Algebra Formulas

Algebra formulas are rules or equations that help with factoring, expanding, and simplifying expressions. We can use these formulas to solve complex algebraic equations efficiently. Here are some algebraic formulas: 

• (a + b)² = a² + 2ab + b²
   
•  (a - b)² = a² - 2ab + b²
   
• (a + b)(a - b) = a² - b²
   
•  (a + b)³ = a³ + 3a²b + 3ab² + b³
   
• (a - b)³ = a³ - 3a²b + 3ab² - b³
   
• a³ + b³ = (a + b) (a² - ab + b²)
   
• a³ - b³ = (a - b) (a² + ab + b²)
   
• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

To solve the expressions involving powers or exponents, we use the exponent rules. These rules are used to simplify expressions with powers. The exponent rules are: 

Logarithms are used to solve multiplication and division of numbers with powers in simple ways. This makes them an effective tool to work with algebraic formulas with exponents. The relationship between the exponent and logarithm is: x^m = a ⇒ log_x a = m

Some commonly used log algebraic formulas are: 

log_a a = 1

log^a 1 = 0

loga (xy) = log_a x+ log_a y

loga (x/y) = log_a x - log_a y 

log_a (x^m) = m log_a x

log_a x = log_cx/log_ca 

a ^log_ax = x
 

The quadratic formula is one of the two methods to solve a quadratic equation. The standard form of a quadratic equation is ax² + bx + c = 0. The value of the variable x can be found by using the formula:     
x = -b ± √b² - 4ac/2a

Progressions are numbers arranged according to specific patterns. Arithmetic progression (AP) and geometric progression (GP) are two main types of progressions. In AP, the difference between two successive terms is called the common difference, which is represented as ‘d.’

Therefore, in an AP, the terms can be written as a, a + d, a + 2d, ….., a + (n - 1)d. On the other hand, the sequence of GP is written by multiplying a constant by the previous term. In GP, the terms are written as a, ar, ar²,....., ar^n-1 

Where n is the number of terms

a is the first term

d is the common difference in an AP

r is the common ratio in a GP

In algebra, permutations and combinations are formulas that help us identify the number of ways something can be arranged. Permutations refer to arrangements of items where the order matters, and combinations are the selection of items where order does not matter. 

Factorial formula: n! = n × (n - 1) × (n - 2) × …. × 3 × 2 × 1
 
Permutations formula: ⁿP_r = n! / (n - r)!

Combination formula: ⁿC_r = n! / (r! (n - r)!)

Binomial theorem: (x + y)ⁿ =  ⁿC₀xⁿy⁰ + ⁿC₁x^n-1y¹ + ⁿC₂x^n-2y² + …. + ⁿC_n-1xy^n-1 + ⁿC_nx⁰yⁿ

The vector algebra formula is used to solve problems related to directions and magnitude. Some important vector formulas are: 

For any three vectors a, b, and c in a 3D space

• The magnitude of a = xi + yj + zk, 
  So, |a| = √x² + y² + z²
   
• The unit vector along a = a/|a|
   
• Dot product: ab = |a| |b| cosθ, where θ is the angle between two vectors a and b
   
• Cross product: a × b = absinθn̂
   
• Scalar triple product: [a b c] = a  (b × c) = (a × b) · c

Algebraic identities are the equations that hold true for all values of the variables involved. It means LHS = RHS of the equation. Some common algebraic identities are - 

•  (a + b)² = a² + 2ab + b²
   
•  (a - b)² = a² - 2ab + b²
   
• (a + b)(a - b) = a² - b²
   
• (x + a) (x + b) = x² + (a + b)x + ab

The algebraic function expresses a relationship between two variables; it is written in the form y = f(x). Where x is the input and y is the output. For example, if x = 4, then f(x) = f(4) = 4² = 16. 

In algebra, fractions that contain variables are called rational expressions. We can add, subtract, multiply, and divide fractions. The algebraic expression of fractions is:

• x /y + z/w = (xw + yz) / (yw)
   
• x /y - z/w = (xw - yz) / (yw)
   
• x /y × z/w = xz / yw
   
• x /y ÷ z/w = xw / yz

In real life, we use algebraic formulas from managing personal finances and cooking to understanding scientific concepts and designing technology. Here are a few applications of algebraic formulas:

• To create budgets and track expenses, we can set up an equation to balance income and spending.

• To calculate the loan payment and compound interest, there are algebraic formulas.

• In cooking and baking, to scale recipes up or down, we can use algebra.  

• In health and fitness, to calculate calorie counting and exercise regimens, we can use algebraic formulas.