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Simplifying 3r2 + 12r2 + -27r + -108 = 0 Reorder the terms: -108 + -27r + 3r2 + 12r2 = 0 Combine like terms: 3r2 + 12r2 = 15r2 -108 + -27r + 15r2 = 0 Solving -108 + -27r + 15r2 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), '3'. 3(-36 + -9r + 5r2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-36 + -9r + 5r2)' equal to zero and attempt to solve: Simplifying -36 + -9r + 5r2 = 0 Solving -36 + -9r + 5r2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -7.2 + -1.8r + r2 = 0 Move the constant term to the right: Add '7.2' to each side of the equation. -7.2 + -1.8r + 7.2 + r2 = 0 + 7.2 Reorder the terms: -7.2 + 7.2 + -1.8r + r2 = 0 + 7.2 Combine like terms: -7.2 + 7.2 = 0.0 0.0 + -1.8r + r2 = 0 + 7.2 -1.8r + r2 = 0 + 7.2 Combine like terms: 0 + 7.2 = 7.2 -1.8r + r2 = 7.2 The r term is -1.8r. Take half its coefficient (-0.9). Square it (0.81) and add it to both sides. Add '0.81' to each side of the equation. -1.8r + 0.81 + r2 = 7.2 + 0.81 Reorder the terms: 0.81 + -1.8r + r2 = 7.2 + 0.81 Combine like terms: 7.2 + 0.81 = 8.01 0.81 + -1.8r + r2 = 8.01 Factor a perfect square on the left side: (r + -0.9)(r + -0.9) = 8.01 Calculate the square root of the right side: 2.83019434 Break this problem into two subproblems by setting (r + -0.9) equal to 2.83019434 and -2.83019434.Subproblem 1
r + -0.9 = 2.83019434 Simplifying r + -0.9 = 2.83019434 Reorder the terms: -0.9 + r = 2.83019434 Solving -0.9 + r = 2.83019434 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.9' to each side of the equation. -0.9 + 0.9 + r = 2.83019434 + 0.9 Combine like terms: -0.9 + 0.9 = 0.0 0.0 + r = 2.83019434 + 0.9 r = 2.83019434 + 0.9 Combine like terms: 2.83019434 + 0.9 = 3.73019434 r = 3.73019434 Simplifying r = 3.73019434Subproblem 2
r + -0.9 = -2.83019434 Simplifying r + -0.9 = -2.83019434 Reorder the terms: -0.9 + r = -2.83019434 Solving -0.9 + r = -2.83019434 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.9' to each side of the equation. -0.9 + 0.9 + r = -2.83019434 + 0.9 Combine like terms: -0.9 + 0.9 = 0.0 0.0 + r = -2.83019434 + 0.9 r = -2.83019434 + 0.9 Combine like terms: -2.83019434 + 0.9 = -1.93019434 r = -1.93019434 Simplifying r = -1.93019434Solution
The solution to the problem is based on the solutions from the subproblems. r = {3.73019434, -1.93019434}Solution
r = {3.73019434, -1.93019434}
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