Answer
Verified
399.9k+ views
Hint: In this question, we first need to take 8 from the left hand side to the right hand side. Then divide both the sides with -8 and apply the cube root on both the sides. Now, equate the left hand side with each of the cube roots of unity and simplify further to get the result.
Complete step-by-step answer:
The Cube Roots of Unity
Cube roots of unity are \[1,\omega ,{{\omega }^{2}}\]
Where \[\omega =\dfrac{-1}{2}+i\dfrac{\sqrt{3}}{2}\]
and \[{{\omega }^{2}}=\dfrac{-1}{2}-i\dfrac{\sqrt{3}}{2}\]
Now, from the given equation in the question we have
\[\Rightarrow {{\left( x-1 \right)}^{3}}+8=0\]
Let us now subtract 8 on both the sides to simplify it further
\[\Rightarrow {{\left( x-1 \right)}^{3}}=-8\]
Let us now divide with -8 on both the sides respectively
\[\Rightarrow \dfrac{{{\left( x-1 \right)}^{3}}}{-8}=1\]
Now, this can be further written in the simplified form to solve further as
\[\Rightarrow {{\left( \dfrac{x-1}{-2} \right)}^{3}}=1\]
Let us now apply the cube root on both the sides for further simplification
\[\Rightarrow {{\left( {{\left( \dfrac{x-1}{-2} \right)}^{3}} \right)}^{\dfrac{1}{3}}}=\sqrt[3]{1}\]
Now, on further simplification the above equation can be further written as
\[\Rightarrow \dfrac{x-1}{-2}=\sqrt[3]{1}\]
As already given in the question that the cube roots of unity are \[1,\omega ,{{\omega }^{2}}\]
Now, on considering each of the cube roots of unity separately we get,
Let us first consider the cube root of unity as 1 and substitute in the above obtained equation
\[\Rightarrow \dfrac{x-1}{-2}=1\]
Now, on multiplying both sides with -2 we get,
\[\Rightarrow x-1=-2\]
Now, on rearranging the terms we get,
\[\therefore x=-1\]
Now, let us consider the cube root of unity as \[\omega \] and substitute in the equation
\[\Rightarrow \dfrac{x-1}{-2}=\omega \]
Now, on multiplying both sides with -2 we get,
\[\Rightarrow x-1=-2\omega \]
Now, on rearranging the terms we get,
\[\therefore x=1-2\omega \]
Now, let us consider the cube root of unity as \[{{\omega }^{2}}\] and substitute in the equation
\[\Rightarrow \dfrac{x-1}{-2}={{\omega }^{2}}\]
Now, on multiplying both sides with -2 we get,
\[\Rightarrow x-1=-2{{\omega }^{2}}\]
Now, on rearranging the terms and simplifying further we get,
\[\therefore x=1-2{{\omega }^{2}}\]
Hence, the roots of the equation \[{{\left( x-1 \right)}^{3}}+8=0\]are \[-1,1-2\omega ,1-2{{\omega }^{2}}\]
Note: It is important to note that after taking -8 to the other side if we apply the cube root there itself and find the value of x then we get only one root. So, we need to write the given equation in terms of the cube root of 1 and then simplify further to get all the three roots of the given cubic equation.
It is also to be noted that while doing the arithmetic operations or rearranging we should not neglect any of the terms or consider the wrong sign. Because it changes the corresponding equation and so the roots.
Complete step-by-step answer:
The Cube Roots of Unity
Cube roots of unity are \[1,\omega ,{{\omega }^{2}}\]
Where \[\omega =\dfrac{-1}{2}+i\dfrac{\sqrt{3}}{2}\]
and \[{{\omega }^{2}}=\dfrac{-1}{2}-i\dfrac{\sqrt{3}}{2}\]
Now, from the given equation in the question we have
\[\Rightarrow {{\left( x-1 \right)}^{3}}+8=0\]
Let us now subtract 8 on both the sides to simplify it further
\[\Rightarrow {{\left( x-1 \right)}^{3}}=-8\]
Let us now divide with -8 on both the sides respectively
\[\Rightarrow \dfrac{{{\left( x-1 \right)}^{3}}}{-8}=1\]
Now, this can be further written in the simplified form to solve further as
\[\Rightarrow {{\left( \dfrac{x-1}{-2} \right)}^{3}}=1\]
Let us now apply the cube root on both the sides for further simplification
\[\Rightarrow {{\left( {{\left( \dfrac{x-1}{-2} \right)}^{3}} \right)}^{\dfrac{1}{3}}}=\sqrt[3]{1}\]
Now, on further simplification the above equation can be further written as
\[\Rightarrow \dfrac{x-1}{-2}=\sqrt[3]{1}\]
As already given in the question that the cube roots of unity are \[1,\omega ,{{\omega }^{2}}\]
Now, on considering each of the cube roots of unity separately we get,
Let us first consider the cube root of unity as 1 and substitute in the above obtained equation
\[\Rightarrow \dfrac{x-1}{-2}=1\]
Now, on multiplying both sides with -2 we get,
\[\Rightarrow x-1=-2\]
Now, on rearranging the terms we get,
\[\therefore x=-1\]
Now, let us consider the cube root of unity as \[\omega \] and substitute in the equation
\[\Rightarrow \dfrac{x-1}{-2}=\omega \]
Now, on multiplying both sides with -2 we get,
\[\Rightarrow x-1=-2\omega \]
Now, on rearranging the terms we get,
\[\therefore x=1-2\omega \]
Now, let us consider the cube root of unity as \[{{\omega }^{2}}\] and substitute in the equation
\[\Rightarrow \dfrac{x-1}{-2}={{\omega }^{2}}\]
Now, on multiplying both sides with -2 we get,
\[\Rightarrow x-1=-2{{\omega }^{2}}\]
Now, on rearranging the terms and simplifying further we get,
\[\therefore x=1-2{{\omega }^{2}}\]
Hence, the roots of the equation \[{{\left( x-1 \right)}^{3}}+8=0\]are \[-1,1-2\omega ,1-2{{\omega }^{2}}\]
Note: It is important to note that after taking -8 to the other side if we apply the cube root there itself and find the value of x then we get only one root. So, we need to write the given equation in terms of the cube root of 1 and then simplify further to get all the three roots of the given cubic equation.
It is also to be noted that while doing the arithmetic operations or rearranging we should not neglect any of the terms or consider the wrong sign. Because it changes the corresponding equation and so the roots.
Recently Updated Pages
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE
Select the correct plural noun from the given singular class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
The sum of three consecutive multiples of 11 is 363 class 7 maths CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
How many squares are there in a chess board A 1296 class 11 maths CBSE