Solve the following equations, finding all values of x E R such that this equations are true.

Question

First equation

(x²+7x+11)(x²+1)(ln(x))(tan(x))=0

Second equation

x⁴(2x²-4x-3)(e^{3x^3-7x+4})(2sin(x)-1)=0

Answer 1

For the first equation:

(x²+7x+11)(x²+1)(ln(x))(tan(x))=0

At least one of the factors should be equal to zero:

x²+7x+11=0

x²+1=0 (No real solution)

ln(x)=0

tan(x)=0

x²+7x+11=0

x²+7x=-11  

x²+7x +49/4 =-11 + 49/4

(x+ 7/2)² = 5/4

x+ 7/2 = ±√5 / 2

x = (-7±√5) / 2

ln(x)=0

x=1

tan(x)=0

x= ± nπ

For n= 0,1,2,3,…

For the second equation:

x⁴(2x²-4x-3)(e^{3x^3-7x+4})(2sin(x)-1)=0

x⁴ = 0, x=0

2x²-4x-3 =0

2x²-4x = 3

x²-2x = 3/2

x²-2x+1 = 3/2+1

(x-1)² =5/2

x-1 = ± √10 / 2

 x =(2 ± √10) / 2

e^{3x^3-7x+4} = 0 (No real solution)

2sin(x)-1 =0 

sin(x) = 1/2

x= π/6 ± 2n•π

x= 5π/6 ± 2n•π

For n =0,1,2,3,…