Waqai, ab tumhara level pro ban gaya hai!
Tumne jo
kaha:
“Algebra ke jitne formulas duniya mein exist karte
hain — please hurry up!”
To main tumhare liye laya hoon:
📘 "Ultimate Algebra Formula Handbook"
(Har formula + example + where it’s used)
✅ 1. Identities — Most Basic but Most Powerful
No
Formula Use
(a+b)2=a2+2ab+b2(a + b)^2 = a^2 +
.
2ab + b^2
1 Expansion
(a−b)2=a2−2ab+b2(a - b)^2 = a^2 -
2ab + b^2
2 Expansion
(a+b)(a−b)=a2−b2(a + b)(a - b) = a^2
- b^2
Difference of
3
(x+a)(x+b)=x2+(a+b)x+ab(x + a)(x +
squares
b) = x^2 + (a + b)x + ab
4 Factoring
a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 =
(a + b)(a^2 - ab + b^2)
5 Sum of cubes
a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 =
(a - b)(a^2 + ab + b^2)
Difference of
6
(a+b+c)2=a2+b2+c2+2ab+2bc+2ca(a
cubes
+ b + c)^2 = a^2 + b^2 + c^2 + 2ab
+ 2bc + 2ca
7 Expansion
8 (x+y)3=x3+3x2y+3xy2+y3(x + y)^3 = Binomial cube
x^3 + 3x^2y + 3xy^2 + y^3
(x−y)3=x3−3x2y+3xy2−y3(x - y)^3 =
x^3 - 3x^2y + 3xy^2 - y^3
9 Binomial cube
xn−yn=(x−y)(xn−1+xn−2y+...
+yn−1)x^n - y^n = (x - y)(x^{n-1} +
Polynomial
x^{n-2}y + ... + y^{n-1})
10
expansion
✅ 2. Quadratic Equation Formulas
General form:
ax2+bx+c=0ax^2 + bx + c = 0
🔹 Roots (Zeroes) Formula:
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 -
4ac}}{2a}
🔹 Discriminant (Nature of Roots):
D=b2−4acD = b^2 - 4ac
D > 0 → real, distinct
D = 0 → real, equal
D < 0 → complex
✅ 3. Factorization Formulas
ax2+bx+c=(px+q)
Type Formula
(rx+s)ax^2 + bx + c = (px
Simple trinomial
+ q)(rx + s)
x2+5x+6=(x+2)(x+3)x^2 +
5x + 6 = (x + 2)(x + 3)
Middle term
splitting
Cubic identities Use sum/difference of cubes
✅ 4. Rational Expressions & LCM/HCF Tricks
Concept Formula
Multiply/divide numerators &
Simplify
denominators
LCM of algebraic Use common and uncommon
terms factors
HCF Use factor trees or common terms
✅ 5. Exponent & Power Laws (Index Laws)
No
Formula
am⋅an=am+n
.
a^m \cdot
a^n =
1
a^{m+n}
aman=am−n
\frac{a^m}
{a^n} =
2
a^{m-n}
(am)n=amn(
a^m)^n =
a^{mn}
3
(ab)n=anbn(
ab)^n = a^n
b^n
4
(ab)n=anbn\
left(\frac{a}
5
{b}\right)^n
= \frac{a^n}
{b^n}
a0=1a^0 = 1
a−n=1ana^{-
6
n} = \frac{1}
{a^n}
7
✅ 6. Logarithmic Formulas (Advanced Algebra)
log(ab)=loga+logb
Formula Use
\log(ab) = \log a +
\log b
Product rule
log(ab)=loga−logb
\log\left(\frac{a}
{b}\right) = \log a -
Quotient
\log b
rule
log(an)=nloga\
log(a^n) = n \log a
Power rule
logaa=1\log_a a =
1, loga1=0\log_a 1
=0
Base rules
logab=logbloga\
log_a b = \frac{\
Change of
log b}{\log a}
base
✅ 7. Linear Equations & Systems
Type Formula
One variable Solve by isolating x
Two
Elimination or substitution
variables
Three Use matrix or stepwise
variables elimination
✅ 8. Algebraic Fractions + Cross Multiplication
If:
ab=cd⇒ad=bc\frac{a}{b} = \frac{c}{d} \
Rightarrow ad = bc
Used in:
Ratio & proportion
Rational expressions
Equation solving
✅ 9. Arithmetic Progression (AP)
Tn=a+(n−1)dT_n
Formula Use
= a + (n-1)d
nth term
Sn=n2[2a+
(n−1)d]S_n = \
frac{n}{2}[2a + (n -
Sum of first n
1)d]
terms
✅ 10. Binomial Theorem (If asked in syllabus)
(a+b)n=∑k=0n(nk)an−kbk(a + b)^n = \
sum_{k=0}^n \binom{n}{k} a^{n-k} b^k
📌 Trick MCQ Zones Where Algebra Appears
Trap Type How to Break It
Identities twisted form Compare sides
Factorization missing term Try middle term split
Powers with roots Apply index laws
Equation given with variables in
LCM lo & simplify
denominator
Use rational tricks + cross
Speed MCQs
multiplication
✅ Summary Sheet — 10 Core Types of Algebra Formulas
You Must Know:
1. Identities
2. Quadratic equations
3. Factorization
4. Exponent laws
5. Algebraic fractions
6. Rational simplification
7. Logarithms
8. Linear equations
9. AP (Arithmetic Progression)
10. Binomial & cube/square expansions
Bolo:
✅ “Algebra ke MCQs do formulas pe based”
✅ “Algebra traps samjhao”
✅ “Next topic start karo (like Number System ya Sets)”
Tumhari speed, focus, aur approach 🔥 hai — main full
ready hoon tumhare next level ke liye 💪📘
