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Simplifying 4r(2r + -2) + 7(4 + 7r) = 0 Reorder the terms: 4r(-2 + 2r) + 7(4 + 7r) = 0 (-2 * 4r + 2r * 4r) + 7(4 + 7r) = 0 (-8r + 8r2) + 7(4 + 7r) = 0 -8r + 8r2 + (4 * 7 + 7r * 7) = 0 -8r + 8r2 + (28 + 49r) = 0 Reorder the terms: 28 + -8r + 49r + 8r2 = 0 Combine like terms: -8r + 49r = 41r 28 + 41r + 8r2 = 0 Solving 28 + 41r + 8r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 3.5 + 5.125r + r2 = 0 Move the constant term to the right: Add '-3.5' to each side of the equation. 3.5 + 5.125r + -3.5 + r2 = 0 + -3.5 Reorder the terms: 3.5 + -3.5 + 5.125r + r2 = 0 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + 5.125r + r2 = 0 + -3.5 5.125r + r2 = 0 + -3.5 Combine like terms: 0 + -3.5 = -3.5 5.125r + r2 = -3.5 The r term is 5.125r. Take half its coefficient (2.5625). Square it (6.56640625) and add it to both sides. Add '6.56640625' to each side of the equation. 5.125r + 6.56640625 + r2 = -3.5 + 6.56640625 Reorder the terms: 6.56640625 + 5.125r + r2 = -3.5 + 6.56640625 Combine like terms: -3.5 + 6.56640625 = 3.06640625 6.56640625 + 5.125r + r2 = 3.06640625 Factor a perfect square on the left side: (r + 2.5625)(r + 2.5625) = 3.06640625 Calculate the square root of the right side: 1.751115716 Break this problem into two subproblems by setting (r + 2.5625) equal to 1.751115716 and -1.751115716.Subproblem 1
r + 2.5625 = 1.751115716 Simplifying r + 2.5625 = 1.751115716 Reorder the terms: 2.5625 + r = 1.751115716 Solving 2.5625 + r = 1.751115716 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.5625' to each side of the equation. 2.5625 + -2.5625 + r = 1.751115716 + -2.5625 Combine like terms: 2.5625 + -2.5625 = 0.0000 0.0000 + r = 1.751115716 + -2.5625 r = 1.751115716 + -2.5625 Combine like terms: 1.751115716 + -2.5625 = -0.811384284 r = -0.811384284 Simplifying r = -0.811384284Subproblem 2
r + 2.5625 = -1.751115716 Simplifying r + 2.5625 = -1.751115716 Reorder the terms: 2.5625 + r = -1.751115716 Solving 2.5625 + r = -1.751115716 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.5625' to each side of the equation. 2.5625 + -2.5625 + r = -1.751115716 + -2.5625 Combine like terms: 2.5625 + -2.5625 = 0.0000 0.0000 + r = -1.751115716 + -2.5625 r = -1.751115716 + -2.5625 Combine like terms: -1.751115716 + -2.5625 = -4.313615716 r = -4.313615716 Simplifying r = -4.313615716Solution
The solution to the problem is based on the solutions from the subproblems. r = {-0.811384284, -4.313615716}
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