2r^4=7r^2-2

Solution for 2r^4=7r^2-2 equation:

Simplifying
2r4 = 7r2 + -2

Reorder the terms:
2r4 = -2 + 7r2

Solving
2r4 = -2 + 7r2

Solving for variable 'r'.

Reorder the terms:
2 + -7r2 + 2r4 = -2 + 7r2 + 2 + -7r2

Reorder the terms:
2 + -7r2 + 2r4 = -2 + 2 + 7r2 + -7r2

Combine like terms: -2 + 2 = 0
2 + -7r2 + 2r4 = 0 + 7r2 + -7r2
2 + -7r2 + 2r4 = 7r2 + -7r2

Combine like terms: 7r2 + -7r2 = 0
2 + -7r2 + 2r4 = 0

Begin completing the square.  Divide all terms by
2 the coefficient of the squared term: 

Divide each side by '2'.
1 + -3.5r2 + r4 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -3.5r2 + -1 + r4 = 0 + -1

Reorder the terms:
1 + -1 + -3.5r2 + r4 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -3.5r2 + r4 = 0 + -1
-3.5r2 + r4 = 0 + -1

Combine like terms: 0 + -1 = -1
-3.5r2 + r4 = -1

The r term is -3.5r2.  Take half its coefficient (-1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
-3.5r2 + 3.0625 + r4 = -1 + 3.0625

Reorder the terms:
3.0625 + -3.5r2 + r4 = -1 + 3.0625

Combine like terms: -1 + 3.0625 = 2.0625
3.0625 + -3.5r2 + r4 = 2.0625

Factor a perfect square on the left side:
(r2 + -1.75)(r2 + -1.75) = 2.0625

Calculate the square root of the right side: 1.436140662

Break this problem into two subproblems by setting 
(r2 + -1.75) equal to 1.436140662 and -1.436140662.

Subproblem 1
r2 + -1.75 = 1.436140662

Simplifying
r2 + -1.75 = 1.436140662

Reorder the terms:
-1.75 + r2 = 1.436140662

Solving
-1.75 + r2 = 1.436140662

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '1.75' to each side of the equation.
-1.75 + 1.75 + r2 = 1.436140662 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + r2 = 1.436140662 + 1.75
r2 = 1.436140662 + 1.75

Combine like terms: 1.436140662 + 1.75 = 3.186140662
r2 = 3.186140662

Simplifying
r2 = 3.186140662

Take the square root of each side:
r = {-1.784976376, 1.784976376}

Subproblem 2
r2 + -1.75 = -1.436140662

Simplifying
r2 + -1.75 = -1.436140662

Reorder the terms:
-1.75 + r2 = -1.436140662

Solving
-1.75 + r2 = -1.436140662

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '1.75' to each side of the equation.
-1.75 + 1.75 + r2 = -1.436140662 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + r2 = -1.436140662 + 1.75
r2 = -1.436140662 + 1.75

Combine like terms: -1.436140662 + 1.75 = 0.313859338
r2 = 0.313859338

Simplifying
r2 = 0.313859338

Take the square root of each side:
r = {-0.560231504, 0.560231504}

Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {-1.784976376, 1.784976376, -0.560231504, 0.560231504}