see attachement
Step-by-step explanation:
Solve for x:
8 x + 26 - 3/x + 6/x^2 = 0
Hint: | Write the left hand side as a single fraction.
Bring 8 x + 26 - 3/x + 6/x^2 together using the common denominator x^2:
(8 x^3 + 26 x^2 - 3 x + 6)/x^2 = 0
Hint: | Multiply both sides by a polynomial to clear fractions.
Multiply both sides by x^2:
8 x^3 + 26 x^2 - 3 x + 6 = 0
Hint: | Look for a simple substitution that eliminates the quadratic term of 8 x^3 + 26 x^2 - 3 x + 6.
Eliminate the quadratic term by substituting y = x + 13/12:
6 - 3 (y - 13/12) + 26 (y - 13/12)^2 + 8 (y - 13/12)^3 = 0
Hint: | Write the cubic polynomial on the left hand side in standard form.
Expand out terms of the left hand side:
8 y^3 - (187 y)/6 + 799/27 = 0
Hint: | Write the cubic equation in standard form.
Divide both sides by 8:
y^3 - (187 y)/48 + 799/216 = 0
Hint: | Perform the substitution y = z + λ/z.
Change coordinates by substituting y = z + λ/z, where λ is a constant value that will be determined later:
799/216 - 187/48 (z + λ/z) + (z + λ/z)^3 = 0
Hint: | Transform the rational equation into a polynomial equation.
Multiply both sides by z^3 and collect in terms of z:
z^6 + z^4 (3 λ - 187/48) + (799 z^3)/216 + z^2 (3 λ^2 - (187 λ)/48) + λ^3 = 0
Hint: | Find an appropriate value for λ in order to make the coefficients of z^2 and z^4 both zero.
Substitute λ = 187/144 and then u = z^3, yielding a quadratic equation in the variable u:
u^2 + (799 u)/216 + 6539203/2985984 = 0
Hint: | Solve for u.
Find the positive solution to the quadratic equation:
u = (17 (9 sqrt(157) - 188))/1728
Hint: | Perform back substitution on u = (17 (9 sqrt(157) - 188))/1728.
Substitute back for u = z^3:
z^3 = (17 (9 sqrt(157) - 188))/1728
Hint: | Take the cube root of both sides.
Taking cube roots gives 1/12 17^(1/3) (-(188 - 9 sqrt(157)))^(1/3) times the third roots of unity:
z = 1/12 17^(1/3) (-(188 - 9 sqrt(157)))^(1/3) or z = -1/12 (-1)^(2/3) 17^(1/3) (188 - 9 sqrt(157))^(1/3) or z = -1/12 17^(1/3) (188 - 9 sqrt(157))^(1/3)
Hint: | Perform back substitution with y = z + 187/(144 z).
Substitute each value of z into y = z + 187/(144 z):
y = 1/12 (-17 (188 - 9 sqrt(157)))^(1/3) - (11 (-17)^(2/3))/(12 (188 - 9 sqrt(157))^(1/3)) or y = 11/12 17^(2/3) ((-1)/(188 - 9 sqrt(157)))^(1/3) - 1/12 (-1)^(2/3) (17 (188 - 9 sqrt(157)))^(1/3) or y = -(11 17^(2/3))/(12 (188 - 9 sqrt(157))^(1/3)) - 1/12 (17 (188 - 9 sqrt(157)))^(1/3)
Hint: | Simplify each solution.
Bring each solution to a common denominator and simplify:
y = 1/12 (153 sqrt(157) - 3196)^(1/3) - (11 (-17)^(2/3))/(12 (188 - 9 sqrt(157))^(1/3)) or y = 1/12 17^(1/3) (11 ((-17)/(188 - 9 sqrt(157)))^(1/3) - (-1)^(2/3) (188 - 9 sqrt(157))^(1/3)) or y = -1/12 17^(1/3) (1/(188 - 9 sqrt(157)))^(1/3) ((188 - 9 sqrt(157))^(2/3) + 11 17^(1/3))
Hint: | Perform back substitution on the three roots.
Substitute back for x = y - 13/12:
x = -13/12 - (11 (-17)^(2/3))/(12 (188 - 9 sqrt(157))^(1/3)) + 1/12 (153 sqrt(157) - 3196)^(1/3) or x = 1/12 17^(1/3) (11 (-17/(188 - 9 sqrt(157)))^(1/3) - (-1)^(2/3) (188 - 9 sqrt(157))^(1/3)) - 13/12 or x = -1/12 (17/(188 - 9 sqrt(157)))^(1/3) (11 17^(1/3) + (188 - 9 sqrt(157))^(2/3)) - 13/12
