x^4-23x^2+49=0

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Solution for x^4-23x^2+49=0 equation:


Simplifying
x4 + -23x2 + 49 = 0

Reorder the terms:
49 + -23x2 + x4 = 0

Solving
49 + -23x2 + x4 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-49' to each side of the equation.
49 + -23x2 + -49 + x4 = 0 + -49

Reorder the terms:
49 + -49 + -23x2 + x4 = 0 + -49

Combine like terms: 49 + -49 = 0
0 + -23x2 + x4 = 0 + -49
-23x2 + x4 = 0 + -49

Combine like terms: 0 + -49 = -49
-23x2 + x4 = -49

The x term is -23x2.  Take half its coefficient (-11.5).
Square it (132.25) and add it to both sides.

Add '132.25' to each side of the equation.
-23x2 + 132.25 + x4 = -49 + 132.25

Reorder the terms:
132.25 + -23x2 + x4 = -49 + 132.25

Combine like terms: -49 + 132.25 = 83.25
132.25 + -23x2 + x4 = 83.25

Factor a perfect square on the left side:
(x2 + -11.5)(x2 + -11.5) = 83.25

Calculate the square root of the right side: 9.124143795

Break this problem into two subproblems by setting 
(x2 + -11.5) equal to 9.124143795 and -9.124143795.

Subproblem 1

x2 + -11.5 = 9.124143795 Simplifying x2 + -11.5 = 9.124143795 Reorder the terms: -11.5 + x2 = 9.124143795 Solving -11.5 + x2 = 9.124143795 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '11.5' to each side of the equation. -11.5 + 11.5 + x2 = 9.124143795 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + x2 = 9.124143795 + 11.5 x2 = 9.124143795 + 11.5 Combine like terms: 9.124143795 + 11.5 = 20.624143795 x2 = 20.624143795 Simplifying x2 = 20.624143795 Take the square root of each side: x = {-4.541381265, 4.541381265}

Subproblem 2

x2 + -11.5 = -9.124143795 Simplifying x2 + -11.5 = -9.124143795 Reorder the terms: -11.5 + x2 = -9.124143795 Solving -11.5 + x2 = -9.124143795 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '11.5' to each side of the equation. -11.5 + 11.5 + x2 = -9.124143795 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + x2 = -9.124143795 + 11.5 x2 = -9.124143795 + 11.5 Combine like terms: -9.124143795 + 11.5 = 2.375856205 x2 = 2.375856205 Simplifying x2 = 2.375856205 Take the square root of each side: x = {-1.541381265, 1.541381265}

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-4.541381265, 4.541381265, -1.541381265, 1.541381265}

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