?
3. Expressions, equations, identities
Three things that look similar and are not.
An expression is a phrase, not a sentence: 3x + 7. It has no verb, claims nothing, and cannot be true or false. You can simplify it or evaluate it at a particular x, but you cannot solve it. "Solve 3x + 7" is not a meaningful instruction, and asking students to do it is one of the small cruelties of maths teaching.
An equation is a sentence with a verb: 3x + 7 = 22. It claims two things are equal. It is true for some values of x and false for others, and solving it means finding which.
An identity is an equation true for every value: (a+b)^2 = a^2 + 2ab + b^2. There is nothing to solve; it is a fact about how the notation works. Some books write \equiv for identities, borrowing the symbol from Chapter 1.6.
The distinction matters because it tells you what question you are being asked. x^2 = 4 has two solutions. x^2 - 4 = (x-2)(x+2) has infinitely many, meaning all of them, meaning it is an identity.
4. The balance: the one rule of solving
An equation is a balance scale. The two sides weigh the same. Anything you do to one side you must do to the other, and the balance still holds.
The legal moves:
- Add the same thing to both sides.
- Subtract the same thing from both sides.
- Multiply both sides by the same thing — unless it is zero.
- Divide both sides by the same thing — unless it is zero.
Why the zero exceptions matter. Multiplying both sides by zero turns any equation into 0 = 0, true but useless — you have destroyed the information. Dividing by zero is meaningless, as Chapter 1.2 settled. And there is a subtler trap: dividing both sides by an expression that might be zero silently throws away solutions.
x^2 = 5x
Divide both sides by x and you get x = 5. But x = 0 also satisfies the original equation, and you have just lost it by dividing by something that was zero. The safe move is always to bring everything to one side and factor:
x^2 - 5x = 0 \quad\Rightarrow\quad x(x - 5) = 0 \quad\Rightarrow\quad x = 0 \text{ or } x = 5
Both solutions survive. This is worth internalising now, because the same mistake, made inside a longer calculation, is nearly invisible.
Solving the shopkeeper's equation
450x - 200 = 3400
Add 200 to both sides — al-jabr, restoration:
450x = 3600
Divide both sides by 450, which is safe because 450 is not zero:
x = 8
Eight shirts. Now check by substituting back: 450 \times 8 - 200 = 3600 - 200 = 3400. Correct.
Always check by substitution. It costs ten seconds and catches every arithmetic slip, and it is the one habit that separates people who trust their algebra from people who do not.
5. The distributive law is the whole of expanding brackets
Chapter 1.1 stated it: a(b+c) = ab + ac. Every bracket expansion in algebra is that law, applied once or repeatedly.
3(x + 4) = 3x + 12
Two brackets means applying it twice. Treat the first bracket as a single lump A = (a+b):
(a+b)(c+d) = A(c+d) = Ac + Ad = (a+b)c + (a+b)d = ac + bc + ad + bd
Every term in the first bracket meets every term in the second. School mnemonics like FOIL name the four products, but the mnemonic only works for two-term brackets and then fails; the rule "everything meets everything" always works.
The three patterns worth recognising instantly, because they appear constantly:
(a+b)^2 = a^2 + 2ab + b^2
(a-b)^2 = a^2 - 2ab + b^2
(a+b)(a-b) = a^2 - b^2
The middle term 2ab in the first is the one people drop. (a+b)^2 is not a^2 + b^2 — check with a = 3, b = 4: the left is 49, the right is 25. The missing 2ab = 24 is the difference.
The third pattern, the difference of two squares, is the most useful of the three because it runs backwards so easily. It also gives you mental arithmetic tricks: 97 \times 103 = (100-3)(100+3) = 10000 - 9 = 9991, computed in your head.
A picture for why (a+b)^2 = a^2 + 2ab + b^2. Draw a square of side a+b. Cut it with one horizontal and one vertical line at distance a from the corner. You get four pieces: an a \times a square, a b \times b square, and two a \times b rectangles. The areas must add to the whole. That is the identity, and it is why the operation is called squaring.
6. Factoring: running the machine backwards
Expanding is mechanical. Factoring — writing an expression as a product — is the reverse, and it is harder because you are searching rather than executing.
