The solutions of x^{3}=1 (x cubed equals 1) are given by x= 1, ω, and ω^{2} where ω = (-1+√3i)/2 is the complex number. In this post, we will learn how to find the root of x^{3}=1.

## Solutions of x^{3}=1

The given equation is x^{3}=1.

**Step 1**: Apply the formula of a^{3}-b^{3}.

⇒ x^{3 }-1 =0

⇒ x^{3 }-1^{3} =0

⇒ (x^{ }-1)(x^{2}+x+1) =0 [applying the formula of a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2}) ]

So either x-1=0 or x^{2}+x+1=0 **…(∗)**

**Step 2**: Now, we will solve the above two equations.

x-1=0 ⇒ x=1.

and solve x^{2}+x+1=0

Comparing this equation with ax^{2} + bx + c = 0, we get that a=1, b=1, c=1.

Using Shreedhara Acharya’s formula, the solutions of x^{2}+x+1=0 are given by

x = $\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$

= $\dfrac{-1 \pm \sqrt{-3}}{2}$

= $\dfrac{-1 \pm \sqrt{3}i}{2}$.

Therefore, the solutions of x^{2}+x+1 = 0 are $\dfrac{-1 + \sqrt{3}i}{2}$ and $\dfrac{-1 – \sqrt{3}i}{2}$.

**Step 3**: Set ω = (-1+√3i)/2.

Then ω^{2} = (-1-√3i)/2.

So the solutions of x^{2}+x+1 = 0 are ω and ω^{2}.

Therefore, from **(∗)**, we conclude that the solutions of x^{3}=1 are 1, ω, and ω^{2} where ω = (-1+√3i)/2.

## Roots of x^{3}=1

As the solutions of the equation x^{3}=1 are the roots of x^{3}=1, from above we deduce that the roots of x^{3}=1 are

1, ω, and ω^{2}

where ω = (-1+√3i)/2 is a complex number.

**ALSO READ:**

How to solve quadratic equations

## FAQs

### Q1: What are the solutions of x^3=1?

Answer: 1, ω, and ω^{2} are the solutions of x^{3}=1, where ω = (-1+√3i)/2 is the complex number.

### Q2: What are the roots of x^3=1?

Answer: 1, ω, and ω^{2} are the roots of the cubic equation x^{3}=1, where ω = (-1+√3i)/2.