t = (5 ((446 sqrt(3188516012553) - 827891226)^(1/3) - 204292 (-1)^(2/3) (3/(413945613 - 223 sqrt(3188516012553)))^(1/3)))/(223 6^(2/3)) + 1125/223 or t = 1125/223 - (5 ((-2)^(1/3) (223 sqrt(3188516012553) - 413945613)^(1/3) - 204292 (-3/(413945613 - 223 sqrt(3188516012553)))^(1/3)))/(223 6^(2/3)) or t = 1125/223 - (5 ((827891226 - 446 sqrt(3188516012553))^(1/3) + 204292 (3/(413945613 - 223 sqrt(3188516012553)))^(1/3)))/(223 6^(2/3))
Step-by-step explanation:
Solve for t over the real numbers:
0.892 t^3 - 13.5 t^2 + 22.3 t + 579 = 0
0.892 t^3 - 13.5 t^2 + 22.3 t + 579 = (223 t^3)/250 - (27 t^2)/2 + (223 t)/10 + 579:
(223 t^3)/250 - (27 t^2)/2 + (223 t)/10 + 579 = 0
Bring (223 t^3)/250 - (27 t^2)/2 + (223 t)/10 + 579 together using the common denominator 250:
1/250 (223 t^3 - 3375 t^2 + 5575 t + 144750) = 0
Multiply both sides by 250:
223 t^3 - 3375 t^2 + 5575 t + 144750 = 0
Eliminate the quadratic term by substituting x = t - 1125/223:
144750 + 5575 (x + 1125/223) - 3375 (x + 1125/223)^2 + 223 (x + 1125/223)^3 = 0
Expand out terms of the left hand side:
223 x^3 - (2553650 x)/223 + 5749244625/49729 = 0
Divide both sides by 223:
x^3 - (2553650 x)/49729 + 5749244625/11089567 = 0
Change coordinates by substituting x = y + λ/y, where λ is a constant value that will be determined later:
5749244625/11089567 - (2553650 (y + λ/y))/49729 + (y + λ/y)^3 = 0
Multiply both sides by y^3 and collect in terms of y:
y^6 + y^4 (3 λ - 2553650/49729) + (5749244625 y^3)/11089567 + y^2 (3 λ^2 - (2553650 λ)/49729) + λ^3 = 0
Substitute λ = 2553650/149187 and then z = y^3, yielding a quadratic equation in the variable z:
z^2 + (5749244625 z)/11089567 + 16652679340752125000/3320419398682203 = 0
Find the positive solution to the quadratic equation:
z = (125 (223 sqrt(3188516012553) - 413945613))/199612206
Substitute back for z = y^3:
y^3 = (125 (223 sqrt(3188516012553) - 413945613))/199612206
Taking cube roots gives (5 (223 sqrt(3188516012553) - 413945613)^(1/3))/(223 2^(1/3) 3^(2/3)) times the third roots of unity:
y = (5 (223 sqrt(3188516012553) - 413945613)^(1/3))/(223 2^(1/3) 3^(2/3)) or y = -(5 (-1/2)^(1/3) (223 sqrt(3188516012553) - 413945613)^(1/3))/(223 3^(2/3)) or y = (5 (-1)^(2/3) (223 sqrt(3188516012553) - 413945613)^(1/3))/(223 2^(1/3) 3^(2/3))
Substitute each value of y into x = y + 2553650/(149187 y):
x = (5 ((223 sqrt(3188516012553) - 413945613)/2)^(1/3))/(223 3^(2/3)) - 510730/223 (-1)^(2/3) (2/(3 (413945613 - 223 sqrt(3188516012553^(1/3) or x = 510730/223 ((-2)/(3 (413945613 - 223 sqrt(3188516012553^(1/3) - (5 ((-1)/2)^(1/3) (223 sqrt(3188516012553) - 413945613)^(1/3))/(223 3^(2/3)) or x = (5 (-1)^(2/3) ((223 sqrt(3188516012553) - 413945613)/2)^(1/3))/(223 3^(2/3)) - 510730/223 (2/(3 (413945613 - 223 sqrt(3188516012553^(1/3)
Bring each solution to a common denominator and simplify:
x = (5 ((446 sqrt(3188516012553) - 827891226)^(1/3) - 204292 (-1)^(2/3) (3/(413945613 - 223 sqrt(3188516012553)))^(1/3)))/(223 6^(2/3)) or x = -(5 ((-2)^(1/3) (223 sqrt(3188516012553) - 413945613)^(1/3) - 204292 ((-3)/(413945613 - 223 sqrt(3188516012553)))^(1/3)))/(223 6^(2/3)) or x = -(5 ((827891226 - 446 sqrt(3188516012553))^(1/3) + 204292 (3/(413945613 - 223 sqrt(3188516012553)))^(1/3)))/(223 6^(2/3))
Substitute back for t = x + 1125/223:
t = (5 ((446 sqrt(3188516012553) - 827891226)^(1/3) - 204292 (-1)^(2/3) (3/(413945613 - 223 sqrt(3188516012553)))^(1/3)))/(223 6^(2/3)) + 1125/223 or t = 1125/223 - (5 ((-2)^(1/3) (223 sqrt(3188516012553) - 413945613)^(1/3) - 204292 (-3/(413945613 - 223 sqrt(3188516012553)))^(1/3)))/(223 6^(2/3)) or t = 1125/223 - (5 ((827891226 - 446 sqrt(3188516012553))^(1/3) + 204292 (3/(413945613 - 223 sqrt(3188516012553)))^(1/3)))/(223 6^(2/3))
