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Simplifying 6r2 + -15r + -90 = 0 Reorder the terms: -90 + -15r + 6r2 = 0 Solving -90 + -15r + 6r2 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), '3'. 3(-30 + -5r + 2r2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-30 + -5r + 2r2)' equal to zero and attempt to solve: Simplifying -30 + -5r + 2r2 = 0 Solving -30 + -5r + 2r2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -15 + -2.5r + r2 = 0 Move the constant term to the right: Add '15' to each side of the equation. -15 + -2.5r + 15 + r2 = 0 + 15 Reorder the terms: -15 + 15 + -2.5r + r2 = 0 + 15 Combine like terms: -15 + 15 = 0 0 + -2.5r + r2 = 0 + 15 -2.5r + r2 = 0 + 15 Combine like terms: 0 + 15 = 15 -2.5r + r2 = 15 The r term is -2.5r. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5r + 1.5625 + r2 = 15 + 1.5625 Reorder the terms: 1.5625 + -2.5r + r2 = 15 + 1.5625 Combine like terms: 15 + 1.5625 = 16.5625 1.5625 + -2.5r + r2 = 16.5625 Factor a perfect square on the left side: (r + -1.25)(r + -1.25) = 16.5625 Calculate the square root of the right side: 4.069705149 Break this problem into two subproblems by setting (r + -1.25) equal to 4.069705149 and -4.069705149.Subproblem 1
r + -1.25 = 4.069705149 Simplifying r + -1.25 = 4.069705149 Reorder the terms: -1.25 + r = 4.069705149 Solving -1.25 + r = 4.069705149 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + r = 4.069705149 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + r = 4.069705149 + 1.25 r = 4.069705149 + 1.25 Combine like terms: 4.069705149 + 1.25 = 5.319705149 r = 5.319705149 Simplifying r = 5.319705149Subproblem 2
r + -1.25 = -4.069705149 Simplifying r + -1.25 = -4.069705149 Reorder the terms: -1.25 + r = -4.069705149 Solving -1.25 + r = -4.069705149 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + r = -4.069705149 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + r = -4.069705149 + 1.25 r = -4.069705149 + 1.25 Combine like terms: -4.069705149 + 1.25 = -2.819705149 r = -2.819705149 Simplifying r = -2.819705149Solution
The solution to the problem is based on the solutions from the subproblems. r = {5.319705149, -2.819705149}Solution
r = {5.319705149, -2.819705149}
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