It matters for one specific reason, the zero product property:
\text{if } AB = 0 \text{ then } A = 0 \text{ or } B = 0
This is true for numbers and it is the only reason factoring is worth doing. It converts a hard question ("for which x does this whole expression equal zero?") into several easy ones ("for which x does this small factor equal zero?").
Note that it works only against zero. If AB = 12 you learn nothing useful, because 12 has many factorisations. So the first step of solving any equation by factoring is always to move everything to one side so the other side is zero.
The techniques, in the order you should try them:
Common factor first, always. 6x^2 + 9x = 3x(2x + 3).
Difference of squares. x^2 - 49 = (x-7)(x+7). Also 16y^2 - 25 = (4y-5)(4y+5).
Simple trinomials. For x^2 + bx + c, find two numbers that multiply to c and add to b. For x^2 + 7x + 12: which pair multiplies to 12 and adds to 7? Four and three. So (x+4)(x+3).
For x^2 - x - 12: multiply to -12, add to -1. The pair is -4 and 3. So (x-4)(x+3).
Grouping, when there are four terms:
x^3 + 2x^2 + 3x + 6 = x^2(x+2) + 3(x+2) = (x^2+3)(x+2)
Take a common factor out of each pair, notice the same bracket appears in both, and pull that out too.
Chapter 2.3 gives the method that always works for quadratics when guessing fails.
7. Rearranging formulas, which is the same skill wearing work clothes
You will use this far more often than you will solve for x. A formula relates several quantities and you want a different one on the left.
Make r the subject of A = \pi r^2. Divide both sides by \pi, then take the square root:
r = \sqrt{\frac{A}{\pi}}
Only the positive root, since a radius cannot be negative — a piece of judgement the algebra cannot supply and you must.
Make C the subject of F = \frac{9}{5}C + 32.
F - 32 = \frac{9}{5}C \quad\Rightarrow\quad C = \frac{5}{9}(F - 32)
Make t the subject of A = P(1+r)^t, the compound interest formula of Chapter 1.3. The unknown is stuck in an exponent, so we need the logarithm from Chapter 1.4 to bring it down:
\frac{A}{P} = (1+r)^t
\ln\!\left(\frac{A}{P}\right) = t \ln(1+r)
t = \frac{\ln(A/P)}{\ln(1+r)}
This is why logarithms exist as a working tool, not just a scale for earthquakes: they are how you solve for a variable that sits in an exponent. How long to double your money at 8%? t = \frac{\ln 2}{\ln 1.08} = \frac{0.693}{0.0770} = 9.0 years — the rule of 72, now derived rather than quoted.
8. Word problems: the translation table
The hardest part of a word problem is never the algebra. It is turning English into symbols. Here is the translation, and the discipline is to do it one phrase at a time rather than trying to see the whole answer at once.
| English | Symbols |
|---|---|
| is, gives, results in | = |
| of (with a fraction or percent) | \times |
| more than, increased by | + |
| less than, reduced by | - |
| per, for each | \div or a rate |
| twice, thrice | 2\times, 3\times |
Watch the order trap: "5 less than x" is x - 5, not 5 - x. English puts the smaller quantity first; algebra does not.
Worked example. A father is three times as old as his son. In 12 years he will be twice as old. How old are they now?
Name the unknown, and name the smaller one to keep the numbers clean. Let the son's age be s. Then the father's age is 3s.
In 12 years the son is s + 12 and the father is 3s + 12. The condition:
3s + 12 = 2(s + 12)
3s + 12 = 2s + 24
s = 12
The son is 12 and the father is 36. Check: in 12 years they will be 24 and 48, and 48 is twice 24. Correct.
Worked example, a rate problem. A boat travels 30 km downstream in 2 hours and returns upstream in 3 hours. Find the boat's speed in still water and the current's speed.
Let b be the boat's speed and c the current's. Downstream the current helps, so the effective speed is b+c; upstream it hinders, so it is b-c. Speed is distance over time:
b + c = \frac{30}{2} = 15, \qquad b - c = \frac{30}{3} = 10
Add the two equations and the c terms cancel: 2b = 25, so b = 12.5\ \text{km/h}. Then c = 2.5\ \text{km/h}.
That trick — adding two equations so one unknown vanishes — is the method Chapter 2.4 develops in general, and Chapter 4.3 turns into a mechanical procedure that works for a hundred equations at once.
