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Simplifying 6r2 + 6r + -112 = 0 Reorder the terms: -112 + 6r + 6r2 = 0 Solving -112 + 6r + 6r2 = 0 Solving for variable 'r'. Factor out the Greatest Common Factor (GCF), '2'. 2(-56 + 3r + 3r2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-56 + 3r + 3r2)' equal to zero and attempt to solve: Simplifying -56 + 3r + 3r2 = 0 Solving -56 + 3r + 3r2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -18.66666667 + r + r2 = 0 Move the constant term to the right: Add '18.66666667' to each side of the equation. -18.66666667 + r + 18.66666667 + r2 = 0 + 18.66666667 Reorder the terms: -18.66666667 + 18.66666667 + r + r2 = 0 + 18.66666667 Combine like terms: -18.66666667 + 18.66666667 = 0.00000000 0.00000000 + r + r2 = 0 + 18.66666667 r + r2 = 0 + 18.66666667 Combine like terms: 0 + 18.66666667 = 18.66666667 r + r2 = 18.66666667 The r term is r. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + r2 = 18.66666667 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + r2 = 18.66666667 + 0.25 Combine like terms: 18.66666667 + 0.25 = 18.91666667 1.25 + r2 = 18.91666667 Factor a perfect square on the left side: (r + 0.5)(r + 0.5) = 18.91666667 Calculate the square root of the right side: 4.349329451 Break this problem into two subproblems by setting (r + 0.5) equal to 4.349329451 and -4.349329451.Subproblem 1
r + 0.5 = 4.349329451 Simplifying r + 0.5 = 4.349329451 Reorder the terms: 0.5 + r = 4.349329451 Solving 0.5 + r = 4.349329451 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = 4.349329451 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = 4.349329451 + -0.5 r = 4.349329451 + -0.5 Combine like terms: 4.349329451 + -0.5 = 3.849329451 r = 3.849329451 Simplifying r = 3.849329451Subproblem 2
r + 0.5 = -4.349329451 Simplifying r + 0.5 = -4.349329451 Reorder the terms: 0.5 + r = -4.349329451 Solving 0.5 + r = -4.349329451 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = -4.349329451 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = -4.349329451 + -0.5 r = -4.349329451 + -0.5 Combine like terms: -4.349329451 + -0.5 = -4.849329451 r = -4.849329451 Simplifying r = -4.849329451Solution
The solution to the problem is based on the solutions from the subproblems. r = {3.849329451, -4.849329451}Solution
r = {3.849329451, -4.849329451}
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