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: 

\(1. \ (a + b)^2 = a^2 + 2ab + b^2 \\[1em]   2. \ (a - b)^2 = a^2 - 2ab + b^2\\[1em] 3. \ (a + b)(a - b) = a^2 - b^2\\[1em]   4. \ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\\[1em] 5. \ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\\[1em] 6. \ a^3 + b^3 = (a + b) (a^2 - ab + b^2)\\[1em] 7. \ a^3 - b^3 = (a - b) (a^2 + ab + b^2)\\[1em] 8. \ (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca\)

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: 
 

• \(log_a a = 1\)
   
• \(log_a 1 = 0\)
   
• \(log_a \ (xy) = log_a x+ log_ay\)
   
• \(log_a \left(\frac{x}{y}\right) = log_a x - log_a y \)
   
• \(log_a (xm) = m log_a x \)
   
• \(log_a x = \frac{log_cx}{log_ca}\)
   
• \(a^ {log_ax} = x\)

The quadratic formula is one of the two methods to solve a quadratic equation.

The standard form of a quadratic equation is given as; 

\(ax^2 + bx + c = 0. \)

The value of the variable x can be found by using the formula:
    
\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

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, \(ar^2\),....., \(ar^{n-1}\) 
 

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: \({}^{n}P_{r} = \frac{n!}{(n - r)!} \)

Combination formula: \({}^{n}C_{r} = \frac{n!}{r!(n - r)!} \)

Binomial theorem:
\((x + y)^n = \binom{n}{0}x^n y^0 + \binom{n}{1}x^{n-1}y^1 + \binom{n}{2}x^{n-2}y^2 + \cdots + \binom{n}{n-1}x^1 y^{n-1} + \binom{n}{n}x^0 y^n \)
 

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, \(|a| = \sqrt {x^2 + y^2 + z^2}\)
   
• The unit vector along \(a = \frac{a}{|a|}\)
   
• Dot product: ab = |a| |b| cosθ, where θ is the angle between two vectors a and b
   
• Cross product: \(a × b = |a| \ |b|sinθn̂\)
   
• Scalar triple product: \([a b c] = a \cdot (b × c) = (a × b) \cdot c\)

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 - 

\((a + b)^2 = a^2 + 2ab + b^2\\[1em] (a - b)^2 = a^2 - 2ab + b^2\\[1em] (a + b)(a - b) = a^2 - b^2\\[1em] (x + a) (x + b) = x^2 + (a + b)x + ab\)

Algebra formulas of functions

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. 

Algebra formulas of fractions 

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

\(\frac{x}{y}+ \frac{z}{w} = \frac{(xw + yz)} {(yw)} \\[1em] \frac{x}{y} - \frac{z}{w}= \frac{(xw - yz)}{(yw)} \\[1em] \frac{x}{y} × \frac{z}{w} = \frac{xz}{yw} \\[1em] \frac{x}{y} ÷ \frac{z}{w} = \frac{xw}{yz}\)

Students often find algebra formulas challenging to memorize. Here are some tips and tricks to help with memorization: 
 

• Break down formulas into smaller parts 
   
• Use mnemonic devices to remember sequences 
   
• Practice regularly with example problems 
   
• Create flashcards for quick recall
   
• Teachers should ask their students to focus on learning the meaning before memorizing the formulas. We should explain to them what a formula represents, not just how it looks.
   
• Parents can help their children by grouping the formulas by the patterns they follow. We can help them by teaching the formulas. Such as,
  
  Square formulas:
  
  \((a+b)^2 \\[1em] (a-b)^2\)
  
  Product formulas:
  
  (a+b)(a-b)
   
• Teachers can utilize visual models and area diagrams to explain the expansions. Use algebra tiles or paper cutouts, as they aid in retention longer than rote learning.
   
• Parents encourage their students to practice mental expansion regularly. Practice daily quick drills like expanding \((x+3)^2.\) It is easier for young learners to work with numbers than with variables, so we can ask them to work on numbers before variables.

• Algebraic Identities: Equations true for all variable values, used for simplifying expressions.

• Quadratic Equation: A polynomial equation of degree two, standard form ax² + bx + c = 0.

• Linear Equation: An equation involving two variables that represents a straight line.

• Discriminant: Part of the quadratic formula under the square root (b² - 4ac), determines the nature of roots.

• Substitution Method: A way to solve systems of equations by replacing one variable with an equivalent expression.