Branch of mathematics

Algebra

Expressions with letters, equations and inequalities, systems and quadratics — how to pin down what you do not know yet: a break-even point, a dimension, a dose.

Topics in this branch

Intermediate
Algebraic expressions
An algebraic expression is a piece of notation where letters stand beside numbers. Learn the variable, the term and the coefficient, collect like terms, multiply brackets out — two binomials included — factor out the common term and evaluate an expression.
Intermediate
Linear equations
A linear equation is an equality with the unknown in the first power. Learn the operations that keep equations equivalent, solve one step by step, check the result, rearrange a formula to make another quantity its subject, and handle an unknown in a denominator.
Advanced
Linear inequalities
A linear inequality has not one solution but a whole set of them. Learn the operations that keep the inequality sign as it is, memorise the single one that flips it, and learn to mark the solution set on a number line.
Advanced
Systems of equations
A system of equations puts two conditions on two unknowns at once. Learn substitution and elimination, see the solution as the point where two lines cross, and learn to recognise inconsistent and dependent systems.
Advanced
Quadratic equations
A quadratic equation has the unknown in the second power. Learn the general form, the discriminant, the formulas for the roots, the shortcuts for incomplete equations, Vieta formulas and how to read the solutions off a parabola.
Intermediate
Special products
Seven identities that turn multiplying out brackets into a single line — and the same seven read right to left, which is factoring. The square of a sum and of a difference, the difference of squares, the cubes and the sum and difference of cubes, proved on a drawing and used for mental arithmetic.
Advanced
Polynomials
A polynomial is a sum of monomials — and everything you do to one is an extension of ordinary arithmetic. Degree and coefficients, adding, subtracting and multiplying, dividing by a binomial with Horner’s scheme, the remainder and factor theorems, integer roots, factoring and biquadratic equations.
Advanced
Rational expressions and equations
A fraction with polynomials where the numbers used to be — and the same four operations as on ordinary fractions, only preceded by factoring. The domain and the excluded numbers, cancelling, multiplying and dividing, adding and subtracting, and rational equations in product form.
Advanced
Quadratic and rational inequalities
Solving an inequality means finding a whole set rather than a single number — and it always takes two steps: the zeros first, then the sign between them. The parabola and the sign of a quadratic, the three discriminant cases, the sign chart, and polynomial and rational inequalities with the trap of a zero denominator.
Advanced
Absolute value equations and inequalities
The modulus measures a distance — and that one sentence solves the whole lesson. The equation |x − a| = r asks for the points r away from a, a strict inequality gives an interval around a, and the other direction gives a union of two. Plus reading the centre and radius back, and the cases with no solution at all.
Advanced
Equations, inequalities and systems with a parameter
A parameter is not a second unknown but a setting — one letter that turns a single problem into a whole family of them. The linear equation and its degenerate case, the discriminant as a condition on the number of roots, Vieta’s formulas, an inequality that holds for every x, and a line meeting a parabola.

Why this branch is worth learning

Every topic here settles a real situation. One example from each lesson:

