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Simplifying -8r = 2r2 + 7 Reorder the terms: -8r = 7 + 2r2 Solving -8r = 7 + 2r2 Solving for variable 'r'. Reorder the terms: -7 + -8r + -2r2 = 7 + 2r2 + -7 + -2r2 Reorder the terms: -7 + -8r + -2r2 = 7 + -7 + 2r2 + -2r2 Combine like terms: 7 + -7 = 0 -7 + -8r + -2r2 = 0 + 2r2 + -2r2 -7 + -8r + -2r2 = 2r2 + -2r2 Combine like terms: 2r2 + -2r2 = 0 -7 + -8r + -2r2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(7 + 8r + 2r2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(7 + 8r + 2r2)' equal to zero and attempt to solve: Simplifying 7 + 8r + 2r2 = 0 Solving 7 + 8r + 2r2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 3.5 + 4r + r2 = 0 Move the constant term to the right: Add '-3.5' to each side of the equation. 3.5 + 4r + -3.5 + r2 = 0 + -3.5 Reorder the terms: 3.5 + -3.5 + 4r + r2 = 0 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + 4r + r2 = 0 + -3.5 4r + r2 = 0 + -3.5 Combine like terms: 0 + -3.5 = -3.5 4r + r2 = -3.5 The r term is 4r. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4r + 4 + r2 = -3.5 + 4 Reorder the terms: 4 + 4r + r2 = -3.5 + 4 Combine like terms: -3.5 + 4 = 0.5 4 + 4r + r2 = 0.5 Factor a perfect square on the left side: (r + 2)(r + 2) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (r + 2) equal to 0.707106781 and -0.707106781.Subproblem 1
r + 2 = 0.707106781 Simplifying r + 2 = 0.707106781 Reorder the terms: 2 + r = 0.707106781 Solving 2 + r = 0.707106781 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 + r = 0.707106781 + -2 Combine like terms: 2 + -2 = 0 0 + r = 0.707106781 + -2 r = 0.707106781 + -2 Combine like terms: 0.707106781 + -2 = -1.292893219 r = -1.292893219 Simplifying r = -1.292893219Subproblem 2
r + 2 = -0.707106781 Simplifying r + 2 = -0.707106781 Reorder the terms: 2 + r = -0.707106781 Solving 2 + r = -0.707106781 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 + r = -0.707106781 + -2 Combine like terms: 2 + -2 = 0 0 + r = -0.707106781 + -2 r = -0.707106781 + -2 Combine like terms: -0.707106781 + -2 = -2.707106781 r = -2.707106781 Simplifying r = -2.707106781Solution
The solution to the problem is based on the solutions from the subproblems. r = {-1.292893219, -2.707106781}Solution
r = {-1.292893219, -2.707106781}
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