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Simplifying 5c + 1 = 2c(c + 11) Reorder the terms: 1 + 5c = 2c(c + 11) Reorder the terms: 1 + 5c = 2c(11 + c) 1 + 5c = (11 * 2c + c * 2c) 1 + 5c = (22c + 2c2) Solving 1 + 5c = 22c + 2c2 Solving for variable 'c'. Combine like terms: 5c + -22c = -17c 1 + -17c + -2c2 = 22c + 2c2 + -22c + -2c2 Reorder the terms: 1 + -17c + -2c2 = 22c + -22c + 2c2 + -2c2 Combine like terms: 22c + -22c = 0 1 + -17c + -2c2 = 0 + 2c2 + -2c2 1 + -17c + -2c2 = 2c2 + -2c2 Combine like terms: 2c2 + -2c2 = 0 1 + -17c + -2c2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -0.5 + 8.5c + c2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 8.5c + 0.5 + c2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 8.5c + c2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 8.5c + c2 = 0 + 0.5 8.5c + c2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 8.5c + c2 = 0.5 The c term is 8.5c. Take half its coefficient (4.25). Square it (18.0625) and add it to both sides. Add '18.0625' to each side of the equation. 8.5c + 18.0625 + c2 = 0.5 + 18.0625 Reorder the terms: 18.0625 + 8.5c + c2 = 0.5 + 18.0625 Combine like terms: 0.5 + 18.0625 = 18.5625 18.0625 + 8.5c + c2 = 18.5625 Factor a perfect square on the left side: (c + 4.25)(c + 4.25) = 18.5625 Calculate the square root of the right side: 4.308421985 Break this problem into two subproblems by setting (c + 4.25) equal to 4.308421985 and -4.308421985.Subproblem 1
c + 4.25 = 4.308421985 Simplifying c + 4.25 = 4.308421985 Reorder the terms: 4.25 + c = 4.308421985 Solving 4.25 + c = 4.308421985 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-4.25' to each side of the equation. 4.25 + -4.25 + c = 4.308421985 + -4.25 Combine like terms: 4.25 + -4.25 = 0.00 0.00 + c = 4.308421985 + -4.25 c = 4.308421985 + -4.25 Combine like terms: 4.308421985 + -4.25 = 0.058421985 c = 0.058421985 Simplifying c = 0.058421985Subproblem 2
c + 4.25 = -4.308421985 Simplifying c + 4.25 = -4.308421985 Reorder the terms: 4.25 + c = -4.308421985 Solving 4.25 + c = -4.308421985 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-4.25' to each side of the equation. 4.25 + -4.25 + c = -4.308421985 + -4.25 Combine like terms: 4.25 + -4.25 = 0.00 0.00 + c = -4.308421985 + -4.25 c = -4.308421985 + -4.25 Combine like terms: -4.308421985 + -4.25 = -8.558421985 c = -8.558421985 Simplifying c = -8.558421985Solution
The solution to the problem is based on the solutions from the subproblems. c = {0.058421985, -8.558421985}
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