  • The electricity bill
    A bill is the expression 39.80 + 1.08u: a fixed charge plus 1.08 for every kilowatt-hour. At 210 kWh you pay 266.60, yet cutting usage to 105 kWh brings the bill down only to 153.20 — 57% of it, not half, because the 39.80 never moves.
    Algebraic expressions
  • The gym
    An unlimited monthly pass costs 129, a single visit 15. Solving 15x = 129 gives x = 8.6, so the pass only starts paying off on the ninth visit of the month — stop at eight and you spend 120 and come out cheaper without it.
    Linear equations
  • Online shopping
    The shop ships free above 199, your basket stands at 164.50 and a pack of coffee costs 12.90. From 164.50 + 12.90n ≥ 199 you get n ≥ 2.68 — and since nobody sells 2.68 packs, you add three and finish at 203.20.
    Linear inequalities
  • A café receipt
    The receipt shows totals only: 3 coffees and 2 pastries for 61, and the day before 2 coffees and 4 pastries for 66. Two equations in two unknowns split those sums into a coffee at 14 and a pastry at 9.50 — one receipt on its own never could.
    Systems of equations
  • The vegetable patch
    You have 38 m of fencing and want to enclose a rectangular patch of 84 m². The two sides add up to 19 m, so x(19 − x) = 84, that is x² − 19x + 84 = 0. The discriminant is 25 and the roots are 7 and 12 — one 7 × 12 m bed, described once by its width and once by its length.
    Quadratic equationsSee the lesson: Area of a rectangle
  • Mental arithmetic at the till
    The product 98 · 102 is (100 − 2)(100 + 2), so 10,000 − 4 = 9,996. In the same way 47 · 53 = 50² − 3² = 2,500 − 9 = 2,491. Any product of two numbers sitting symmetrically around a round hundred or fifty collapses into a single subtraction.
    Special products
  • A box folded from a sheet of card
    Cut squares of side x from the corners of a 30 × 20 cm sheet and fold the edges up. The volume is the polynomial V(x) = x(30 − 2x)(20 − 2x) = 4x³ − 100x² + 600x. At x = 4 cm that is 4 · 22 · 12 = 1,056 cm³, while x = 3 cm gives only 3 · 24 · 14 = 1,008 cm³ — one centimetre in the cut is almost 5% of the capacity.
    Polynomials
  • A round trip there and back
    The same route driven out at 60 km/h and back at 40 km/h averages 2 / (1/60 + 1/40) = 48 km/h, not 50 km/h. The slower leg takes longer and therefore weighs more in the average — and that only becomes visible once the two fractions are put over a common denominator.
    Rational expressions and equationsSee the lesson: Speed, distance and time
  • A thrown ball and time above a height
    A ball thrown straight up at 20 m/s is at height h(t) = 20t − 5t². Asking how long it stays above 15 m is the inequality 20t − 5t² ≥ 15, that is t² − 4t + 3 ≤ 0, whose solution is [1, 3]. The ball spends exactly 2 seconds above that height.
    Quadratic and rational inequalities
  • A parcel and its weight limit
    A carrier accepts parcels of 5 kg with a 200 g tolerance, which is the single condition |m − 5| ≤ 0.2 and means the range from 4.8 to 5.2 kg. A parcel weighing 5.25 kg falls into the next price band — over 50 grams nobody can see by eye.
    Absolute value equations and inequalities
  • A phone tariff and the price per minute
    Tariff A costs 30 a month plus m per minute, tariff B costs 50 plus 0.10 per minute. Equal cost means 30 + mt = 50 + 0.1t, that is (m − 0.1)t = 20. At m = 0.10 the coefficient of t vanishes and the equation has no solution: the tariffs differ by a flat 20 whatever the usage. At m = 0.15 they meet at t = 400 minutes.
    Equations, inequalities and systems with a parameter

Branch formulas

Branch: Algebra

Algebraic expressions

  • Collecting like terms

    ax+bx=(a+b)xax + bx = (a + b)x

    add the coefficients, the letter stays as it is

  • The distributive law

    a(b+c)=ab+aca(b + c) = ab + ac

    multiply every term inside the bracket

  • Distributing over a difference

    a(b−c)=ab−aca(b - c) = ab - ac

    the same rule, with a minus sign

  • A minus in front of a bracket

    −(b+c)=−b−c-(b + c) = -b - c

    flips the sign of every term inside

  • Multiplying two binomials

    (a+b)(c+d)=ac+ad+bc+bd(a + b)(c + d) = ac + ad + bc + bd

    every term of the first bracket by every term of the second

  • Factoring out the common term

    ab+ac=a(b+c)ab + ac = a(b + c)

    the distributive law read right to left

  • Adding vs. multiplying letters

    x⋅x=x2,x+x=2xx \cdot x = x^2, \quad x + x = 2x

    two different operations — do not mix them up

Linear equations

  • A linear equation in one unknown

    ax+b=c(a≠0)ax + b = c \quad (a \neq 0)

