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Count the real roots of a reciprocal equation (in e^x or x) by putting t = y + 1/y or y - 1/y

This type of question, asked 5 times: 2021 Paper 17, 28 Jun 2022, 24 Jan 2023, 31 Jan 2023, 11 Apr 2023. All 5 questions below, oldest first.

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  1. Question 1

    2021 Paper 17

    86
    The number of real roots of the equation e4x−e3x−4e2x−ex+1=0e^{4x}-e^{3x}- 4e^{2x}-e^{x}+ 1 = 0 is equal to

  2. Question 2

    28 Jun 2022

    82
    The number of real solutions of the equation e4x+4e3x−58e2x+4ex+1=0e^{4x}+ 4e^{3x}- 58e^{2x}+ 4e^{x}+ 1 = 0 is

  3. Question 3

    24 Jan 2023

    157
    The number of real solutions of the equation 3(x2+1x2)−2(x+1x)+5=03\left( x^{2}+\frac{1}{x^{2}} \right)- 2\left( x +\frac{1}{x} \right)+ 5 = 0, is

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  4. Question 4

    31 Jan 2023

    151
    The equation e4x+8e3x+13e2x−8ex+1=0,x∈Re^{4x}+ 8e^{3x}+ 13e^{2x}- 8e^{x}+ 1 = 0,x \in R has:

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  5. Question 5

    11 Apr 2023

    171
    The number of points, where the curve f(x)=e8x−e6x−3e4x−e2x+1,x∈Rf(x) =e^{8x}-e^{6x}- 3e^{4x}-e^{2x}+ 1,x \in R cuts xx-axis, is equal to