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Simplifying 13 + 3r(4 + -1r) = 5r + -2 13 + (4 * 3r + -1r * 3r) = 5r + -2 13 + (12r + -3r2) = 5r + -2 Reorder the terms: 13 + 12r + -3r2 = -2 + 5r Solving 13 + 12r + -3r2 = -2 + 5r Solving for variable 'r'. Reorder the terms: 13 + 2 + 12r + -5r + -3r2 = -2 + 5r + 2 + -5r Combine like terms: 13 + 2 = 15 15 + 12r + -5r + -3r2 = -2 + 5r + 2 + -5r Combine like terms: 12r + -5r = 7r 15 + 7r + -3r2 = -2 + 5r + 2 + -5r Reorder the terms: 15 + 7r + -3r2 = -2 + 2 + 5r + -5r Combine like terms: -2 + 2 = 0 15 + 7r + -3r2 = 0 + 5r + -5r 15 + 7r + -3r2 = 5r + -5r Combine like terms: 5r + -5r = 0 15 + 7r + -3r2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -5 + -2.333333333r + r2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -2.333333333r + 5 + r2 = 0 + 5 Reorder the terms: -5 + 5 + -2.333333333r + r2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -2.333333333r + r2 = 0 + 5 -2.333333333r + r2 = 0 + 5 Combine like terms: 0 + 5 = 5 -2.333333333r + r2 = 5 The r term is -2.333333333r. Take half its coefficient (-1.166666667). Square it (1.361111112) and add it to both sides. Add '1.361111112' to each side of the equation. -2.333333333r + 1.361111112 + r2 = 5 + 1.361111112 Reorder the terms: 1.361111112 + -2.333333333r + r2 = 5 + 1.361111112 Combine like terms: 5 + 1.361111112 = 6.361111112 1.361111112 + -2.333333333r + r2 = 6.361111112 Factor a perfect square on the left side: (r + -1.166666667)(r + -1.166666667) = 6.361111112 Calculate the square root of the right side: 2.522124325 Break this problem into two subproblems by setting (r + -1.166666667) equal to 2.522124325 and -2.522124325.Subproblem 1
r + -1.166666667 = 2.522124325 Simplifying r + -1.166666667 = 2.522124325 Reorder the terms: -1.166666667 + r = 2.522124325 Solving -1.166666667 + r = 2.522124325 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + r = 2.522124325 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + r = 2.522124325 + 1.166666667 r = 2.522124325 + 1.166666667 Combine like terms: 2.522124325 + 1.166666667 = 3.688790992 r = 3.688790992 Simplifying r = 3.688790992Subproblem 2
r + -1.166666667 = -2.522124325 Simplifying r + -1.166666667 = -2.522124325 Reorder the terms: -1.166666667 + r = -2.522124325 Solving -1.166666667 + r = -2.522124325 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + r = -2.522124325 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + r = -2.522124325 + 1.166666667 r = -2.522124325 + 1.166666667 Combine like terms: -2.522124325 + 1.166666667 = -1.355457658 r = -1.355457658 Simplifying r = -1.355457658Solution
The solution to the problem is based on the solutions from the subproblems. r = {3.688790992, -1.355457658}
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