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Simplifying -18r2 + 130r + 72 = 0 Reorder the terms: 72 + 130r + -18r2 = 0 Solving 72 + 130r + -18r2 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), '2'. 2(36 + 65r + -9r2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(36 + 65r + -9r2)' equal to zero and attempt to solve: Simplifying 36 + 65r + -9r2 = 0 Solving 36 + 65r + -9r2 = 0 Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -4 + -7.222222222r + r2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + -7.222222222r + 4 + r2 = 0 + 4 Reorder the terms: -4 + 4 + -7.222222222r + r2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + -7.222222222r + r2 = 0 + 4 -7.222222222r + r2 = 0 + 4 Combine like terms: 0 + 4 = 4 -7.222222222r + r2 = 4 The r term is -7.222222222r. Take half its coefficient (-3.611111111). Square it (13.04012346) and add it to both sides. Add '13.04012346' to each side of the equation. -7.222222222r + 13.04012346 + r2 = 4 + 13.04012346 Reorder the terms: 13.04012346 + -7.222222222r + r2 = 4 + 13.04012346 Combine like terms: 4 + 13.04012346 = 17.04012346 13.04012346 + -7.222222222r + r2 = 17.04012346 Factor a perfect square on the left side: (r + -3.611111111)(r + -3.611111111) = 17.04012346 Calculate the square root of the right side: 4.127968442 Break this problem into two subproblems by setting (r + -3.611111111) equal to 4.127968442 and -4.127968442.Subproblem 1
r + -3.611111111 = 4.127968442 Simplifying r + -3.611111111 = 4.127968442 Reorder the terms: -3.611111111 + r = 4.127968442 Solving -3.611111111 + r = 4.127968442 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.611111111' to each side of the equation. -3.611111111 + 3.611111111 + r = 4.127968442 + 3.611111111 Combine like terms: -3.611111111 + 3.611111111 = 0.000000000 0.000000000 + r = 4.127968442 + 3.611111111 r = 4.127968442 + 3.611111111 Combine like terms: 4.127968442 + 3.611111111 = 7.739079553 r = 7.739079553 Simplifying r = 7.739079553Subproblem 2
r + -3.611111111 = -4.127968442 Simplifying r + -3.611111111 = -4.127968442 Reorder the terms: -3.611111111 + r = -4.127968442 Solving -3.611111111 + r = -4.127968442 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.611111111' to each side of the equation. -3.611111111 + 3.611111111 + r = -4.127968442 + 3.611111111 Combine like terms: -3.611111111 + 3.611111111 = 0.000000000 0.000000000 + r = -4.127968442 + 3.611111111 r = -4.127968442 + 3.611111111 Combine like terms: -4.127968442 + 3.611111111 = -0.516857331 r = -0.516857331 Simplifying r = -0.516857331Solution
The solution to the problem is based on the solutions from the subproblems. r = {7.739079553, -0.516857331}Solution
r = {7.739079553, -0.516857331}
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