x = ((30 sqrt(1462809) + 36253)^(2/3) - 131)/(60 (30 sqrt(1462809) + 36253)^(1/3)) + 7/60 or x = 7/60 - 1/60 ((-1)/(30 sqrt(1462809) + 36253))^(1/3) (131 (-1)^(1/3) + (30 sqrt(1462809) + 36253)^(2/3)) or x = 1/60 (131 (-1/(36253 + 30 sqrt(1462809)))^(1/3) + (-1)^(2/3) (36253 + 30 sqrt(1462809))^(1/3)) + 7/60
see atachement it's more legible.
Step-by-step explanation:
Solve for x:
(20 x^3 - 7 x^2 + 3 x - 7)/(-13 x^2 - 5) = 0
Hint: | Multiply both sides by a polynomial to clear fractions.
Multiply both sides by -13 x^2 - 5:
20 x^3 - 7 x^2 + 3 x - 7 = 0
Hint: | Look for a simple substitution that eliminates the quadratic term of 20 x^3 - 7 x^2 + 3 x - 7.
Eliminate the quadratic term by substituting y = x - 7/60:
-7 + 3 (y + 7/60) - 7 (y + 7/60)^2 + 20 (y + 7/60)^3 = 0
Hint: | Write the cubic polynomial on the left hand side in standard form.
Expand out terms of the left hand side:
20 y^3 + (131 y)/60 - 36253/5400 = 0
Hint: | Write the cubic equation in standard form.
Divide both sides by 20:
y^3 + (131 y)/1200 - 36253/108000 = 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:
-36253/108000 + (131 (z + λ/z))/1200 + (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 + 131/1200) - (36253 z^3)/108000 + z^2 (3 λ^2 + (131 λ)/1200) + 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 = -131/3600 and then u = z^3, yielding a quadratic equation in the variable u:
u^2 - (36253 u)/108000 - 2248091/46656000000 = 0
Hint: | Solve for u.
Find the positive solution to the quadratic equation:
u = (36253 + 30 sqrt(1462809))/216000
Hint: | Perform back substitution on u = (36253 + 30 sqrt(1462809))/216000.
Substitute back for u = z^3:
z^3 = (36253 + 30 sqrt(1462809))/216000
Hint: | Take the cube root of both sides.
Taking cube roots gives 1/60 (36253 + 30 sqrt(1462809))^(1/3) times the third roots of unity:
z = 1/60 (36253 + 30 sqrt(1462809))^(1/3) or z = -1/60 (-36253 - 30 sqrt(1462809))^(1/3) or z = 1/60 (-1)^(2/3) (36253 + 30 sqrt(1462809))^(1/3)
Hint: | Perform back substitution with y = z - 131/(3600 z).
Substitute each value of z into y = z - 131/(3600 z):
y = 1/60 (30 sqrt(1462809) + 36253)^(1/3) - 131/(60 (30 sqrt(1462809) + 36253)^(1/3)) or y = -1/60 (-30 sqrt(1462809) - 36253)^(1/3) - (131 (-1)^(2/3))/(60 (30 sqrt(1462809) + 36253)^(1/3)) or y = 131/60 ((-1)/(30 sqrt(1462809) + 36253))^(1/3) + 1/60 (-1)^(2/3) (30 sqrt(1462809) + 36253)^(1/3)
Hint: | Simplify each solution.
Bring each solution to a common denominator and simplify:
y = ((30 sqrt(1462809) + 36253)^(2/3) - 131)/(60 (36253 + 30 sqrt(1462809))^(1/3)) or y = -1/60 (-1/(36253 + 30 sqrt(1462809)))^(1/3) ((30 sqrt(1462809) + 36253)^(2/3) + 131 (-1)^(1/3)) or y = 1/60 (131 ((-1)/(30 sqrt(1462809) + 36253))^(1/3) + (-1)^(2/3) (30 sqrt(1462809) + 36253)^(1/3))
Hint: | Perform back substitution on the three roots.
Substitute back for x = y + 7/60:
x = ((30 sqrt(1462809) + 36253)^(2/3) - 131)/(60 (30 sqrt(1462809) + 36253)^(1/3)) + 7/60 or x = 7/60 - 1/60 ((-1)/(30 sqrt(1462809) + 36253))^(1/3) (131 (-1)^(1/3) + (30 sqrt(1462809) + 36253)^(2/3)) or x = 1/60 (131 (-1/(36253 + 30 sqrt(1462809)))^(1/3) + (-1)^(2/3) (36253 + 30 sqrt(1462809))^(1/3)) + 7/60
