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Simplifying 2r2 = 4 + 4r Solving 2r2 = 4 + 4r Solving for variable 'r'. Reorder the terms: -4 + -4r + 2r2 = 4 + 4r + -4 + -4r Reorder the terms: -4 + -4r + 2r2 = 4 + -4 + 4r + -4r Combine like terms: 4 + -4 = 0 -4 + -4r + 2r2 = 0 + 4r + -4r -4 + -4r + 2r2 = 4r + -4r Combine like terms: 4r + -4r = 0 -4 + -4r + 2r2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-2 + -2r + r2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-2 + -2r + r2)' equal to zero and attempt to solve: Simplifying -2 + -2r + r2 = 0 Solving -2 + -2r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '2' to each side of the equation. -2 + -2r + 2 + r2 = 0 + 2 Reorder the terms: -2 + 2 + -2r + r2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -2r + r2 = 0 + 2 -2r + r2 = 0 + 2 Combine like terms: 0 + 2 = 2 -2r + r2 = 2 The r term is -2r. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2r + 1 + r2 = 2 + 1 Reorder the terms: 1 + -2r + r2 = 2 + 1 Combine like terms: 2 + 1 = 3 1 + -2r + r2 = 3 Factor a perfect square on the left side: (r + -1)(r + -1) = 3 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (r + -1) equal to 1.732050808 and -1.732050808.Subproblem 1
r + -1 = 1.732050808 Simplifying r + -1 = 1.732050808 Reorder the terms: -1 + r = 1.732050808 Solving -1 + r = 1.732050808 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + r = 1.732050808 + 1 Combine like terms: -1 + 1 = 0 0 + r = 1.732050808 + 1 r = 1.732050808 + 1 Combine like terms: 1.732050808 + 1 = 2.732050808 r = 2.732050808 Simplifying r = 2.732050808Subproblem 2
r + -1 = -1.732050808 Simplifying r + -1 = -1.732050808 Reorder the terms: -1 + r = -1.732050808 Solving -1 + r = -1.732050808 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + r = -1.732050808 + 1 Combine like terms: -1 + 1 = 0 0 + r = -1.732050808 + 1 r = -1.732050808 + 1 Combine like terms: -1.732050808 + 1 = -0.732050808 r = -0.732050808 Simplifying r = -0.732050808Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.732050808, -0.732050808}Solution
r = {2.732050808, -0.732050808}
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