Expert: Sherman D. - 3/9/2005

Question
-------------------------
Followup To
Question -
A and B are  the roots 4x^2 + 3x -5 = 0

im trying to find the  a value for (A-3B)(B-3A)

A and B ARE ALPHA  AND BETA

I DID IT AND GOT  - 347 / 16 IS THIS  CORRECT
 
Answer -
4x^2 + 3x - 5 = 0
add 5 to both sides
4x^2 + 3x = 5
x^2 + (3/4)x = (5/4)

find half of (3/4), which is (3/8), square it, which is (9/64), add to both sides

x^2 + (3/4)x + (9/64) = (89/64)
factor left side into a perfect square
(x + (3/8))^2 = (89/64)
sqrt both sides
x + (3/8) = ±(1/8)sqrt(89)
subtract (3/8) from both sides
x = (-3/8) ± (1/8)sqrt(89)

beta = ((n - 2)/n)pi radians = 180°(n - 2)/n
alpha = (2/n)pi radians = 360°/n

A = (-3/8) + (1/8)sqrt(89)
B = (-3/8) - (1/8)sqrt(89)

(A - 3B)(B - 3A), then

(((-3/8) + (1/8)sqrt(89)) - 3((-3/8) - (1/8)sqrt(89))) * ((-3/8) - (1/8)sqrt(89)) - 3((-3/8) + (1/8)sqrt(89))

((-3/8) + (1/8)sqrt(89) + (9/8) + (3/8)sqrt(89))*((-3/8) - (1/8)sqrt(89) + (9/8) - (3/8)sqrt(89))

(((-3/8) + (9/8)) + ((1/8) + (3/8))sqrt(89)) * (((-3/8) + (9/8)) + ((-1/8) - (3/8))sqrt(89))

((6/8) + (4/8)sqrt(89)) * ((6/8) + (-4/8)sqrt(89))

((3/4) + (1/2)sqrt(89)) * ((3/4) - (1/2)sqrt(89))

(9/16) - (3/8)sqrt(89) + (3/8)sqrt(89) - (1/4)(sqrt(89)^2)

(9/16) - (1/4)(89)

(9/16) - (89/4)



(9/16) - (356/16)

(-347/16)
equation  = 3x^2 -5x +1 = 0
how  can i fimd an equation with roots (A^2- B^2)

so according to what i got, you are 100% correct.

Answer
3x^2 - 5x + 1 = 0

I will use the quadratic formula, which is nothing more than the conclusion of completing the square.

x = (-b ± sqrt(b^2 - 4ac))/2a
whereas
a = 3
b = -5
c = 1

x = (-(-5) ± sqrt((-5)^2 - 4(3)(1)))/2(3)
x = (5 ± sqrt(25 - 12))/6
x = (5 ± sqrt(13))/6
A = (5/6) + (1/6)sqrt(13)
B = (5/6) - (1/6)sqrt(13)

A^2 - B^2
((5 + sqrt(13))/6)^2 - ((5/6) - (1/6)sqrt(13))^2

(((5 + sqrt(13))*((5 + sqrt(13))/36) -
(((5 - sqrt(13))*((5 - sqrt(13))/36)

((25 + 5sqrt(13) + 5sqrt(13) + (sqrt(13))^2)/36) -
((25 - 5sqrt(13) - 5sqrt(13) + (sqrt(13))^2)/36)

((25 + (5 + 5)sqrt(13) + 13)/36) -
((25 + (-5 - 5)sqrt(13) + 13)/36)

((25 + 10sqrt(13) + 13)/36) -
((25 + (-10)sqrt(13) + 13)/36)

((25 + 13) + 10sqrt(13))/36) - ((25 + 13) - 10sqrt(13))/36)

((38 + 10sqrt(13))/36) - ((38 - 10sqrt(13))/36)

(38 + 10sqrt(13) - (38 - 10sqrt(13))/36
(38 + 10sqrt(13) - 38 + 10sqrt(13))/36
((38 - 38) + (10 + 10)sqrt(13))/36
(20sqrt(13))/36
(20/36)sqrt(13)
(10/18)sqrt(13)
(5/9)sqrt(13)

(5sqrt(13))/9

Let me know if that is what you got as well.