Find the number of real root and the number of imaginary root in the following equation e^x=x^2 =?

1 Answer
Cesareo R.
Jul 12, 2016

x = -0.703467

Explanation:

e^x = (e^{x/2})^2

so

e^x - x^2= (e^{x/2})^2-x^2 = (e^{x/2}+x) (e^{x/2}-x)=0

Analyzing the solutions

1) e^{x/2}+x = 0->e^{x/2}=-x
but e^{x/2} > 0, forall x in RR so the solution must be a negative number. Solving iteratively we obtain
x = -0.703467

2) e^{x/2}-x=0->e^{x/2}=x
but e^{x/2}>x, forall x in RR so no intersection between y = e^{x/2} and y = x and consequently, no solution.

Concluding. The solution is x = -0.703467

Attached a figure showing:
y = e^{x/2} green
y = x red
y = -x blue
and the solution point in black

enter image source here

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