Alpha Math

Introduction to Alpha Math:

     Alpha is the first letter of the Greek alphabet and it is written as α . The word "mathematics" comes from the Greek, which means learning, study, science, and additionally came to have the narrower and more technical meaning "Mathematical Study", Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.


Example problems of Alpha math:


Here we will discuss about the alpha math,

Alpha math – Example problem: 1

  Prove that the condition that one root of ax2 + bx + c = 0 may be the square of the other is m3 + l2 n + ln2 = 3lmn

Proof:

Let the roots of the given equation be  α and α2

then,                 α +  α2 = `(m)/(-alpha)`

 α+  α2  =  α3 = `(n)/(alpha)`

Cubing both sides of above equation (α+ α2)3 =- (m3 / l3)

 α3 +  α6 + 3α3 (α+ α2 )= - (m3 / l3 )

           n/l + (n/l)2 +3n/l (-m/l)= -m3 / l3

that is m3 + l2 n + ln2 = 3lmn

Alpha math – Example problem: 2

  We form the quadratic equation whose roots are α and γ .

Proof:

Then x = α ,

           y = γ are the roots

Therefore x - α  = 0 and

              y - γ = 0

        (x -α) (y -γ) = 0


Some more example problems of Alpha math:


Alpha math – Example problem: 3

F the roots of x4 - 6x3 + 13x2 - 12x + 4 = 0 are α,α,γ,γ  then the values of  α,  γ= ?

Solution:

        2α + 2γ = 6

              α + γ  = 3

            α 2γ 2 = 4

                    αγ = 2

    α= 2  and γ = 1

Alpha math – Example problem: 4

Prove that sin2α + sin2 ( α + 60) + sin2 (  α - 60)  =`(3)/(2)`

L. H. S = sin2α + [ sin  ( α + 60) ]2 + [ sin (α - 60)]2

sin2α + [ sinαcos 60 + cos2αsin 60) ]2+ sinα cos60 - cosα sin60]2

sin2α+ 2( sin2αcos2 60 + cos2α sin2 60 )

sin2α+ 2 [`(1)/(4)`sin2α+ `(3)/(4)` cos2α ]

sin2α+ [ (sin2 α) / 2 ] + [ (3cos2α ) / 2 ]

`(3)/(4)` [ sin2α + cos2α ] = = R. H. S

Here we proved the L.H.S is equal to R.H.S.