Branch of mathematics

Algebra

Expressions with letters, equations and inequalities, systems and quadratics — the language of relationships, not just arithmetic.

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 and evaluate an expression for a given number.
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, and learn to spot contradictory and identity equations.
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.

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

  • 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)

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

Related articles

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Algebra

What x really means — unknown, variable and parameter

The letter x does not always mean the same thing. Sometimes it is a number you are hunting for, sometimes a quantity that varies continuously, sometimes a blank waiting for any number at all. Here is where algebraic notation came from, why it took two thousand years to arrive, what exactly separates an unknown from a variable and a parameter, and why an inequality flips when you multiply by a negative number.

11 min