Every formula above, built from scratch
None of the results in this chapter are worth memorising, because each one can be rebuilt in under a minute from something simpler. What follows is that rebuilding, one result at a time, so the formula and the reason for it sit on the same page as the explanation that needed them.
The square of a sum, seen as an actual square
(a+b)^2 = a^2 + 2ab + b^2
Read it aloud. "a plus b, all squared, equals a squared plus two a b plus b squared."
Where it comes from, by algebra. (a+b)^2 means (a+b)(a+b). Multiply every term in the first bracket by every term in the second:
(a+b)(a+b) = a\cdot a + a\cdot b + b\cdot a + b\cdot b = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2
The 2ab appears because there are two ways to pick one a and one b — once with the a from the front bracket, once with the a from the back.
Where it comes from, by picture. Draw a square of side a+b and cut it at the point that divides the side into a and b. You get four pieces: a square of area a^2, a square of area b^2, and two identical rectangles each of area ab.
The mistake this kills. (a+b)^2 \ne a^2 + b^2. The two ab rectangles are real area; dropping them is dropping most of the square. With a=3, b=4: the truth is 49, the mistake gives 25, and the missing 24 is exactly 2ab.
Changing the sign of b gives the companion identity for free, since (-b)^2 = b^2:
(a-b)^2 = a^2 - 2ab + b^2
The difference of two squares
a^2 - b^2 = (a+b)(a-b)
Where it comes from. Expand the right-hand side and watch the middle collapse:
(a+b)(a-b) = a^2 - ab + ba - b^2 = a^2 - b^2
The cross terms -ab and +ba are equal in size and opposite in sign, so they annihilate. That cancellation is the entire content of the identity.
Why it is the most useful identity in arithmetic. It turns a subtraction into a multiplication, and multiplication of nearby numbers is easy. To compute 97 \times 103, notice both sit either side of 100:
97 \times 103 = (100-3)(100+3) = 100^2 - 3^2 = 10000 - 9 = 9991
That is a two-second mental calculation. The same identity is how you rationalise a denominator:
\frac{1}{\sqrt5 - \sqrt3} = \frac{1}{\sqrt5-\sqrt3} \times \frac{\sqrt5+\sqrt3}{\sqrt5+\sqrt3} = \frac{\sqrt5+\sqrt3}{5 - 3} = \frac{\sqrt5+\sqrt3}{2}
Multiplying by the conjugate — the same expression with the middle sign flipped — clears the roots from the bottom, because the cross terms cancel and only squares survive.
Exponential and logarithmic equations
The whole method is one rule: to free a variable, apply the operation that undoes what is trapping it.
b^x = y \;\Leftrightarrow\; x = \log_b y \qquad\qquad e^x = y \;\Leftrightarrow\; x = \ln y
Two standard shapes:
a\,b^{kx} = c \quad \Rightarrow \quad x = \frac{\ln(c/a)}{k\ln b}
Derivation. Divide by a to isolate the power: b^{kx} = c/a. Take \ln of both sides: kx\ln b = \ln(c/a). Divide by k \ln b.
\log_b(x) + \log_b(x - 3) = 1 \quad \Rightarrow \quad \log_b\big(x(x-3)\big) = 1 \quad \Rightarrow \quad x(x-3) = b
The product law folds two logarithms into one, after which the definition of a logarithm removes it entirely and leaves a quadratic. Always check the answers: a solution that makes any logarithm's input zero or negative must be thrown away, because such a logarithm does not exist. This is the one place in algebra where a correct calculation can produce a wrong answer.
9. Where this shows up in your life
Every spreadsheet formula you have ever written is an algebraic expression with cell references instead of letters. =B2*1.18 is 1.18x.
Every unit conversion. Rearranging a formula for a different subject is exactly Section 7.
Every loan calculation. Working out how long a debt takes to clear at a given payment is solving for an exponent, needing logarithms.
Every line of code with a variable in it. A programming variable is closer to algebra's variable than to its unknown — it holds whatever you put in it. But x = x + 1, which is nonsense as an equation, is perfectly sensible as an instruction. That mismatch is why some languages use := for assignment and reserve = for equality, and it is a genuine source of bugs when a language uses = for both.
We have solved equations where the unknown appears to the first power. The moment it appears squared, the balance method alone stops being enough. The next chapter builds the machinery, and the one after that solves the squared case completely.