r^4-5r^2+5=0

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Solution for r^4-5r^2+5=0 equation:


Simplifying
r4 + -5r2 + 5 = 0

Reorder the terms:
5 + -5r2 + r4 = 0

Solving
5 + -5r2 + r4 = 0

Solving for variable 'r'.

Begin completing the square.

Move the constant term to the right:

Add '-5' to each side of the equation.
5 + -5r2 + -5 + r4 = 0 + -5

Reorder the terms:
5 + -5 + -5r2 + r4 = 0 + -5

Combine like terms: 5 + -5 = 0
0 + -5r2 + r4 = 0 + -5
-5r2 + r4 = 0 + -5

Combine like terms: 0 + -5 = -5
-5r2 + r4 = -5

The r term is -5r2.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5r2 + 6.25 + r4 = -5 + 6.25

Reorder the terms:
6.25 + -5r2 + r4 = -5 + 6.25

Combine like terms: -5 + 6.25 = 1.25
6.25 + -5r2 + r4 = 1.25

Factor a perfect square on the left side:
(r2 + -2.5)(r2 + -2.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(r2 + -2.5) equal to 1.118033989 and -1.118033989.

Subproblem 1

r2 + -2.5 = 1.118033989 Simplifying r2 + -2.5 = 1.118033989 Reorder the terms: -2.5 + r2 = 1.118033989 Solving -2.5 + r2 = 1.118033989 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + r2 = 1.118033989 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + r2 = 1.118033989 + 2.5 r2 = 1.118033989 + 2.5 Combine like terms: 1.118033989 + 2.5 = 3.618033989 r2 = 3.618033989 Simplifying r2 = 3.618033989 Take the square root of each side: r = {-1.902113033, 1.902113033}

Subproblem 2

r2 + -2.5 = -1.118033989 Simplifying r2 + -2.5 = -1.118033989 Reorder the terms: -2.5 + r2 = -1.118033989 Solving -2.5 + r2 = -1.118033989 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + r2 = -1.118033989 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + r2 = -1.118033989 + 2.5 r2 = -1.118033989 + 2.5 Combine like terms: -1.118033989 + 2.5 = 1.381966011 r2 = 1.381966011 Simplifying r2 = 1.381966011 Take the square root of each side: r = {-1.175570504, 1.175570504}

Solution

The solution to the problem is based on the solutions from the subproblems. r = {-1.902113033, 1.902113033, -1.175570504, 1.175570504}

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