We are tasked with solving the equation:

$x^4 + x^2 - x - 1 = 1 - x - x^2 - x^4$

Step 1: Combine like terms

Combine terms on both sides to simplify the equation: $x^4 + x^2 - x - 1 - (1 - x - x^2 - x^4) = 0$

Expanding and rearranging: $x^4 + x^2 - x - 1 - 1 + x + x^2 + x^4 = 0$

Simplify: $2x^4 + 2x^2 - 2 = 0$

Step 2: Factor out common terms

Factor out $2$: $2(x^4 + x^2 - 1) = 0$

This simplifies to: $x^4 + x^2 - 1 = 0$

Step 3: Substitute and solve

Let $y = x^2$. Then the equation becomes: $y^2 + y - 1 = 0$

This is a quadratic equation. Solve using the quadratic formula: $y = \frac{-1 \pm \sqrt{1^2 - 4(1)(-1)}}{2(1)}$

$y = \frac{-1 \pm \sqrt{1 + 4}}{2}$

$y = \frac{-1 \pm \sqrt{5}}{2}$

Thus: $y = \frac{-1 + \sqrt{5}}{2} \quad \text{or} \quad y = \frac{-1 - \sqrt{5}}{2}$

Since $y = x^2 \geq 0$, we discard $y = \frac{-1 - \sqrt{5}}{2}$ as it is negative. Therefore: $y = \frac{-1 + \sqrt{5}}{2}$

Step 4: Back-substitute $y = x^2$

Now solve: $x^2 = \frac{-1 + \sqrt{5}}{2}$

Take the square root of both sides: $x = \pm \sqrt{\frac{-1 + \sqrt{5}}{2}}$

Final Answer:

The possible values of $x$ are: $x = \sqrt{\frac{-1 + \sqrt{5}}{2}} \quad \text{and} \quad x = -\sqrt{\frac{-1 + \sqrt{5}}{2}}$

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Would you like a more detailed breakdown of any steps?

Related Questions:

1. How do we verify the solutions in the original equation?
2. Can this equation have complex roots if we include the negative value of $y$?
3. How do we graph $x^4 + x^2 - 1 = 0$ to confirm these solutions visually?
4. What is the significance of the golden ratio in this solution process?
5. How does the discriminant help determine the nature of roots in a quadratic equation?

Tip:

Always check for extraneous solutions when solving equations involving radicals or substitutions.