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Simplifying 2r(-5 + -1r) = 13 + 2r (-5 * 2r + -1r * 2r) = 13 + 2r (-10r + -2r2) = 13 + 2r Solving -10r + -2r2 = 13 + 2r Solving for variable 'r'. Reorder the terms: -13 + -10r + -2r + -2r2 = 13 + 2r + -13 + -2r Combine like terms: -10r + -2r = -12r -13 + -12r + -2r2 = 13 + 2r + -13 + -2r Reorder the terms: -13 + -12r + -2r2 = 13 + -13 + 2r + -2r Combine like terms: 13 + -13 = 0 -13 + -12r + -2r2 = 0 + 2r + -2r -13 + -12r + -2r2 = 2r + -2r Combine like terms: 2r + -2r = 0 -13 + -12r + -2r2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(13 + 12r + 2r2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(13 + 12r + 2r2)' equal to zero and attempt to solve: Simplifying 13 + 12r + 2r2 = 0 Solving 13 + 12r + 2r2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 6.5 + 6r + r2 = 0 Move the constant term to the right: Add '-6.5' to each side of the equation. 6.5 + 6r + -6.5 + r2 = 0 + -6.5 Reorder the terms: 6.5 + -6.5 + 6r + r2 = 0 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + 6r + r2 = 0 + -6.5 6r + r2 = 0 + -6.5 Combine like terms: 0 + -6.5 = -6.5 6r + r2 = -6.5 The r term is 6r. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6r + 9 + r2 = -6.5 + 9 Reorder the terms: 9 + 6r + r2 = -6.5 + 9 Combine like terms: -6.5 + 9 = 2.5 9 + 6r + r2 = 2.5 Factor a perfect square on the left side: (r + 3)(r + 3) = 2.5 Calculate the square root of the right side: 1.58113883 Break this problem into two subproblems by setting (r + 3) equal to 1.58113883 and -1.58113883.Subproblem 1
r + 3 = 1.58113883 Simplifying r + 3 = 1.58113883 Reorder the terms: 3 + r = 1.58113883 Solving 3 + r = 1.58113883 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + r = 1.58113883 + -3 Combine like terms: 3 + -3 = 0 0 + r = 1.58113883 + -3 r = 1.58113883 + -3 Combine like terms: 1.58113883 + -3 = -1.41886117 r = -1.41886117 Simplifying r = -1.41886117Subproblem 2
r + 3 = -1.58113883 Simplifying r + 3 = -1.58113883 Reorder the terms: 3 + r = -1.58113883 Solving 3 + r = -1.58113883 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + r = -1.58113883 + -3 Combine like terms: 3 + -3 = 0 0 + r = -1.58113883 + -3 r = -1.58113883 + -3 Combine like terms: -1.58113883 + -3 = -4.58113883 r = -4.58113883 Simplifying r = -4.58113883Solution
The solution to the problem is based on the solutions from the subproblems. r = {-1.41886117, -4.58113883}Solution
r = {-1.41886117, -4.58113883}
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