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Simplifying 4c2 + 9c + -7 + 12c2 + -5c + 5 + c + -2 = 0 Reorder the terms: -7 + 5 + -2 + 9c + -5c + c + 4c2 + 12c2 = 0 Combine like terms: -7 + 5 = -2 -2 + -2 + 9c + -5c + c + 4c2 + 12c2 = 0 Combine like terms: -2 + -2 = -4 -4 + 9c + -5c + c + 4c2 + 12c2 = 0 Combine like terms: 9c + -5c = 4c -4 + 4c + c + 4c2 + 12c2 = 0 Combine like terms: 4c + c = 5c -4 + 5c + 4c2 + 12c2 = 0 Combine like terms: 4c2 + 12c2 = 16c2 -4 + 5c + 16c2 = 0 Solving -4 + 5c + 16c2 = 0 Solving for variable 'c'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.25 + 0.3125c + c2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + 0.3125c + 0.25 + c2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + 0.3125c + c2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + 0.3125c + c2 = 0 + 0.25 0.3125c + c2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 0.3125c + c2 = 0.25 The c term is 0.3125c. Take half its coefficient (0.15625). Square it (0.0244140625) and add it to both sides. Add '0.0244140625' to each side of the equation. 0.3125c + 0.0244140625 + c2 = 0.25 + 0.0244140625 Reorder the terms: 0.0244140625 + 0.3125c + c2 = 0.25 + 0.0244140625 Combine like terms: 0.25 + 0.0244140625 = 0.2744140625 0.0244140625 + 0.3125c + c2 = 0.2744140625 Factor a perfect square on the left side: (c + 0.15625)(c + 0.15625) = 0.2744140625 Calculate the square root of the right side: 0.523845457 Break this problem into two subproblems by setting (c + 0.15625) equal to 0.523845457 and -0.523845457.Subproblem 1
c + 0.15625 = 0.523845457 Simplifying c + 0.15625 = 0.523845457 Reorder the terms: 0.15625 + c = 0.523845457 Solving 0.15625 + c = 0.523845457 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-0.15625' to each side of the equation. 0.15625 + -0.15625 + c = 0.523845457 + -0.15625 Combine like terms: 0.15625 + -0.15625 = 0.00000 0.00000 + c = 0.523845457 + -0.15625 c = 0.523845457 + -0.15625 Combine like terms: 0.523845457 + -0.15625 = 0.367595457 c = 0.367595457 Simplifying c = 0.367595457Subproblem 2
c + 0.15625 = -0.523845457 Simplifying c + 0.15625 = -0.523845457 Reorder the terms: 0.15625 + c = -0.523845457 Solving 0.15625 + c = -0.523845457 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-0.15625' to each side of the equation. 0.15625 + -0.15625 + c = -0.523845457 + -0.15625 Combine like terms: 0.15625 + -0.15625 = 0.00000 0.00000 + c = -0.523845457 + -0.15625 c = -0.523845457 + -0.15625 Combine like terms: -0.523845457 + -0.15625 = -0.680095457 c = -0.680095457 Simplifying c = -0.680095457Solution
The solution to the problem is based on the solutions from the subproblems. c = {0.367595457, -0.680095457}
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