%5E2%2B6x%3D1%2B9x%5E2.jpeg "(2x^2-x+1)^2+6x=1+9x^2")
Solving the Equation: (2x^2 - x + 1)^2 + 6x = 1 + 9x^2
This article explores the process of solving the equation (2x^2 - x + 1)^2 + 6x = 1 + 9x^2. We will utilize algebraic manipulations to isolate the variable 'x' and arrive at the solution(s).
Expanding and Simplifying
First, we need to expand the square on the left-hand side of the equation:
(2x^2 - x + 1)^2 = (2x^2 - x + 1)(2x^2 - x + 1)
This expansion results in:
4x^4 - 4x^3 + 6x^2 - 2x + 1
Now, let's substitute this back into the original equation:
4x^4 - 4x^3 + 6x^2 - 2x + 1 + 6x = 1 + 9x^2
Combining like terms, we get:
4x^4 - 4x^3 - 3x^2 + 4x = 0
Factoring and Solving
To solve for 'x', we can factor out a common factor of 4x:
4x(x^3 - x^2 - (3/4)x + 1) = 0
For the entire expression to be equal to zero, either 4x = 0 or the expression inside the parenthesis must equal zero.
Solving for 4x = 0, we get x = 0.
To solve the cubic equation x^3 - x^2 - (3/4)x + 1 = 0, we can use methods like the Rational Root Theorem and synthetic division. However, finding the roots of this cubic equation may not be straightforward and might require numerical methods.
Conclusion
In conclusion, we have found one solution for the equation: x = 0. The other possible solutions would require further analysis of the cubic equation x^3 - x^2 - (3/4)x + 1 = 0. Remember that a polynomial equation of degree 'n' can have up to 'n' distinct solutions (real or complex).