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Simplifying 8r2 + -16 = 4r Reorder the terms: -16 + 8r2 = 4r Solving -16 + 8r2 = 4r Solving for variable 'r'. Reorder the terms: -16 + -4r + 8r2 = 4r + -4r Combine like terms: 4r + -4r = 0 -16 + -4r + 8r2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-4 + -1r + 2r2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-4 + -1r + 2r2)' equal to zero and attempt to solve: Simplifying -4 + -1r + 2r2 = 0 Solving -4 + -1r + 2r2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2 + -0.5r + r2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -0.5r + 2 + r2 = 0 + 2 Reorder the terms: -2 + 2 + -0.5r + r2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -0.5r + r2 = 0 + 2 -0.5r + r2 = 0 + 2 Combine like terms: 0 + 2 = 2 -0.5r + r2 = 2 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 = 2 + 0.0625 Reorder the terms: 0.0625 + -0.5r + r2 = 2 + 0.0625 Combine like terms: 2 + 0.0625 = 2.0625 0.0625 + -0.5r + r2 = 2.0625 Factor a perfect square on the left side: (r + -0.25)(r + -0.25) = 2.0625 Calculate the square root of the right side: 1.436140662 Break this problem into two subproblems by setting (r + -0.25) equal to 1.436140662 and -1.436140662.Subproblem 1
r + -0.25 = 1.436140662 Simplifying r + -0.25 = 1.436140662 Reorder the terms: -0.25 + r = 1.436140662 Solving -0.25 + r = 1.436140662 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 = 1.436140662 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + r = 1.436140662 + 0.25 r = 1.436140662 + 0.25 Combine like terms: 1.436140662 + 0.25 = 1.686140662 r = 1.686140662 Simplifying r = 1.686140662Subproblem 2
r + -0.25 = -1.436140662 Simplifying r + -0.25 = -1.436140662 Reorder the terms: -0.25 + r = -1.436140662 Solving -0.25 + r = -1.436140662 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 = -1.436140662 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + r = -1.436140662 + 0.25 r = -1.436140662 + 0.25 Combine like terms: -1.436140662 + 0.25 = -1.186140662 r = -1.186140662 Simplifying r = -1.186140662Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.686140662, -1.186140662}Solution
r = {1.686140662, -1.186140662}
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