Solving Equations with Radicals and Polynomials: A Comprehensive Guide with Examples

Algebra is a fundamental branch of mathematics that deals with the study of equations, simplification of expressions, and manipulation of symbols. In this article, we'll explore how to solve the equation x^2 - sqrt{2}x - 1 when the condition x^4 / x^4 14 is given. This case will be further broken down into multiple scenarios involving radicals and polynomials. Let's delve into the solution process and understand the concepts involved.

Understanding the Given Condition and Its Implications

We start with the given condition:

x^4 / x^4 14

First, simplify the equation:

x^4 - 1 / x^4 14

This can be rewritten as:

(x^2 1/x^2) 14

Next, we set (x^2 1/x^2) k^2 for some real number k. Thus, we have:

(x^2 1/x^2) 16

Since:

(x^2 - 1/x^2) pm 4

Evaluating the Sub-Conditions

Case 1: When (x^2 1/x^2) 4

Substitute:

(x - 1/x)^2 - 2 4

This implies:

(x - 1/x) pm sqrt{6}

Case 1a: When (x - 1/x) sqrt{6}

Solving the quadratic equation:

x^2 - sqrt{6}x - 1 0

The solutions are:

x frac{sqrt{6} pm sqrt{6 4}}{2} frac{sqrt{6} pm sqrt{10}}{2}

Substitute these solutions back into the original equation:

y (frac{sqrt{6} pm sqrt{10}}{2})(sqrt{6} - sqrt{2})

After simplification, we get:

y 2 pm 2sqrt{3}

Case 1b: When (x - 1/x) -sqrt{6}

Solving the quadratic equation:

x^2 sqrt{6}x - 1 0

The solutions are:

x frac{-sqrt{6} pm sqrt{6 4}}{2} frac{-1 pm sqrt{3}}{sqrt{2}}

Substitute these solutions back into the original equation:

y (frac{-1 pm sqrt{3}}{sqrt{2}})(-sqrt{2} - sqrt{2})

After simplification, we get:

y -2 pm 2sqrt{3}

Case 2: When (x^2 1/x^2) -4

Substitute:

(x - 1/x)^2 - 2 -4

This implies:

(x - 1/x) pm isqrt{6}

Case 2a: When (x - 1/x) isqrt{6}

Solving the quadratic equation:

x^2 - isqrt{6}x - 1 0

The solutions are:

x frac{isqrt{6} pm sqrt{-6 4}}{2} frac{isqrt{3} pm 1}{sqrt{2}}

Substitute these solutions back into the original equation:

y (frac{isqrt{3} pm 1}{sqrt{2}})(isqrt{6} - sqrt{2})

After simplification, we get:

y -3 pm sqrt{3} - i(1 pm sqrt{3})

Case 2b: When (x - 1/x) -isqrt{6}

Solving the quadratic equation:

x^2 isqrt{6}x - 1 0

The solutions are:

x frac{-isqrt{6} pm sqrt{-6 4}}{2} frac{isqrt{3} pm 1}{sqrt{2}}

Substitute these solutions back into the original equation:

y (frac{isqrt{3} pm 1}{sqrt{2}})(isqrt{6} - sqrt{2})

After simplification, we get:

y 3 pm sqrt{3} - i(pm 1 sqrt{3})

Combining All Possible Values for y

Combining all possible values from the cases above, we have:

y boxed{0, 2 pm 2sqrt{3}, -3 pm sqrt{3} - i(1 pm sqrt{3}), 3 pm sqrt{3} - i(pm 1 pm sqrt{3})}

This solution provides a detailed breakdown of the mathematical problem, showcasing the application of algebraic techniques, polynomial manipulations, and the handling of complex numbers.