Important Formulas: Algebra - SSC CGL PDF Download

Definition of Algebra

Algebra, within the realm of mathematics, is a discipline concerned with symbols and the principles governing their manipulation to solve equations and explore relationships between different quantities.

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Uses of Algebra

Algebraic formulas serve varied and indispensable purposes in numerous domains, emerging as essential instruments for both problem-solving and analysis. Within engineering, these formulas contribute to the design of structures and the analysis of intricate systems. In finance, they play a crucial role in budgeting, investment calculations, and risk assessments. In the realm of computer science, algebraic expressions are pivotal for coding and modeling algorithms.

Algebraic Expressions

Here we have listed some of the most important Formulas For Algebra:-

• a² – b² = (a – b)(a + b)
• (a+b)² = a² + 2ab + b²
• a² + b² = (a – b)² + 2ab
• (a – b)² = a² – 2ab + b²
• (a + b + c)² = a² + b² + c² + 2ab + 2ac + 2bc
• (a – b – c)² = a² + b² + c² – 2ab – 2ac + 2bc
• (a + b)³ = a³ + 3a²b + 3ab² + b³  , which can be also denoted as – (a + b)³ = a³ + b³ + 3ab(a + b)
• (a – b)³ = a³ – 3a²b + 3ab² – b³
• a³ – b³ = (a – b)(a² + ab + b²)
• a³ + b³ = (a + b)(a² – ab + b²)
• (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴)
• (a – b)⁴ = a⁴ – 4a³b + 6a²b² – 4ab³ + b⁴)
• a⁴ – b⁴ = (a – b)(a + b)(a² + b²)
• a⁵ – b⁵ = (a – b)(a⁴ + a³b + a²b² + ab³ + b⁴)

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Properties and Formulas For Algebra

1. Commutative Property of Addition:
The sum of the expression does not changes if the order of the elements are changed. The elements can be expressions or numbers.
a + b = b + a

2. Commutative Property of Multiplication:
When the order of the factors are changed, the product does not change. These factors can be numbers or expressions.
a x b = b x a

3. Associative Property of Addition:
The property defines that when two or more numbers are grouped together which are performing basic arithmetic addition, their order does not play a significant role in the result.
(a + b)+ c = a + (b + c)

4. Associative property of Multiplication:
Associative property states that when two or more elements are grouped together in the basic arithmetical multiplication, their order does not change the final result. Also, in this case, the grouping is usually done by parenthesis.
(a x b) xc = a x (b x c)

5. Distributive properties of Addition and Multiplication:
The distributive property says that the product of a single term and a sum or a difference of two or more terms present in the bracket is same as multiplying each of the element by a single term and them adding and subtracting the products.
a × (b + c) = a × b + a × c
and
(a + b) × c = a × c + b × c

6. Rule of multiplication over subtraction:
If p, q, and r are any numbers, then, p (q-r) = p*q – p*r. Similarly, in the addition rule, the distribution for multiplication over subtraction can be done in left distribution and right distribution.
If p* (q-r) = (p * q) – (p*r)- Left distributive law
and
If (p-q)*r = (p*r) – (q*r)- Right distributive law

Algebra Questions and Answers

Q1: Kobe Bryant wants to know that If a×b=2a–3b+ab , then 3×5+5×3 is equal to :
(a) 22
(b) 24
(c) 26
(d) 28
Ans: (a)
a×∗b = 2a – 3b + ab

⇒  3 × 5 = 2 × 3 – 3 × 5 + 3 × 5 = 6
5 × 3 = 2 × 5 – 3 × 3 + 3 × 5
= 10 – 9 + 15 = 16
Therefore,  3 × 5 + 5 × 3
= 6 + 16 = 22

Q2: Varun Dhawan wants to know that if two numbers x and y (x > y) are such that their sum is equal to three times their difference. Then value of  will be:
(a) 2/ 3
(b) 1
(c) 1 1 /2
(d) 1 2 /3
Ans: (b)
(x + y) = 3 (x – y) = 3x – 3y
⇒ 3y + y = 3x – x
⇒ 2x = 4y
⇒ x = 2y
⇒ x/y = 2/1

= 6/6 = 1

Q3: Ranveer Singh wants to know that If a * b = 2 (a + b), then 5 * 2 is equal to :
(a) 3
(b) 10
(c) 14
(d) 20
Ans: (c)
a * b = 2 (a + b)
∴ 5 * 2 = 2 (5 + 2)
= 2 *7 = 14

Q4: Kareena Kapoor wants to know that if , then the value of 8×2 is :
(a) 6
(b) 10
(c) 14
(d) 16
Ans: (c)


=10 + 4 = 14

Q5: Siddharth Malhotra wants to know the value of  expression:
 
(a) (1 + 1/(x+4))
(b) x+4
(c) 1/ x
(d)  x + 4/x
Ans: (d)