    the unknown appears in the first power

  • The solution

    x=c−bax = \frac{c - b}{a}

    subtract the constant, divide by the coefficient

  • Adding a number to both sides

    a=b  ⟺  a+c=b+ca = b \iff a + c = b + c

    an equivalent equation — same solutions

  • Multiplying both sides

    a=b  ⟺  a⋅c=b⋅c(c≠0)a = b \iff a \cdot c = b \cdot c \quad (c \neq 0)

    multiply and divide only by a non-zero number

  • Special cases

    0⋅x=0vs.0⋅x=c≠00 \cdot x = 0 \quad \text{vs.} \quad 0 \cdot x = c \neq 0

    an identity (every x) and a contradiction (no solutions)

  • Rearranging a formula

    s=vt  ⟺  t=sv(v≠0)s = vt \iff t = \frac{s}{v} \quad (v \neq 0)

    isolate one letter, treat the rest as numbers

  • An unknown in the denominator

    px+b=q  ⟺  p=q(x+b)(x≠−b)\frac{p}{x + b} = q \iff p = q(x + b) \quad (x \neq -b)

    state the domain first, multiply afterwards

Linear inequalities

  • A linear inequality

    ax+b>c(a≠0)ax + b > c \quad (a \neq 0)

    one of <, >, ≤, ≥ in place of the equals sign

  • Adding a number to both sides

    a>b  ⟺  a+c>b+ca > b \iff a + c > b + c

    the inequality sign stays as it is

  • Multiplying by a positive number

    a>b  ⟺  ac>bc(c>0)a > b \iff ac > bc \quad (c > 0)

    the inequality sign stays as it is

  • Multiplying by a negative number

    a>b  ⟺  ac<bc(c<0)a > b \iff ac < bc \quad (c < 0)

    the inequality sign FLIPS

  • Interval notation

    x>3  ⟺  x∈(3,∞)x > 3 \iff x \in (3, \infty)

    a round bracket — the endpoint is not included

  • A non-strict inequality

    x≥3  ⟺  x∈[3,∞)x \ge 3 \iff x \in [3, \infty)

    a square bracket — the endpoint is included

Systems of equations

  • A system of two linear equations

    {a1x+b1y=c1a2x+b2y=c2\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}

    the solution is a pair (x, y) satisfying both equations

  • Condition for a determinate system

    a1b2−a2b1≠0a_1 b_2 - a_2 b_1 \neq 0

    exactly one solution — the lines cross at a single point

  • Substitution method

    y=c1−a1xb1  ⟶  a2x+b2⋅c1−a1xb1=c2y = \frac{c_1 - a_1 x}{b_1} \;\longrightarrow\; a_2 x + b_2 \cdot \frac{c_1 - a_1 x}{b_1} = c_2

    express one unknown and put it into the other equation

  • Elimination method

    m⋅(a1x+b1y)+n⋅(a2x+b2y)=mc1+nc2m \cdot (a_1 x + b_1 y) + n \cdot (a_2 x + b_2 y) = m c_1 + n c_2

    choose m and n so that one unknown cancels out

  • Determinant formulas

    W=a1b2−a2b1,x=WxW,y=WyWW = a_1 b_2 - a_2 b_1, \quad x = \frac{W_x}{W}, \quad y = \frac{W_y}{W}

    W_x and W_y come from W by replacing the matching column with the constant terms

  • Inconsistent system

    a1a2=b1b2≠c1c2\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}

    parallel lines — no solutions; when all three ratios are equal the system is dependent

Quadratic equations

  • General form of a quadratic equation

    ax2+bx+c=0(a≠0)ax^2 + bx + c = 0 \quad (a \neq 0)

    the condition a ≠ 0 is what tells it from a linear one

  • The discriminant

    Δ=b2−4ac\Delta = b^2 - 4ac

    its sign decides how many roots there are

  • Roots when Δ > 0

    x1=−b−Δ2a,x2=−b+Δ2ax_1 = \frac{-b - \sqrt{\Delta}}{2a}, \quad x_2 = \frac{-b + \sqrt{\Delta}}{2a}

