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Simplifying -1r2 + 23r + -110 = 0 Reorder the terms: -110 + 23r + -1r2 = 0 Solving -110 + 23r + -1r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 110 + -23r + r2 = 0 Move the constant term to the right: Add '-110' to each side of the equation. 110 + -23r + -110 + r2 = 0 + -110 Reorder the terms: 110 + -110 + -23r + r2 = 0 + -110 Combine like terms: 110 + -110 = 0 0 + -23r + r2 = 0 + -110 -23r + r2 = 0 + -110 Combine like terms: 0 + -110 = -110 -23r + r2 = -110 The r term is -23r. Take half its coefficient (-11.5). Square it (132.25) and add it to both sides. Add '132.25' to each side of the equation. -23r + 132.25 + r2 = -110 + 132.25 Reorder the terms: 132.25 + -23r + r2 = -110 + 132.25 Combine like terms: -110 + 132.25 = 22.25 132.25 + -23r + r2 = 22.25 Factor a perfect square on the left side: (r + -11.5)(r + -11.5) = 22.25 Calculate the square root of the right side: 4.716990566 Break this problem into two subproblems by setting (r + -11.5) equal to 4.716990566 and -4.716990566.Subproblem 1
r + -11.5 = 4.716990566 Simplifying r + -11.5 = 4.716990566 Reorder the terms: -11.5 + r = 4.716990566 Solving -11.5 + r = 4.716990566 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '11.5' to each side of the equation. -11.5 + 11.5 + r = 4.716990566 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + r = 4.716990566 + 11.5 r = 4.716990566 + 11.5 Combine like terms: 4.716990566 + 11.5 = 16.216990566 r = 16.216990566 Simplifying r = 16.216990566Subproblem 2
r + -11.5 = -4.716990566 Simplifying r + -11.5 = -4.716990566 Reorder the terms: -11.5 + r = -4.716990566 Solving -11.5 + r = -4.716990566 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '11.5' to each side of the equation. -11.5 + 11.5 + r = -4.716990566 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + r = -4.716990566 + 11.5 r = -4.716990566 + 11.5 Combine like terms: -4.716990566 + 11.5 = 6.783009434 r = 6.783009434 Simplifying r = 6.783009434Solution
The solution to the problem is based on the solutions from the subproblems. r = {16.216990566, 6.783009434}
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