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$\exponential{(x)}{3} = 1 $
Solve for x (complex solution)
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x^{3}-1=0
Subtract 1 from both sides.
±1
By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -1 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.
x=1
Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.
x^{2}+x+1=0
By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-1 by x-1 to get x^{2}+x+1. Solve the equation where the result equals to 0.
x=\frac{-1±\sqrt{1^{2}-4\times 1\times 1}}{2}
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, 1 for b, and 1 for c in the quadratic formula.
x=\frac{-1±\sqrt{-3}}{2}
Do the calculations.
x=\frac{-\sqrt{3}i-1}{2} x=\frac{-1+\sqrt{3}i}{2}
Solve the equation x^{2}+x+1=0 when ± is plus and when ± is minus.
x=1 x=\frac{-\sqrt{3}i-1}{2} x=\frac{-1+\sqrt{3}i}{2}
List all found solutions.
x^{3}-1=0
Subtract 1 from both sides.
±1
By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -1 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.
x=1
Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.
x^{2}+x+1=0
By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-1 by x-1 to get x^{2}+x+1. Solve the equation where the result equals to 0.
x=\frac{-1±\sqrt{1^{2}-4\times 1\times 1}}{2}
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, 1 for b, and 1 for c in the quadratic formula.
x=\frac{-1±\sqrt{-3}}{2}
Do the calculations.
x\in \emptyset
Since the square root of a negative number is not defined in the real field, there are no solutions.
x=1
List all found solutions.