Answer:
X = ((56 sqrt(3) + 97)^(2/3) + 1)/(4 (56 sqrt(3) + 97)^(1/3)) + 1/4 or X = ((-1)^(2/3) - (-1)^(1/3) (56 sqrt(3) + 97)^(2/3))/(4 (56 sqrt(3) + 97)^(1/3)) + 1/4 or X = 1/4 ((-1)^(2/3) (97 + 56 sqrt(3))^(1/3) - (56 sqrt(3) - 97)^(1/3)) + 1/4
Explanation:
Solve for X:
X^3 - (3 X^2)/4 - 3 = 0
Bring X^3 - (3 X^2)/4 - 3 together using the common denominator 4:
1/4 (4 X^3 - 3 X^2 - 12) = 0
Multiply both sides by 4:
4 X^3 - 3 X^2 - 12 = 0
Eliminate the quadratic term by substituting x = X - 1/4:
-12 - 3 (x + 1/4)^2 + 4 (x + 1/4)^3 = 0
Expand out terms of the left hand side:
4 x^3 - (3 x)/4 - 97/8 = 0
Divide both sides by 4:
x^3 - (3 x)/16 - 97/32 = 0
Change coordinates by substituting x = y + λ/y, where λ is a constant value that will be determined later:
-97/32 - 3/16 (y + λ/y) + (y + λ/y)^3 = 0
Multiply both sides by y^3 and collect in terms of y:
y^6 + y^4 (3 λ - 3/16) - (97 y^3)/32 + y^2 (3 λ^2 - (3 λ)/16) + λ^3 = 0
Substitute λ = 1/16 and then z = y^3, yielding a quadratic equation in the variable z:
z^2 - (97 z)/32 + 1/4096 = 0
Find the positive solution to the quadratic equation:
z = 1/64 (97 + 56 sqrt(3))
Substitute back for z = y^3:
y^3 = 1/64 (97 + 56 sqrt(3))
Taking cube roots gives 1/4 (97 + 56 sqrt(3))^(1/3) times the third roots of unity:
y = 1/4 (97 + 56 sqrt(3))^(1/3) or y = -1/4 (-97 - 56 sqrt(3))^(1/3) or y = 1/4 (-1)^(2/3) (97 + 56 sqrt(3))^(1/3)
Substitute each value of y into x = y + 1/(16 y):
x = 1/(4 (56 sqrt(3) + 97)^(1/3)) + 1/4 (56 sqrt(3) + 97)^(1/3) or x = (-1)^(2/3)/(4 (56 sqrt(3) + 97)^(1/3)) - 1/4 (-56 sqrt(3) - 97)^(1/3) or x = 1/4 (-1)^(2/3) (56 sqrt(3) + 97)^(1/3) - 1/4 ((-1)/(56 sqrt(3) + 97))^(1/3)
Bring each solution to a common denominator and simplify:
x = ((56 sqrt(3) + 97)^(2/3) + 1)/(4 (97 + 56 sqrt(3))^(1/3)) or x = ((-1)^(2/3) - (-1)^(1/3) (56 sqrt(3) + 97)^(2/3))/(4 (97 + 56 sqrt(3))^(1/3)) or x = 1/4 ((-1)^(2/3) (56 sqrt(3) + 97)^(1/3) - (56 sqrt(3) - 97)^(1/3))
Substitute back for X = x + 1/4:
Answer: X = ((56 sqrt(3) + 97)^(2/3) + 1)/(4 (56 sqrt(3) + 97)^(1/3)) + 1/4 or X = ((-1)^(2/3) - (-1)^(1/3) (56 sqrt(3) + 97)^(2/3))/(4 (56 sqrt(3) + 97)^(1/3)) + 1/4 or X = 1/4 ((-1)^(2/3) (97 + 56 sqrt(3))^(1/3) - (56 sqrt(3) - 97)^(1/3)) + 1/4