MANIPULATING EXPRESSIONS INVOLVING ALPHA NAD BETA

α2=  (α+β)2 - 2αβ

α-β  =  (α+β)- 4αβ

α33  =  (α-β)+ 3αβ(α-β)

α44  =  (α22)- 2α2β2

Question 1 :

If α and β are the roots of the equation

3x2-5x+2  =  0

then find the values of

(i) (α/β) + (β/α)

(ii) α - β

(iii) (α²/β) + (β²/α)

Solution :

By comparing the given quadratic equation with the general form of quadratic equation, we get

a  =  3, b  =  -5 and c = 2

Sum of roots :

α+β  =  -b/a

α+β  =  -(-5)/3

α+β  =  5/3

Product of roots :

αβ  =  c/a  

αβ  =  2/3

(i)  (α/β)+(β/α)

By combining the above fractions, we get

(α/β)+(β/α)  =  (α22)/αβ  -----(1)

α22  =  (α+β)2-2αβ

=  (5/3)2-2(2/3)

=  (25/9)-(4/3)

=  (25-12)/9

α22  =  13/9

By applying the values in (1), we get

 (α/β) + (β/α)  =  (α22)/αβ

=  (13/9)/(2/3)

 (α/β) + (β/α)  =  13/6

(ii) α - β

α-β  =  √(α+β)2 - 4αβ

=  √(5/3)2-4(2/3)

=  √(25/9)-(8/3)

=  √1/9

α-β  =  ± 1/3

(iii)  (α2/β) + (β2/α)

By combining the given fractions, we get

2/β) + (β2/α)  =  (α33)/αβ  ----(1)

α33  =  (α-β)+ 3αβ(α-β)

=  (5/3)3-3(2/3)(5/3)

=  (125/27)-(10/9)

α33  =  95/27

By applying the values in (1), we get

2/β) + (β2/α)  =  (95/27)/(2/3)

2/β) + (β2/α)  =  95/18

Question 2 :

If α and β are the roots of

3x2-6x+4  =  0

find the value of α22

Solution :

 a = 3  b = - 6 and c = 4

α22  =  (α+β)- 2αβ  ----(1)

Sum of roots :

α+β  =  -b/a

=  -(-6)/3

α+β  =  2

Product of roots :

αβ  =  c/a  

αβ  = 4/3

By applying the values in (1), we get

=  22-2(4/3)

=  4-(8/3)

=  4/3

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