x= -3 and ![x=\frac{-3+/-3i\sqrt{3} }{2}](/tex.php?f=x=\frac{-3+/-3i\sqrt{3} }{2})
step-by-step explanation:
to find the roots of a cubic, use division or factoring to find the factors then set equal to 0. as this is a difference of cubes, it factors into
![(x^3-a^3)=](/tex.php?f=(x^3-a^3)=)
where a is the cube root of
.
here this is 27 and the cube root of 27 is 3. so we write using the form:
.
set each factor to 0 and solve for x.
x-3=0
x=3
the quadratic expression does not factor and must be solved using the quadratic formula.
![x=\frac{-b+/-\sqrt{b^2-4ac} }{2a}](/tex.php?f=x=\frac{-b+/-\sqrt{b^2-4ac} }{2a})
where a = 1, b=3, and c=9
![x=\frac{-3+/-\sqrt{3^2-4(1)(9)} }{2(1)} \\x=\frac{-3+/-\sqrt{9-36)} }{2} \\x=\frac{-3+/-\sqrt{-27)} }{2}\\x=\frac{-3+/-3i\sqrt{3} }{2}](/tex.php?f=x=\frac{-3+/-\sqrt{3^2-4(1)(9)} }{2(1)} \\x=\frac{-3+/-\sqrt{9-36)} }{2} \\x=\frac{-3+/-\sqrt{-27)} }{2}\\x=\frac{-3+/-3i\sqrt{3} }{2})
this cubic has two complex roots.