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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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