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Simplifying 11r2 + 24r + -94 = -12 Reorder the terms: -94 + 24r + 11r2 = -12 Solving -94 + 24r + 11r2 = -12 Solving for variable 'r'. Reorder the terms: -94 + 12 + 24r + 11r2 = -12 + 12 Combine like terms: -94 + 12 = -82 -82 + 24r + 11r2 = -12 + 12 Combine like terms: -12 + 12 = 0 -82 + 24r + 11r2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -7.454545455 + 2.181818182r + r2 = 0 Move the constant term to the right: Add '7.454545455' to each side of the equation. -7.454545455 + 2.181818182r + 7.454545455 + r2 = 0 + 7.454545455 Reorder the terms: -7.454545455 + 7.454545455 + 2.181818182r + r2 = 0 + 7.454545455 Combine like terms: -7.454545455 + 7.454545455 = 0.000000000 0.000000000 + 2.181818182r + r2 = 0 + 7.454545455 2.181818182r + r2 = 0 + 7.454545455 Combine like terms: 0 + 7.454545455 = 7.454545455 2.181818182r + r2 = 7.454545455 The r term is 2.181818182r. Take half its coefficient (1.090909091). Square it (1.190082645) and add it to both sides. Add '1.190082645' to each side of the equation. 2.181818182r + 1.190082645 + r2 = 7.454545455 + 1.190082645 Reorder the terms: 1.190082645 + 2.181818182r + r2 = 7.454545455 + 1.190082645 Combine like terms: 7.454545455 + 1.190082645 = 8.6446281 1.190082645 + 2.181818182r + r2 = 8.6446281 Factor a perfect square on the left side: (r + 1.090909091)(r + 1.090909091) = 8.6446281 Calculate the square root of the right side: 2.940174842 Break this problem into two subproblems by setting (r + 1.090909091) equal to 2.940174842 and -2.940174842.Subproblem 1
r + 1.090909091 = 2.940174842 Simplifying r + 1.090909091 = 2.940174842 Reorder the terms: 1.090909091 + r = 2.940174842 Solving 1.090909091 + r = 2.940174842 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.090909091' to each side of the equation. 1.090909091 + -1.090909091 + r = 2.940174842 + -1.090909091 Combine like terms: 1.090909091 + -1.090909091 = 0.000000000 0.000000000 + r = 2.940174842 + -1.090909091 r = 2.940174842 + -1.090909091 Combine like terms: 2.940174842 + -1.090909091 = 1.849265751 r = 1.849265751 Simplifying r = 1.849265751Subproblem 2
r + 1.090909091 = -2.940174842 Simplifying r + 1.090909091 = -2.940174842 Reorder the terms: 1.090909091 + r = -2.940174842 Solving 1.090909091 + r = -2.940174842 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.090909091' to each side of the equation. 1.090909091 + -1.090909091 + r = -2.940174842 + -1.090909091 Combine like terms: 1.090909091 + -1.090909091 = 0.000000000 0.000000000 + r = -2.940174842 + -1.090909091 r = -2.940174842 + -1.090909091 Combine like terms: -2.940174842 + -1.090909091 = -4.031083933 r = -4.031083933 Simplifying r = -4.031083933Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.849265751, -4.031083933}
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