What is the solution of x^4 + 6x^2 + 5 = 0

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The solutions of the equation x^4 + 6x^2 + 5 = 0 have to be determined. It can be seen that the highest power of x in the equation is 4. As a result the equation has 4 solutions.

x^4 + 6x^2 + 5 = 0

Let x^2 = y

=> y^2 + 6y + 5 = 0

=> y^2 + 5y + y + 5 = 0

=> y(y + 5) + 1(y + 5) = 0

=> (y + 1)(y + 5) = 0

y + 1 = 0

=> y = -1

y + 5 = 0

=> y = -5

As y = x^2

x^2 = -1

=> x = i and x = -i

x^2 = -5

=> x = `i*sqrt 5` and x = `-sqrt 5*i`

The solutions of the equation x^4 + 6x^2 + 5 = 0 are `{-i, -sqrt 5*i, sqrt 5*i, i}`

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vaaruni | High School Teacher | (Level 1) Salutatorian

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Given :- x^4 + 6x^2 + 5 = 0

       => (x^2)^2 + 6(x^2) + 5 = 0

   Let x^2 be = a ,   then the given equation

 (x^2)^2 + 6(x^2) + 5 = a^2 + 6a + 5 = 0

 Taking the equation :-   a^2 + 6a + 5 =0

 => a^2 + 5a + a + 5 = 0

=> a(a+5)+1(a+5) = 0

=> (a+5)(a+1) = 0

case I :

If  a+5 = 0  then,   a = -5  

                  Or,   x^2= -5  [ substituting value of 'a' ]

                  Or,   x = sqrt(-5)

                  x = √5 i   and  x = - √5 i [ x will be imaginary number]

case II :

 If   a + 1 =0   then ,  a = -1

                      Or,   x^2 = -1

                      Or,   x = sqrt(-1)

                    Or,  x = i  and  x = -i  [ x will be imaginary number]      Hence ,

 x = [ i,-i , √5 i, -√5 i ]   <--- Answer 

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