To solve this equation, we first notice that we can rewrite the absolute value expression as two separate cases: when x^2 + x is greater than or equal to 0, and when x^2 + x is less than 0.
Case 1: x^2 + x >= In this case, the absolute value expression simplifies to x^2 + x. So the equation becomes (x^2 + x)^2 + (x^2 + x) - 2 = 0.
Expanding the left side (x^2 + x)^2 = x^4 + 2x^3 + x^ (x^2 + x) = x^2 + x
Adding these together (x^4 + 2x^3 + x^2) + (x^2 + x) - 2 = x^4 + 2x^3 + 2x^2 + x - 2 = 0
This is a quartic equation that can be difficult to solve without the use of a calculator or computer algebra system.
Case 2: x^2 + x < In this case, the absolute value expression simplifies to -(x^2 + x). So the equation becomes (x^2 + x)^2 - (x^2 + x) - 2 = 0.
Expanding the left side (x^2 + x)^2 = x^4 + 2x^3 + x^ -(x^2 + x) = -x^2 - x
Adding these together (x^4 + 2x^3 + x^2) - (x^2 + x) - 2 = x^4 + 2x^3 + 2x^2 - x - 2 = 0
This is also a quartic equation that can be difficult to solve without a calculator or computer algebra system.
In general, solving quartic equations can be quite challenging and often requires numerical methods or other advanced techniques.
To solve this equation, we first notice that we can rewrite the absolute value expression as two separate cases: when x^2 + x is greater than or equal to 0, and when x^2 + x is less than 0.
Case 1: x^2 + x >=
In this case, the absolute value expression simplifies to x^2 + x. So the equation becomes (x^2 + x)^2 + (x^2 + x) - 2 = 0.
Expanding the left side
(x^2 + x)^2 = x^4 + 2x^3 + x^
(x^2 + x) = x^2 + x
Adding these together
(x^4 + 2x^3 + x^2) + (x^2 + x) - 2 =
x^4 + 2x^3 + 2x^2 + x - 2 = 0
This is a quartic equation that can be difficult to solve without the use of a calculator or computer algebra system.
Case 2: x^2 + x <
In this case, the absolute value expression simplifies to -(x^2 + x). So the equation becomes (x^2 + x)^2 - (x^2 + x) - 2 = 0.
Expanding the left side
(x^2 + x)^2 = x^4 + 2x^3 + x^
-(x^2 + x) = -x^2 - x
Adding these together
(x^4 + 2x^3 + x^2) - (x^2 + x) - 2 =
x^4 + 2x^3 + 2x^2 - x - 2 = 0
This is also a quartic equation that can be difficult to solve without a calculator or computer algebra system.
In general, solving quartic equations can be quite challenging and often requires numerical methods or other advanced techniques.