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Simplifying 7(7c + 1) + -4c = 13c(3c + -2) Reorder the terms: 7(1 + 7c) + -4c = 13c(3c + -2) (1 * 7 + 7c * 7) + -4c = 13c(3c + -2) (7 + 49c) + -4c = 13c(3c + -2) Combine like terms: 49c + -4c = 45c 7 + 45c = 13c(3c + -2) Reorder the terms: 7 + 45c = 13c(-2 + 3c) 7 + 45c = (-2 * 13c + 3c * 13c) 7 + 45c = (-26c + 39c2) Solving 7 + 45c = -26c + 39c2 Solving for variable 'c'. Combine like terms: 45c + 26c = 71c 7 + 71c + -39c2 = -26c + 39c2 + 26c + -39c2 Reorder the terms: 7 + 71c + -39c2 = -26c + 26c + 39c2 + -39c2 Combine like terms: -26c + 26c = 0 7 + 71c + -39c2 = 0 + 39c2 + -39c2 7 + 71c + -39c2 = 39c2 + -39c2 Combine like terms: 39c2 + -39c2 = 0 7 + 71c + -39c2 = 0 Begin completing the square. Divide all terms by -39 the coefficient of the squared term: Divide each side by '-39'. -0.1794871795 + -1.820512821c + c2 = 0 Move the constant term to the right: Add '0.1794871795' to each side of the equation. -0.1794871795 + -1.820512821c + 0.1794871795 + c2 = 0 + 0.1794871795 Reorder the terms: -0.1794871795 + 0.1794871795 + -1.820512821c + c2 = 0 + 0.1794871795 Combine like terms: -0.1794871795 + 0.1794871795 = 0.0000000000 0.0000000000 + -1.820512821c + c2 = 0 + 0.1794871795 -1.820512821c + c2 = 0 + 0.1794871795 Combine like terms: 0 + 0.1794871795 = 0.1794871795 -1.820512821c + c2 = 0.1794871795 The c term is -1.820512821c. Take half its coefficient (-0.9102564105). Square it (0.8285667329) and add it to both sides. Add '0.8285667329' to each side of the equation. -1.820512821c + 0.8285667329 + c2 = 0.1794871795 + 0.8285667329 Reorder the terms: 0.8285667329 + -1.820512821c + c2 = 0.1794871795 + 0.8285667329 Combine like terms: 0.1794871795 + 0.8285667329 = 1.0080539124 0.8285667329 + -1.820512821c + c2 = 1.0080539124 Factor a perfect square on the left side: (c + -0.9102564105)(c + -0.9102564105) = 1.0080539124 Calculate the square root of the right side: 1.004018881 Break this problem into two subproblems by setting (c + -0.9102564105) equal to 1.004018881 and -1.004018881.Subproblem 1
c + -0.9102564105 = 1.004018881 Simplifying c + -0.9102564105 = 1.004018881 Reorder the terms: -0.9102564105 + c = 1.004018881 Solving -0.9102564105 + c = 1.004018881 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '0.9102564105' to each side of the equation. -0.9102564105 + 0.9102564105 + c = 1.004018881 + 0.9102564105 Combine like terms: -0.9102564105 + 0.9102564105 = 0.0000000000 0.0000000000 + c = 1.004018881 + 0.9102564105 c = 1.004018881 + 0.9102564105 Combine like terms: 1.004018881 + 0.9102564105 = 1.9142752915 c = 1.9142752915 Simplifying c = 1.9142752915Subproblem 2
c + -0.9102564105 = -1.004018881 Simplifying c + -0.9102564105 = -1.004018881 Reorder the terms: -0.9102564105 + c = -1.004018881 Solving -0.9102564105 + c = -1.004018881 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '0.9102564105' to each side of the equation. -0.9102564105 + 0.9102564105 + c = -1.004018881 + 0.9102564105 Combine like terms: -0.9102564105 + 0.9102564105 = 0.0000000000 0.0000000000 + c = -1.004018881 + 0.9102564105 c = -1.004018881 + 0.9102564105 Combine like terms: -1.004018881 + 0.9102564105 = -0.0937624705 c = -0.0937624705 Simplifying c = -0.0937624705Solution
The solution to the problem is based on the solutions from the subproblems. c = {1.9142752915, -0.0937624705}
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