Solving the Quartic Equation (x_1x_2x_3x_4 -1): Methods and Solutions
In this article, we explore how to solve the given quartic equation (x_1x_2x_3x_4 -1). This problem involves advanced algebra and requires careful manipulation of polynomial expressions.
Introduction to the Problem
Consider the equation:
(x_1x_2x_3x_4 -1)
Algebraic Manipulation
One of the given equations can be rewritten as a matrix multiplication:
(left[x_1x_4right]cdotleft[x_2x_3right] -1)
Additionally, another related equation is:
(x^{25}x^4x^{25}x^6 -1)
Substitution and Simplification
Let us define a new variable (y x^{25}x^5). This simplifies our problem as follows:
(y-1y1 -1)
After further simplification:
(y^2 - 1 -1 implies y^2 0 implies y 0)
Substitute back the value of (y) into the original variable definition:
(x^{25}x^5 0)
From which we derive:
(x^{25}x -5)
Multiplying both sides by (x^{25}x^{25/4} -5^{25/4}), we get:
(left(frac{5}{2}right)^2 frac{25 - 20}{4} implies x^{5/2} pm frac{sqrt{5}}{2})
Therefore, the solutions are:
(x frac{-5 pm sqrt{5}}{2})
Another Approach using Polynomial Manipulation
Consider the equation:
(x_1x_4x_2x_3 -1)
We can see that:
(x_1x_2x_3x_4 -1)
Let us substitute
(d x^{5/2})
We now have:
(d - frac{3}{2}d - frac{1}{2}d^{1/2}d^{3/2} -1)
Which simplifies to:
(d^2 - frac{1}{4}d^2 - frac{9}{4} -1)
Further simplification gives:
(d^4 - frac{5}{2}d^2 frac{25}{16} 0)
This implies:
(d^2 - frac{5}{4} 0 implies d^2 frac{5}{4} implies d pm frac{sqrt{5}}{2})
Substituting back, we get:
(x^{5/2} pm frac{sqrt{5}}{2} implies x -frac{5}{2} pm frac{sqrt{5}}{2})
Thus, the solutions are:
(x -phisqrt{5} text{ and } x -frac{sqrt{5}}{phi})
Alternative Simplification and Solution
From the equation:
(x_1x_4x_2x_3 -1)
Divide both sides by (x_4x_2x_3), we get:
(x_1 -frac{1}{x_2x_3x_4})
Now, substituting (x_1) back into the original equation, we simplify to:
(x_1x_2x_3x_4 -1)
This equation can be simplified to:
(x^{25}x^4x^{25}x^6 -1)
Substituting (y x^{25}x^5), we get:
(y^2 - 2y - 1 0)
((y - 1)^2 0 implies y 1)
Thus, the solutions are:
(x frac{-5 pm sqrt{5}}{2})
Conclusion
The solutions to the quartic equation (x_1x_2x_3x_4 -1) are given by:
(x frac{-5 pm sqrt{5}}{2})
This demonstrates the power of algebraic manipulation and substitution in solving complex polynomial equations.