Basic Algebra Formula


Algebra is a branch of mathematics in which we substitute letters for numbers. In the basic algebra formula, the constants are Integers, the operators are the basic arithmetic operators (+,−,∗,/), and the variables are usually represented by the alphabet x and y. Algebra also includes real numbers, complex numbers, matrices, vectors, etc.

What is the Basic Algebra Formula?

Here is the list of basic algebra formulas used in mathematics:

  • 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 + 2bc + 2ca
  • (a – b – c)2 = a2 + b2 + c2 – 2ab + 2bc – 2ca
  • (a + b)3 = a3 + 3a2b + 3ab2 + b3 ; (a + b)3 = a3 + b3 + 3ab(a + b)
  • (a – b)3 = a3 – 3a2b + 3ab2 – b3 = a3 – b3 – 3ab(a – b)
  • 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)
  • If n is a natural number an – bn = (a – b)(an-1 + an-2b+…+ bn-2a + bn-1)
  • If n is even (n = 2k), an + bn = (a + b)(an-1 – an-2b +…+ bn-2a – bn-1)
  • If n is odd (n = 2k + 1), an + bn = (a + b)(an-1 – an-2b +an-3b2…- bn-2a + bn-1)
  • (a + b + c + …)2 = a2 + b2 + c2 + … + 2(ab + ac + bc + ….)
  • Laws of Exponents (am)(an) = am+n ; (ab)m = ambn ; (am)n = amn
  • Fractional Exponents a0 = 1 ; am/an=am-n ; am = 1/a-m ; a-m = 1/am
  • Quadratic Equation Roots:
    • For a quadratic equation ax2 + bx + c = 0 where a ≠ 0, the roots will be given by the equation as \( x = \frac{−b \pm \sqrt{b^2−4ac}}{2a}\)
      Δ = b− 4ac is called the discriminant
    • For real and distinct roots, Δ > 0
    • For real and coincident roots, Δ = 0
    • For non-real roots, Δ < 0
    • If α and β are the two roots of the equation ax2 + bx + c = 0 then, α + β = (-b / a) and α × β = (c / a).
    • If the roots of a quadratic equation are α and β, the equation will be (x − α)(x − β) = 0
  • Factorials:
    • n! = (1).(2).(3)…..(n − 1).n
    • n! = n(n − 1)! = n(n − 1)(n − 2)! = ….
    • 0! = 1
    • (a+b)n=a+ nan-1b + n(n−1)/2! an-2b+ n(n−1)(n−2)/3! an-3b+….+ bn, where, n>1.

Solved Examples Using Basic Algebra Formulas

Example 1: 

Calculate the value of 42-32.

Solution:     

To find: 42-32

Using difference of squares formula,

a2 – b2 = (a – b)(a + b)

42-32= (4-3)(4+3)

=7

Answer: 42-3= 7

 

Example 2: 

53 × 52 = ?

Solution:     

To find: 53 × 52

Using exponential formula,

(am)(an) = am+n

53 × 52= 53+2

= 57

= 78,125

Answer: 53 × 52= 78,125