Find all real numbers a that satisfy
1/(a^3 + 7) - 7 = -a^3/(a^3 + 7) + 5.
Solve for a:
1/(a^3 + 7) - 7 = 5 - a^3/(a^3 + 7)
Bring 1/(a^3 + 7) - 7 together using the common denominator a^3 + 7. Bring 5 - a^3/(a^3 + 7) together using the common denominator a^3 + 7:
(-7 a^3 - 48)/(a^3 + 7) = (4 a^3 + 35)/(a^3 + 7)
Multiply both sides by a^3 + 7:
-7 a^3 - 48 = 4 a^3 + 35
Subtract 4 a^3 - 48 from both sides:
-11 a^3 = 83
Divide both sides by -11:
a^3 = -83/11
Taking cube roots gives (-83/11)^(1/3) times the third roots of unity:
a = (-83/11)^(1/3) or a = -(83/11)^(1/3) or a = -(-1)^(2/3) (83/11)^(1/3)
