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Simplifying r4 + -4r2 + 2 = 0 Reorder the terms: 2 + -4r2 + r4 = 0 Solving 2 + -4r2 + r4 = 0 Solving for variable 'r'. Begin completing the square. Move the constant term to the right: Add '-2' to each side of the equation. 2 + -4r2 + -2 + r4 = 0 + -2 Reorder the terms: 2 + -2 + -4r2 + r4 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + -4r2 + r4 = 0 + -2 -4r2 + r4 = 0 + -2 Combine like terms: 0 + -2 = -2 -4r2 + r4 = -2 The r term is -4r2. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4r2 + 4 + r4 = -2 + 4 Reorder the terms: 4 + -4r2 + r4 = -2 + 4 Combine like terms: -2 + 4 = 2 4 + -4r2 + r4 = 2 Factor a perfect square on the left side: (r2 + -2)(r2 + -2) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (r2 + -2) equal to 1.414213562 and -1.414213562.Subproblem 1
r2 + -2 = 1.414213562 Simplifying r2 + -2 = 1.414213562 Reorder the terms: -2 + r2 = 1.414213562 Solving -2 + r2 = 1.414213562 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + r2 = 1.414213562 + 2 Combine like terms: -2 + 2 = 0 0 + r2 = 1.414213562 + 2 r2 = 1.414213562 + 2 Combine like terms: 1.414213562 + 2 = 3.414213562 r2 = 3.414213562 Simplifying r2 = 3.414213562 Take the square root of each side: r = {-1.847759065, 1.847759065}Subproblem 2
r2 + -2 = -1.414213562 Simplifying r2 + -2 = -1.414213562 Reorder the terms: -2 + r2 = -1.414213562 Solving -2 + r2 = -1.414213562 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + r2 = -1.414213562 + 2 Combine like terms: -2 + 2 = 0 0 + r2 = -1.414213562 + 2 r2 = -1.414213562 + 2 Combine like terms: -1.414213562 + 2 = 0.585786438 r2 = 0.585786438 Simplifying r2 = 0.585786438 Take the square root of each side: r = {-0.765366865, 0.765366865}Solution
The solution to the problem is based on the solutions from the subproblems. r = {-1.847759065, 1.847759065, -0.765366865, 0.765366865}
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