Question

Please help me answer this question genius

Answer 1

x ∈ {-0.465, 1.014}

Explanation:

The equation can be cast in the form f(x) = 0, and solved easily using a graphing calculator. That shows x ≈ -0.465 and x ≈ 1.014. The same calculator can iterate the roots to full calculator precision.

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The equation can be made a quadratic by the substitution ...

z = e^(2x)

Then we have ...

`2\cosh(2x)-\sinh(2x)=4\\\\2\cdot(e^(2x)+e^(-2x))/(2)-(e^(2x)-e^(-2x))/(2)-4=0\qquad\text{use exponential identities}\\\\2z+2z^(-1)-z+z^(-1)-8=0\qquad\text{multiply by 2, substitute z}\\\\z^2 -8z+3=0\qquad\text{multiply by z}\\\\z=(8\pm√((-8)^2-4(1)(3)))/(2(1))=4\pm√(13)\\\\x=(\ln(z))/(2)=(\ln(4\pm√(13)))/(2)\approx\{-0465133,1.014439\}`