Hello from MrBillDoesMath!
Answer: -1/3 + i* (1/3) * sqrt(14) and -1/3 - i* (1/3) * sqrt(14)
Discussion :
Recall the solutions of the quadratic equation "ax^2 + bx + c = 0" are given by the formula
x = ( - b +/- sqrt (b^2 - 4ac) ) / 2a
Rewriting the given equation in this for gives
3x^2 + 2x + 5 = 0
so a = 3, b = 2, and c = 5. Substituting the values in the formula gives
x = (- 2 + sqrt ( 2^2 - 4*3*5) ) / 2a #1
and
x = (- 2 - sqrt ( 2^2 - 4*3*5) ) / 2a #2
Simplifying, #1 = ( -2 + sqrt(4-60) )/ (2 * 3) =
-2/(2*3) + (sqrt(-56)) / (2*3) =
-1/3 + i * (sqrt (56)) /6 =
-1/3 + i * (sqrt(4 *14)) /6 =
-1/3 + i * (sqrt(4)/6) * sqrt(14) =
-1/3 + i * (2/6) * sqrt(14) =
-1/3 + i* (1/3) * sqrt(14)
Similarly, the solution of # 2= -1/3 - i* (1/3) * sqrt(14). The only difference between the two solutions is the minus sign boldfaced above.
Regards, MrB