Answer:
[tex]x=\dfrac{-1 \pm \sqrt{5}}{2}[/tex]
Step-by-step explanation:
Given the equation:
[tex]x^2e^x + xe^x - e^x = 0[/tex]
We can solve for x by factoring and using the zero product property.
↓ factoring out an [tex]e^x[/tex] from every term
[tex]e^x(x^2 + x - 1) = 0[/tex]
↓ splitting into two equations using the zero product property
1) [tex]e^x \ne 0[/tex]
[tex]\implies \text{extraneous solution}[/tex]
_____
2) [tex]x^2 + x - 1 = 0[/tex]
↓ plugging into the quadratic formula
[tex]x=\dfrac{-1 \pm \sqrt{1^2 - 4(1)(-1)}}{2(1)}[/tex]
↓ simplifying the square root
[tex]\boxed{x=\dfrac{-1 \pm \sqrt{5}}{2}}[/tex]
