Question

(x^2+x)^2+|x^2+x|-2=0

Answer 1

x^2 + x >= 0

(x^2 + x)^2 + x^2 + x - 2 = 0

x^2 + x + (x^2 + x)^2 - 2 = 0

x1 = 1/2 + √5/2

x2 = - 1/2 - √7i/2

x^2 + x < 0

-1 < x ^ x < 0

-x^2 - x + (x^2 + x)^2 - 2 = 0

x5 = -2

x6 = 1

x7 = -1/2 - √3i/2

x8 = -1/2 + √3i/2

x1 = - 1/2 + √5/2

x2 = - √5/2 + 1/2

x1 = -1.61803398875

x2 = 0.61803398875

Answer 2

To solve the equation $(x^2+x)^2+|x^2+x|-2=0$, we'll first simplify it step by step and then find the solutions for $x$.

Step 1: Simplify the equation

The term $|x^2+x|$ can be split into two cases: $x^2+x$ if $x^2+x \geq 0$ and $-(x^2+x)$ if $x^2+x < 0$.

Case 1: $x^2+x \geq 0$ In this case, $|x^2+x| = x^2+x$. So the equation becomes: $(x^2+x)^2 + (x^2+x) - 2 = 0$.

Case 2: $x^2+x < 0$ In this case, $|x^2+x| = -(x^2+x) = -x^2 - x$. So the equation becomes: $(x^2+x)^2 - (x^2+x) - 2 = 0$.

Step 2: Solve for $x$ in each case

Case 1: $(x^2+x)^2 + (x^2+x) - 2 = 0$ Let's expand $(x^2+x)^2$: $(x^2+x)^2 = (x^2+x)(x^2+x) = x^4 + 2x^3 + x^2$.

Now the equation becomes: $x^4 + 2x^3 + x^2 + x^2 + x - 2 = 0$.

Combine like terms: $x^4 + 2x^3 + 2x^2 + x - 2 = 0$.

Unfortunately, this polynomial equation does not have an elementary closed-form solution. You would need to use numerical methods to approximate the solutions.

Case 2: $(x^2+x)^2 - (x^2+x) - 2 = 0$ Let's expand $(x^2+x)^2$: $(x^2+x)^2 = (x^2+x)(x^2+x) = x^4 + 2x^3 + x^2$.

Now the equation becomes: $x^4 + 2x^3 + x^2 - x^2 - x - 2 = 0$.

Combine like terms: $x^4 + 2x^3 - x - 2 = 0$.

Again, this polynomial equation does not have an elementary closed-form solution, so you would need to use numerical methods to approximate the solutions.

In summary, the solutions to the original equation $(x^2+x)^2+|x^2+x|-2=0$ involve finding the roots of either $x^4 + 2x^3 + 2x^2 + x - 2 = 0$