    two distinct real solutions

  • Root when Δ = 0

    x0=−b2ax_0 = \frac{-b}{2a}

    one (double) solution

  • No roots

    Δ<0  ⟹  x∈∅\Delta < 0 \implies x \in \varnothing

    the equation has no real solutions

  • Vieta formulas

    x1+x2=−ba,x1⋅x2=cax_1 + x_2 = -\frac{b}{a}, \quad x_1 \cdot x_2 = \frac{c}{a}

    the sum and product of the roots without computing them

  • Factored form

    ax2+bx+c=a(x−x1)(x−x2)ax^2 + bx + c = a(x - x_1)(x - x_2)

    it exists only when Δ ≥ 0

Special products

  • Square of a sum

    (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2

    the middle term is twice the product, not the product

  • Square of a difference

    (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2

    both squares positive, the minus sits on the middle term only

  • Difference of squares

    (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2

    the only one of these where the middle term disappears

  • Cube of a sum

    (a+b)3=a3+3a2b+3ab2+b3(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

    coefficients 1, 3, 3, 1

  • Cube of a difference

    (a−b)3=a3−3a2b+3ab2−b3(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3

    alternating signs, starting with a plus

  • Difference of cubes

    a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)

    a plus on the middle term of the bracket

  • Sum of cubes

    a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)

    a minus on the middle term of the bracket

Polynomials

  • General form

    W(x)=anxn+an−1xn−1+⋯+a1x+a0W(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0

    the degree n is the highest power with a non-zero coefficient

  • Degree of a product

    deg⁡(V⋅W)=deg⁡V+deg⁡W\deg (V \cdot W) = \deg V + \deg W

    adding can drop the degree, multiplying never does

  • Division with remainder

    W(x)=P(x)⋅Q(x)+R(x),deg⁡R<deg⁡PW(x) = P(x) \cdot Q(x) + R(x), \quad \deg R < \deg P

    the remainder has a lower degree than the divisor

  • Remainder theorem

    W(x)=(x−a)⋅Q(x)+W(a)W(x) = (x - a) \cdot Q(x) + W(a)

    the remainder is simply the value of the polynomial at a

  • Factor theorem

    (x−a)∣W(x)  ⟺  W(a)=0(x - a) \mid W(x) \iff W(a) = 0

    a root and a linear factor are the same piece of information

  • Integer roots

    p∣a0for every integer root pp \mid a_0 \quad \text{for every integer root } p

    they can only be divisors of the constant term

  • Biquadratic equation

    ax4+bx2+c=0 → t=x2  at2+bt+c=0ax^4 + bx^2 + c = 0 \ \xrightarrow{\ t = x^2\ } \ at^2 + bt + c = 0

    substitute t = x², then take square roots to get back to x

Rational expressions and equations

  • Domain

    D={x∈R:Q(x)≠0}D = \{ x \in \mathbb{R} : Q(x) \neq 0 \}

    the domain first, the arithmetic second

  • Cancelling

    P(x)⋅S(x)Q(x)⋅S(x)=P(x)Q(x)\frac{P(x) \cdot S(x)}{Q(x) \cdot S(x)} = \frac{P(x)}{Q(x)}

    only a common FACTOR cancels, never a term

  • Multiplying

    AB⋅CD=ACBD\frac{A}{B} \cdot \frac{C}{D} = \frac{AC}{BD}

    numerator by numerator, denominator by denominator

  • Dividing

    AB:CD=AB⋅DC\frac{A}{B} : \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C}

    multiply by the reciprocal; the condition C ≠ 0 joins the domain

  • Adding

    AB+CD=AD+CBBD\frac{A}{B} + \frac{C}{D} = \frac{AD + CB}{BD}

    the common denominator is the least common multiple, not always the product

  • Rational equation

    P(x)Q(x)=0  ⟺  P(x)=0 and Q(x)≠0\frac{P(x)}{Q(x)} = 0 \iff P(x) = 0 \ \text{and} \ Q(x) \neq 0

