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Simplifying 12r2 + -36r + -144 = 0 Reorder the terms: -144 + -36r + 12r2 = 0 Solving -144 + -36r + 12r2 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), '12'. 12(-12 + -3r + r2) = 0 Ignore the factor 12.Subproblem 1
Set the factor '(-12 + -3r + r2)' equal to zero and attempt to solve: Simplifying -12 + -3r + r2 = 0 Solving -12 + -3r + r2 = 0 Begin completing the square. Move the constant term to the right: Add '12' to each side of the equation. -12 + -3r + 12 + r2 = 0 + 12 Reorder the terms: -12 + 12 + -3r + r2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + -3r + r2 = 0 + 12 -3r + r2 = 0 + 12 Combine like terms: 0 + 12 = 12 -3r + r2 = 12 The r term is -3r. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3r + 2.25 + r2 = 12 + 2.25 Reorder the terms: 2.25 + -3r + r2 = 12 + 2.25 Combine like terms: 12 + 2.25 = 14.25 2.25 + -3r + r2 = 14.25 Factor a perfect square on the left side: (r + -1.5)(r + -1.5) = 14.25 Calculate the square root of the right side: 3.774917218 Break this problem into two subproblems by setting (r + -1.5) equal to 3.774917218 and -3.774917218.Subproblem 1
r + -1.5 = 3.774917218 Simplifying r + -1.5 = 3.774917218 Reorder the terms: -1.5 + r = 3.774917218 Solving -1.5 + r = 3.774917218 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + r = 3.774917218 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + r = 3.774917218 + 1.5 r = 3.774917218 + 1.5 Combine like terms: 3.774917218 + 1.5 = 5.274917218 r = 5.274917218 Simplifying r = 5.274917218Subproblem 2
r + -1.5 = -3.774917218 Simplifying r + -1.5 = -3.774917218 Reorder the terms: -1.5 + r = -3.774917218 Solving -1.5 + r = -3.774917218 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + r = -3.774917218 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + r = -3.774917218 + 1.5 r = -3.774917218 + 1.5 Combine like terms: -3.774917218 + 1.5 = -2.274917218 r = -2.274917218 Simplifying r = -2.274917218Solution
The solution to the problem is based on the solutions from the subproblems. r = {5.274917218, -2.274917218}Solution
r = {5.274917218, -2.274917218}
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