### Algebra

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→ If ax2 + bx + c = 0, then x =

→ If = 2, then x = 1

→ If = –2, then x = –1

→ If = 1 , then x3 = –1

→ If = , then x6 + 1 = 0

→ If ax + by = m and bx – ay = n, then

(a2 + b2) (x2 + y2) = (m2 + n2)

→ If (x – a)2 + (y – b)2 + (z – c)2 = 0

then x – a = 0 ⇒ x = a

y – a = 0 ⇒ y = b

z – c = 0 ⇒ z = c

→ If Y =

and x = n × (n + 1), then y = (n + 1)

→ If Y =

and x = n × (n + 1),  then Y = n

→ If x = , then = 2

→ If x = , then = 2

→ If ax2 + bx + c = 0 is a quadratic equation,

and α, β are the roots of this equation

then

i)   α + β =  –b/a

ii) αβ = c/a

iii) α2 + β2 = (α + β)2 -  2αβ =

→ If

then

Concept-1

Same Basic Formulae -

1. (A + B)2 = A2 + B2+ 2AB

2. (A – B)2 = A2 + B2 – 2AB

3. (A + B + C)2 = A2 + B2 + C2 + 2 (AB + BC + CA)

4. (A + B)3 = A3 + B3 + 3AB (A + B)

5. (A – B)3 = A3 – B3 – 3AB (A – B)

6. A3 + B3 = (A + B) (A2 + B2 – AB)

7. A3 – B3 = (A – B) (A2 + B2 + AB)

8. (A + B)2 = (A – B)2 +4AB

9. (A – B)2 = (A + B)2 – 4AB.

10. (A + B + C)3 = A3 + B3 + C3 + 3(A + B) (B + C) (C + A) and if A + B + C = 0  then A3 + B3 + C3 = 3ABC.

11. A3 + B3 + C3 – 3ABC = [(A + B + C) (A2 + B2 + C2 – AB – BC – CA)]

=

12. A2 (B +C) + B2 (C + A) + C2 (B + A) + 3ABC = (A + B + C) (AB + BC + CA)

• Law of Indices

1. xa × xb = xa+b

2.       = xa–b

3. (xa)b  = (x)ab

4.

5. (xy)a = xa ya

• Rules of i [iota]

(i) The value of i =

(ii) i2 = – 1

(iii)

(iv) It is an imaginary value.

• Laws of divisibility

1. Term (An + Bn) will divisible by (A + B), if n=odd.

2. Term (An + Bn) will never be divisible by (A + B), if n=even.

3. (An – Bn) will always be divisible by (A – B) whether n = odd or even.

4. (An – Bn)  will be divisible by (A + B) if n = even.

Simplification

Order of signs to follow

* BODMAS rule

B → Bracket

O → of

D → Division

M → Multiplication

S → Subtraction

Order of Brackets

* (  ) ← {  } ← [  ]

Vernaculum → When an expression contains Vernaculum (bar),it needed to be solved before all other symbolic expressions.

System of linear equations →

Let two linear equations are

a1x + b1 y + c1 = 0 and

a2 x + b2 y + c2 = 0

Conditions                                                    Result

(i) Lines Intersecting at a point                    Unique solution

(ii) Lines parallel to each other                   No solutions

(iii) Coincident lines                                    Infinite solution

Or

(i) If then there will be only one solution.

(ii) If then there will be infinite solution.

(iii) If then no solution

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