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Simplifying 27r = 4r2 + -15 Reorder the terms: 27r = -15 + 4r2 Solving 27r = -15 + 4r2 Solving for variable 'r'. Reorder the terms: 15 + 27r + -4r2 = -15 + 4r2 + 15 + -4r2 Reorder the terms: 15 + 27r + -4r2 = -15 + 15 + 4r2 + -4r2 Combine like terms: -15 + 15 = 0 15 + 27r + -4r2 = 0 + 4r2 + -4r2 15 + 27r + -4r2 = 4r2 + -4r2 Combine like terms: 4r2 + -4r2 = 0 15 + 27r + -4r2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -3.75 + -6.75r + r2 = 0 Move the constant term to the right: Add '3.75' to each side of the equation. -3.75 + -6.75r + 3.75 + r2 = 0 + 3.75 Reorder the terms: -3.75 + 3.75 + -6.75r + r2 = 0 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + -6.75r + r2 = 0 + 3.75 -6.75r + r2 = 0 + 3.75 Combine like terms: 0 + 3.75 = 3.75 -6.75r + r2 = 3.75 The r term is -6.75r. Take half its coefficient (-3.375). Square it (11.390625) and add it to both sides. Add '11.390625' to each side of the equation. -6.75r + 11.390625 + r2 = 3.75 + 11.390625 Reorder the terms: 11.390625 + -6.75r + r2 = 3.75 + 11.390625 Combine like terms: 3.75 + 11.390625 = 15.140625 11.390625 + -6.75r + r2 = 15.140625 Factor a perfect square on the left side: (r + -3.375)(r + -3.375) = 15.140625 Calculate the square root of the right side: 3.891095604 Break this problem into two subproblems by setting (r + -3.375) equal to 3.891095604 and -3.891095604.Subproblem 1
r + -3.375 = 3.891095604 Simplifying r + -3.375 = 3.891095604 Reorder the terms: -3.375 + r = 3.891095604 Solving -3.375 + r = 3.891095604 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.375' to each side of the equation. -3.375 + 3.375 + r = 3.891095604 + 3.375 Combine like terms: -3.375 + 3.375 = 0.000 0.000 + r = 3.891095604 + 3.375 r = 3.891095604 + 3.375 Combine like terms: 3.891095604 + 3.375 = 7.266095604 r = 7.266095604 Simplifying r = 7.266095604Subproblem 2
r + -3.375 = -3.891095604 Simplifying r + -3.375 = -3.891095604 Reorder the terms: -3.375 + r = -3.891095604 Solving -3.375 + r = -3.891095604 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '3.375' to each side of the equation. -3.375 + 3.375 + r = -3.891095604 + 3.375 Combine like terms: -3.375 + 3.375 = 0.000 0.000 + r = -3.891095604 + 3.375 r = -3.891095604 + 3.375 Combine like terms: -3.891095604 + 3.375 = -0.516095604 r = -0.516095604 Simplifying r = -0.516095604Solution
The solution to the problem is based on the solutions from the subproblems. r = {7.266095604, -0.516095604}
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