Solve: x^2e^x+x e^x-e^x=0

Factor

x^2e^x+x e^x-e^x=0
(x²+x-1)e^x=0
since e^x can never be zero, so
Solve for
(x²+x-1)=0 using the quadratic formula or completing the squares.

To solve the given equation x^2e^x + xe^x - e^x = 0, we can use factoring and the zero-product property. However, this equation is not factorable in a conventional way. Instead, we can use a different approach by factoring out the common factor of e^x.

Step 1:
Rearrange the terms to isolate the common factor of e^x:
e^x(x^2 + x - 1) = 0

Step 2:
Now, set each factor equal to zero and solve for x.

a) e^x = 0
This equation does not have a real solution because e^x is never equal to zero.

b) x^2 + x - 1 = 0
To solve this quadratic equation, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 1, b = 1, and c = -1.

x = (-1 ± √(1^2 - 4(1)(-1))) / (2(1))
x = (-1 ± √(1 + 4)) / 2
x = (-1 ± √5) / 2

So, the solutions to the equation x^2e^x + xe^x - e^x = 0 are x = (-1 + √5) / 2 and x = (-1 - √5) / 2.