Algebra
🔒 Log in to trackhigh importance~3 Q in Tier 143 formulas⚡ 21 shortcuts6 subtopics
Every formula in this topic, grouped by subtopic. Print it and pin it above your desk.
Basic algebraic identities
Square of sum / difference
(a\pm b)^2=a^2\pm 2ab+b^2
Difference of squares
a^2-b^2=(a+b)(a-b)
Sum & difference of the two squares
(a+b)^2+(a-b)^2=2(a^2+b^2),\quad (a+b)^2-(a-b)^2=4ab
Cube of sum
(a+b)^3=a^3+b^3+3ab(a+b)
Cube of difference
(a-b)^3=a^3-b^3-3ab(a-b)
Sum of cubes
a^3+b^3=(a+b)(a^2-ab+b^2)=(a+b)^3-3ab(a+b)
Difference of cubes
a^3-b^3=(a-b)(a^2+ab+b^2)=(a-b)^3+3ab(a-b)
Square of trinomial
(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)
Fourth-power factorisation
a^4+a^2b^2+b^4=(a^2+ab+b^2)(a^2-ab+b^2)
Fourth powers from sum and product
a^4+b^4=(a^2+b^2)^2-2a^2b^2
$x+\frac{1}{x}$ type expressions
Square
x^2+\frac{1}{x^2}=\left(x+\frac1x\right)^2-2=\left(x-\frac1x\right)^2+2
Cube (sum)
x^3+\frac{1}{x^3}=\left(x+\frac1x\right)^3-3\left(x+\frac1x\right)
if $x+\frac1x=k$ then it is $k^3-3k$
Cube (difference)
x^3-\frac{1}{x^3}=\left(x-\frac1x\right)^3+3\left(x-\frac1x\right)
Fourth power
x^4+\frac{1}{x^4}=\left(x^2+\frac{1}{x^2}\right)^2-2
Fifth power
x^5+\frac{1}{x^5}=\left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)-\left(x+\frac1x\right)
Link between sum and difference
\left(x+\frac1x\right)^2-\left(x-\frac1x\right)^2=4
Quadratic to reciprocal form
ax^2-bx+a=0\ \Rightarrow\ x+\frac1x=\frac ba
Mixed form
\left(px+\frac{1}{qx}\right)^2=p^2x^2+\frac{1}{q^2x^2}+\frac{2p}{q}
$a^3+b^3+c^3-3abc$ and conditional identities
Main identity
a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)
Half form
a^3+b^3+c^3-3abc=\tfrac12(a+b+c)\left[(a-b)^2+(b-c)^2+(c-a)^2\right]
Zero-sum case
a+b+c=0\ \Rightarrow\ a^3+b^3+c^3=3abc
Equal-variables case
a^2+b^2+c^2=ab+bc+ca\ \Rightarrow\ a=b=c
Pairwise products
ab+bc+ca=\frac{(a+b+c)^2-(a^2+b^2+c^2)}{2}
Cyclic differences
(a-b)^3+(b-c)^3+(c-a)^3=3(a-b)(b-c)(c-a)
Surds: rationalisation and square roots of surds
Rationalising
\frac{1}{\sqrt a+\sqrt b}=\frac{\sqrt a-\sqrt b}{a-b}
Conjugate product
(\sqrt a+\sqrt b)(\sqrt a-\sqrt b)=a-b
Square root of a surd
\sqrt{a+2\sqrt b}=\sqrt m+\sqrt n,\ \text{where } m+n=a,\ mn=b
Square root (minus)
\sqrt{a-2\sqrt b}=\sqrt m-\sqrt n\ \ (m>n)
Product-1 pair
x=p+\sqrt q,\ p^2-q=1\ \Rightarrow\ \frac1x=p-\sqrt q,\ x+\frac1x=2p
Telescoping sum
\sum_{n=1}^{N-1}\frac{1}{\sqrt n+\sqrt{n+1}}=\sqrt N-1
Linear equations, graphs and polynomials
Unique solution
\frac{a_1}{a_2}\neq\frac{b_1}{b_2}
lines intersect
No solution
\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq\frac{c_1}{c_2}
parallel lines
Infinitely many solutions
\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}
coincident lines
Intercept form
\frac xp+\frac yq=1
Area with the axes
\text{Area}=\frac12\cdot\left|\frac ca\right|\cdot\left|\frac cb\right|=\frac{c^2}{2|ab|}
Slope of ax+by+c=0
m=-\frac ab
Remainder theorem
f(x)\div(x-a)\Rightarrow R=f(a);\quad f(x)\div(px-q)\Rightarrow R=f\!\left(\tfrac qp\right)
Roots of a quadratic
\alpha+\beta=-\frac ba,\quad \alpha\beta=\frac ca
Maxima and minima (AM ≥ GM, quadratics)
AM–GM
\frac{a+b}{2}\ge\sqrt{ab}\quad(a,b>0)
Min of ax + b/x
ax+\frac bx\ge 2\sqrt{ab},\ \text{at } x=\sqrt{\tfrac ba}
Vertex of a quadratic
x=-\frac{b}{2a},\quad \text{extreme value}=\frac{4ac-b^2}{4a}
Fixed sum
x+y=S\ \Rightarrow\ xy\le\frac{S^2}{4}
Fixed product
xy=P\ \Rightarrow\ x+y\ge 2\sqrt P