    the numerator vanishes — but only inside the domain

Quadratic and rational inequalities

  • Factored form

    ax2+bx+c=a(x−x1)(x−x2),Δ>0ax^2 + bx + c = a(x - x_1)(x - x_2), \quad \Delta > 0

    the starting point of every quadratic inequality

  • Sign of a quadratic

    Δ>0  ⟹  sign of a outside the roots, −a between them\Delta > 0 \implies \text{sign of } a \ \text{outside the roots, } -a \ \text{between them}

    the parabola crosses the axis twice

  • Zero discriminant

    Δ=0  ⟹  sign of a everywhere except x0\Delta = 0 \implies \text{sign of } a \ \text{everywhere except } x_0

    the parabola touches the axis at one point

  • Negative discriminant

    Δ<0  ⟹  sign of a for every x\Delta < 0 \implies \text{sign of } a \ \text{for every } x

    the parabola never meets the axis

  • Rational inequality

    P(x)Q(x)≥0  ⟺  P(x)⋅Q(x)≥0 and Q(x)≠0\frac{P(x)}{Q(x)} \ge 0 \iff P(x) \cdot Q(x) \ge 0 \ \text{and} \ Q(x) \neq 0

    multiply by the SQUARE of the denominator; the domain survives

Absolute value equations and inequalities

  • Equation with a modulus

    ∣x∣=c  ⟺  x=c or x=−c,c>0|x| = c \iff x = c \ \text{or} \ x = -c, \quad c > 0

    for c = 0 one solution, for c < 0 none

  • Distance equal to r

    ∣x−a∣=r  ⟺  x=a−r or x=a+r|x - a| = r \iff x = a - r \ \text{or} \ x = a + r

    two points r away from a

  • Closer than r

    ∣x−a∣<r  ⟺  x∈(a−r,a+r)|x - a| < r \iff x \in (a - r, a + r)

    one interval around the centre a

  • Further than r

    ∣x−a∣>r  ⟺  x∈(−∞,a−r)∪(a+r,+∞)|x - a| > r \iff x \in (-\infty, a - r) \cup (a + r, +\infty)

    a union of two intervals, never one

  • Modulus of a linear expression

    ∣ax+b∣=c  ⟺  ax+b=c or ax+b=−c|ax + b| = c \iff ax + b = c \ \text{or} \ ax + b = -c

    two branches, each solved on its own

  • Centre and radius of an interval

    a=p+q2,r=q−p2a = \frac{p + q}{2}, \qquad r = \frac{q - p}{2}

    writing the interval [p, q] as the single condition |x − a| ≤ r

Equations, inequalities and systems with a parameter

  • Linear equation with a parameter

    ax=b: a≠0  ⟹  x=ba; a=0,b≠0  ⟹  ∅; a=b=0  ⟹  x∈Rax = b: \ a \neq 0 \implies x = \tfrac{b}{a}; \ a = 0, b \neq 0 \implies \varnothing; \ a = b = 0 \implies x \in \mathbb{R}

    three cases, settled by the coefficient of the unknown

  • Degree condition

    ax2+bx+c=0 is quadratic  ⟺  a≠0ax^2 + bx + c = 0 \ \text{is quadratic} \iff a \neq 0

    the first question in every parameter problem with an x² in it

  • Discriminant and the number of roots

    Δ>0  ⟹  two roots,Δ=0  ⟹  one,Δ<0  ⟹  none\Delta > 0 \implies \text{two roots}, \quad \Delta = 0 \implies \text{one}, \quad \Delta < 0 \implies \text{none}

    the condition on the parameter comes out of an inequality on the discriminant

  • Vieta’s formulas

    x1+x2=−ba,x1⋅x2=cax_1 + x_2 = -\frac{b}{a}, \qquad x_1 \cdot x_2 = \frac{c}{a}

    a condition on the roots without computing the roots

  • An always-true inequality

    ax2+bx+c>0 for every x  ⟺  a>0 and Δ<0ax^2 + bx + c > 0 \ \text{for every } x \iff a > 0 \ \text{and} \ \Delta < 0

    both conditions at once — the discriminant alone is not enough

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11 min