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Simplifying 5 = 4r(2r + 3) Reorder the terms: 5 = 4r(3 + 2r) 5 = (3 * 4r + 2r * 4r) 5 = (12r + 8r2) Solving 5 = 12r + 8r2 Solving for variable 'r'. Reorder the terms: 5 + -12r + -8r2 = 12r + -12r + 8r2 + -8r2 Combine like terms: 12r + -12r = 0 5 + -12r + -8r2 = 0 + 8r2 + -8r2 5 + -12r + -8r2 = 8r2 + -8r2 Combine like terms: 8r2 + -8r2 = 0 5 + -12r + -8r2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -0.625 + 1.5r + r2 = 0 Move the constant term to the right: Add '0.625' to each side of the equation. -0.625 + 1.5r + 0.625 + r2 = 0 + 0.625 Reorder the terms: -0.625 + 0.625 + 1.5r + r2 = 0 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + 1.5r + r2 = 0 + 0.625 1.5r + r2 = 0 + 0.625 Combine like terms: 0 + 0.625 = 0.625 1.5r + r2 = 0.625 The r term is 1.5r. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5r + 0.5625 + r2 = 0.625 + 0.5625 Reorder the terms: 0.5625 + 1.5r + r2 = 0.625 + 0.5625 Combine like terms: 0.625 + 0.5625 = 1.1875 0.5625 + 1.5r + r2 = 1.1875 Factor a perfect square on the left side: (r + 0.75)(r + 0.75) = 1.1875 Calculate the square root of the right side: 1.089724736 Break this problem into two subproblems by setting (r + 0.75) equal to 1.089724736 and -1.089724736.Subproblem 1
r + 0.75 = 1.089724736 Simplifying r + 0.75 = 1.089724736 Reorder the terms: 0.75 + r = 1.089724736 Solving 0.75 + r = 1.089724736 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + r = 1.089724736 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + r = 1.089724736 + -0.75 r = 1.089724736 + -0.75 Combine like terms: 1.089724736 + -0.75 = 0.339724736 r = 0.339724736 Simplifying r = 0.339724736Subproblem 2
r + 0.75 = -1.089724736 Simplifying r + 0.75 = -1.089724736 Reorder the terms: 0.75 + r = -1.089724736 Solving 0.75 + r = -1.089724736 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + r = -1.089724736 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + r = -1.089724736 + -0.75 r = -1.089724736 + -0.75 Combine like terms: -1.089724736 + -0.75 = -1.839724736 r = -1.839724736 Simplifying r = -1.839724736Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.339724736, -1.839724736}
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