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Simplifying 8r2 + 4r + -6 = 0 Reorder the terms: -6 + 4r + 8r2 = 0 Solving -6 + 4r + 8r2 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), '2'. 2(-3 + 2r + 4r2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-3 + 2r + 4r2)' equal to zero and attempt to solve: Simplifying -3 + 2r + 4r2 = 0 Solving -3 + 2r + 4r2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.75 + 0.5r + r2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 0.5r + 0.75 + r2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 0.5r + r2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 0.5r + r2 = 0 + 0.75 0.5r + r2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 0.5r + r2 = 0.75 The r term is 0.5r. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5r + 0.0625 + r2 = 0.75 + 0.0625 Reorder the terms: 0.0625 + 0.5r + r2 = 0.75 + 0.0625 Combine like terms: 0.75 + 0.0625 = 0.8125 0.0625 + 0.5r + r2 = 0.8125 Factor a perfect square on the left side: (r + 0.25)(r + 0.25) = 0.8125 Calculate the square root of the right side: 0.901387819 Break this problem into two subproblems by setting (r + 0.25) equal to 0.901387819 and -0.901387819.Subproblem 1
r + 0.25 = 0.901387819 Simplifying r + 0.25 = 0.901387819 Reorder the terms: 0.25 + r = 0.901387819 Solving 0.25 + r = 0.901387819 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + r = 0.901387819 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + r = 0.901387819 + -0.25 r = 0.901387819 + -0.25 Combine like terms: 0.901387819 + -0.25 = 0.651387819 r = 0.651387819 Simplifying r = 0.651387819Subproblem 2
r + 0.25 = -0.901387819 Simplifying r + 0.25 = -0.901387819 Reorder the terms: 0.25 + r = -0.901387819 Solving 0.25 + r = -0.901387819 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + r = -0.901387819 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + r = -0.901387819 + -0.25 r = -0.901387819 + -0.25 Combine like terms: -0.901387819 + -0.25 = -1.151387819 r = -1.151387819 Simplifying r = -1.151387819Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.651387819, -1.151387819}Solution
r = {0.651387819, -1.151387819}
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