Given the equation 2x3 - 54 = 0, find the roots.
First, factor it:
2(x3 - 27) = 0
Divide both sides by 2:
x3 - 27 = 0
Factor:
(x - 3)(x2 + 3x + 9) = 0
Obviously, one of the roots is 3, but since the given equation had a high power of 3 as in x3, we need 3 roots. Since one of the factors is in the from ax2 + bx + c = 0, we can use the Quadratic Formula to solve for x2 + 3x + 9 = 0 in order to find the other roots:
x =-b ∓ √b2 - 4ac
2a
Fill in teh values and calaculate:
x =-3 ∓ √32 - 4(1)(9)
2(1)
-3 ∓ √9 - 36
2
-3 ∓ √-27
2
-3 ∓ √-27
2
-3 ∓ 3i√3
2
So the 3 roots are as follows:
3
x =-3 + 3i√3
2
-3 - 3i√3
2
For more help on determining possible roots, consider looking into the Rational Root Theorem.