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Algebra Formula Sheet

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Formula Sheet • Key Rules & Equations

Algebra Formula Sheet

Every key algebra formula you need — expanding brackets, factorising, solving quadratic & simultaneous equations, sequences, and the 6 index laws.

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Expanding Brackets

Single Bracket (Distributive)
a(b+c)=ab+aca(b + c) = ab + ac

Multiply the outer term a across every term inside the bracket.

Worked Example
3(x+4)=3(x)+3(4)=3x+12\displaystyle 3(x + 4) = 3(x) + 3(4) = 3x + 12
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Double Brackets (FOIL)
(a+b)(c+d)=ac+ad+bc+bd(a+b)(c+d) = ac + ad + bc + bd

Multiply First, Outside, Inside, and Last terms.

Worked Example
(x+2)(x+3)=x2+3x+2x+6=x2+5x+6\displaystyle (x+2)(x+3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6
Equation Solver →
Perfect Square (Addition)
(a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2

Square the first, double the product of both terms, square the last.

Worked Example
(x+3)2=x2+2(x)(3)+32=x2+6x+9\displaystyle (x + 3)^2 = x^2 + 2(x)(3) + 3^2 = x^2 + 6x + 9
Equation Solver →
Perfect Square (Subtraction)
(a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2

Identical to addition, but the middle term is subtracted (−2ab).

Worked Example
(x−5)2=x2−2(x)(5)+52=x2−10x+25\displaystyle (x - 5)^2 = x^2 - 2(x)(5) + 5^2 = x^2 - 10x + 25
Equation Solver →
Difference of Two Squares (DOTS)
(a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2

Cross-terms cancel exactly: +ab − ab = 0, leaving a² − b².

Worked Example
(x+5)(x−5)=x2−52=x2−25\displaystyle (x + 5)(x - 5) = x^2 - 5^2 = x^2 - 25
Equation Solver →

Factorising

Highest Common Factor (HCF)
ab+ac=a(b+c)ab + ac = a(b + c)

Find the highest common numeric/algebraic factor and extract outside.

Worked Example
6x2+9x=3x(2x+3)\displaystyle 6x^2 + 9x = 3x(2x + 3)
Equation Solver →
Simple Quadratic (a = 1)
x2+(p+q)x+pq=(x+p)(x+q)x^2 + (p+q)x + pq = (x+p)(x+q)

Find two numbers p and q that add to middle term and multiply to constant.

Worked Example
x2+5x+6=(x+2)(x+3)\displaystyle x^2 + 5x + 6 = (x + 2)(x + 3)
Equation Solver →
Completing the Square
x2+bx=(x+b2)2−(b2)2x^2 + bx = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2

Halve coefficient b, put inside square, and subtract (b/2)².

Worked Example
x2+6x=(x+3)2−32=(x+3)2−9\displaystyle x^2 + 6x = (x + 3)^2 - 3^2 = (x + 3)^2 - 9
Quadratic Calculator →

Solving Equations

Linear Equation
ax+b=c  ⟹  x=c−baax + b = c \implies x = \frac{c - b}{a}

Isolate x by balancing inverse operations: subtract b, then divide by a.

Worked Example
2x+3=11  ⟹  x=11−32=4\displaystyle 2x + 3 = 11 \implies x = \frac{11 - 3}{2} = 4
Equation Solver →
Quadratic Formula
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Solves any quadratic equation in standard form ax² + bx + c = 0.

Worked Example
x2−5x+6=0  ⟹  x=5±25−242=3 or 2\displaystyle x^2 - 5x + 6 = 0 \implies x = \frac{5 \pm \sqrt{25 - 24}}{2} = 3 \text{ or } 2
Quadratic Calculator →
The Discriminant (Root Nature)
Δ=b2−4ac\Delta = b^2 - 4ac

Δ > 0: two real roots • Δ = 0: one repeated root • Δ < 0: no real roots

Worked Example
x2+4x+5=0  ⟹  42−4(1)(5)=−4<0\displaystyle x^2 + 4x + 5 = 0 \implies 4^2 - 4(1)(5) = -4 < 0
Quadratic Calculator →
Simultaneous Equations (Elimination)
{a1x+b1y=c1a2x+b2y=c2\begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}

Scale equations so coefficients match; add or subtract to eliminate one variable.

Worked Example
x+y=10,  x−y=4  ⟹  2x=14  ⟹  x=7,  y=3\displaystyle x + y = 10, \; x - y = 4 \implies 2x = 14 \implies x = 7, \; y = 3
Simultaneous Equations Solver →

Sequences & Series

Arithmetic nth Term
un=a+(n−1)du_n = a + (n - 1)d

a = first term · d = common difference · n = term index

Worked Example
3,7,11,15⋯  ⟹  un=3+(n−1)4=4n−1\displaystyle 3, 7, 11, 15\dots \implies u_n = 3 + (n-1)4 = 4n - 1
Nth Term Calculator →
Geometric nth Term
un=a rn−1u_n = a \, r^{n - 1}

a = first term · r = common multiplier ratio · n = term index

Worked Example
2,6,18,54⋯  ⟹  un=2×3n−1\displaystyle 2, 6, 18, 54\dots \implies u_n = 2 \times 3^{n-1}
Nth Term Calculator →
Arithmetic Series Sum (Sn)
Sn=n2(2a+(n−1)d)S_n = \frac{n}{2}\big(2a + (n-1)d\big)

Sum of the first n terms in an arithmetic progression.

Worked Example
Sum of 10 terms of 3,7,11⋯=102(6+36)=210\displaystyle \text{Sum of 10 terms of } 3,7,11\dots = \frac{10}{2}(6 + 36) = 210
Nth Term Calculator →

The 6 Index Laws

Product Rule
am×an=am+na^m \times a^n = a^{m + n}

When multiplying identical bases, add the exponents.

Worked Example
x3×x4=x3+4=x7\displaystyle x^3 \times x^4 = x^{3 + 4} = x^7
Quotient Rule
aman=am−n\frac{a^m}{a^n} = a^{m - n}

When dividing identical bases, subtract the bottom exponent from top.

Worked Example
x6x2=x6−2=x4\displaystyle \frac{x^6}{x^2} = x^{6 - 2} = x^4
Power of a Power
(am)n=am×n(a^m)^n = a^{m \times n}

When raising an existing power to an outer power, multiply exponents.

Worked Example
(x3)4=x3×4=x12\displaystyle (x^3)^4 = x^{3 \times 4} = x^{12}
Zero Exponent
a0=1(a≠0)a^0 = 1 \quad (a \neq 0)

Any non-zero base raised to the power of zero equals strictly 1.

Worked Example
70=1,(3x2)0=1\displaystyle 7^0 = 1, \quad (3x^2)^0 = 1
Negative Exponent
a−n=1ana^{-n} = \frac{1}{a^n}

A negative power creates the reciprocal of the positive power.

Worked Example
2−3=123=18\displaystyle 2^{-3} = \frac{1}{2^3} = \frac{1}{8}
Fractional Exponent
amn=amn=(an)ma^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m

Denominator n is root degree; numerator m is power.

Worked Example
823=(83)2=22=4\displaystyle 8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 2^2 = 4
Curriculum & Revision Note

About These Algebra Formulas

Algebra is the backbone of secondary school and pre-university maths. This formula sheet covers expanding single and double brackets, factorising, arithmetic and geometric sequences, completing the square, the quadratic formula, and the 6 core index laws.

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