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Write an equation using “x” and then solve the equation.
x= -3 and 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)=(x-a)(x^2+ax+a^2)
where a is the cube root of a^3.
here this is 27 and the cube root of 27 is 3. so we write using the form:
(x-3)(x^2+(3)x+3^2)\\(x-3)(x^2+3x+9).
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}
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}
this cubic has two complex roots.
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