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Simplifying (-5 + -2c)(-6c) + 7 = 0 Remove parenthesis around (-6c) (-5 + -2c) * -6c + 7 = 0 Reorder the terms for easier multiplication: -6c(-5 + -2c) + 7 = 0 (-5 * -6c + -2c * -6c) + 7 = 0 (30c + 12c2) + 7 = 0 Reorder the terms: 7 + 30c + 12c2 = 0 Solving 7 + 30c + 12c2 = 0 Solving for variable 'c'. Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. 0.5833333333 + 2.5c + c2 = 0 Move the constant term to the right: Add '-0.5833333333' to each side of the equation. 0.5833333333 + 2.5c + -0.5833333333 + c2 = 0 + -0.5833333333 Reorder the terms: 0.5833333333 + -0.5833333333 + 2.5c + c2 = 0 + -0.5833333333 Combine like terms: 0.5833333333 + -0.5833333333 = 0.0000000000 0.0000000000 + 2.5c + c2 = 0 + -0.5833333333 2.5c + c2 = 0 + -0.5833333333 Combine like terms: 0 + -0.5833333333 = -0.5833333333 2.5c + c2 = -0.5833333333 The c term is 2.5c. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5c + 1.5625 + c2 = -0.5833333333 + 1.5625 Reorder the terms: 1.5625 + 2.5c + c2 = -0.5833333333 + 1.5625 Combine like terms: -0.5833333333 + 1.5625 = 0.9791666667 1.5625 + 2.5c + c2 = 0.9791666667 Factor a perfect square on the left side: (c + 1.25)(c + 1.25) = 0.9791666667 Calculate the square root of the right side: 0.989528507 Break this problem into two subproblems by setting (c + 1.25) equal to 0.989528507 and -0.989528507.Subproblem 1
c + 1.25 = 0.989528507 Simplifying c + 1.25 = 0.989528507 Reorder the terms: 1.25 + c = 0.989528507 Solving 1.25 + c = 0.989528507 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + c = 0.989528507 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + c = 0.989528507 + -1.25 c = 0.989528507 + -1.25 Combine like terms: 0.989528507 + -1.25 = -0.260471493 c = -0.260471493 Simplifying c = -0.260471493Subproblem 2
c + 1.25 = -0.989528507 Simplifying c + 1.25 = -0.989528507 Reorder the terms: 1.25 + c = -0.989528507 Solving 1.25 + c = -0.989528507 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + c = -0.989528507 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + c = -0.989528507 + -1.25 c = -0.989528507 + -1.25 Combine like terms: -0.989528507 + -1.25 = -2.239528507 c = -2.239528507 Simplifying c = -2.239528507Solution
The solution to the problem is based on the solutions from the subproblems. c = {-0.260471493, -2.239528507}
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