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Simplifying 2c2 + -6c = -3 Reorder the terms: -6c + 2c2 = -3 Solving -6c + 2c2 = -3 Solving for variable 'c'. Reorder the terms: 3 + -6c + 2c2 = -3 + 3 Combine like terms: -3 + 3 = 0 3 + -6c + 2c2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 1.5 + -3c + c2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + -3c + -1.5 + c2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + -3c + c2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + -3c + c2 = 0 + -1.5 -3c + c2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 -3c + c2 = -1.5 The c term is -3c. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3c + 2.25 + c2 = -1.5 + 2.25 Reorder the terms: 2.25 + -3c + c2 = -1.5 + 2.25 Combine like terms: -1.5 + 2.25 = 0.75 2.25 + -3c + c2 = 0.75 Factor a perfect square on the left side: (c + -1.5)(c + -1.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (c + -1.5) equal to 0.866025404 and -0.866025404.Subproblem 1
c + -1.5 = 0.866025404 Simplifying c + -1.5 = 0.866025404 Reorder the terms: -1.5 + c = 0.866025404 Solving -1.5 + c = 0.866025404 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + c = 0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + c = 0.866025404 + 1.5 c = 0.866025404 + 1.5 Combine like terms: 0.866025404 + 1.5 = 2.366025404 c = 2.366025404 Simplifying c = 2.366025404Subproblem 2
c + -1.5 = -0.866025404 Simplifying c + -1.5 = -0.866025404 Reorder the terms: -1.5 + c = -0.866025404 Solving -1.5 + c = -0.866025404 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + c = -0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + c = -0.866025404 + 1.5 c = -0.866025404 + 1.5 Combine like terms: -0.866025404 + 1.5 = 0.633974596 c = 0.633974596 Simplifying c = 0.633974596Solution
The solution to the problem is based on the solutions from the subproblems. c = {2.366025404, 0.633974596}
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