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Simplifying 5(3c + -2) * 7c = 40 + -2c Reorder the terms: 5(-2 + 3c) * 7c = 40 + -2c Reorder the terms for easier multiplication: 5 * 7c(-2 + 3c) = 40 + -2c Multiply 5 * 7 35c(-2 + 3c) = 40 + -2c (-2 * 35c + 3c * 35c) = 40 + -2c (-70c + 105c2) = 40 + -2c Solving -70c + 105c2 = 40 + -2c Solving for variable 'c'. Reorder the terms: -40 + -70c + 2c + 105c2 = 40 + -2c + -40 + 2c Combine like terms: -70c + 2c = -68c -40 + -68c + 105c2 = 40 + -2c + -40 + 2c Reorder the terms: -40 + -68c + 105c2 = 40 + -40 + -2c + 2c Combine like terms: 40 + -40 = 0 -40 + -68c + 105c2 = 0 + -2c + 2c -40 + -68c + 105c2 = -2c + 2c Combine like terms: -2c + 2c = 0 -40 + -68c + 105c2 = 0 Begin completing the square. Divide all terms by 105 the coefficient of the squared term: Divide each side by '105'. -0.380952381 + -0.6476190476c + c2 = 0 Move the constant term to the right: Add '0.380952381' to each side of the equation. -0.380952381 + -0.6476190476c + 0.380952381 + c2 = 0 + 0.380952381 Reorder the terms: -0.380952381 + 0.380952381 + -0.6476190476c + c2 = 0 + 0.380952381 Combine like terms: -0.380952381 + 0.380952381 = 0.000000000 0.000000000 + -0.6476190476c + c2 = 0 + 0.380952381 -0.6476190476c + c2 = 0 + 0.380952381 Combine like terms: 0 + 0.380952381 = 0.380952381 -0.6476190476c + c2 = 0.380952381 The c term is -0.6476190476c. Take half its coefficient (-0.3238095238). Square it (0.1048526077) and add it to both sides. Add '0.1048526077' to each side of the equation. -0.6476190476c + 0.1048526077 + c2 = 0.380952381 + 0.1048526077 Reorder the terms: 0.1048526077 + -0.6476190476c + c2 = 0.380952381 + 0.1048526077 Combine like terms: 0.380952381 + 0.1048526077 = 0.4858049887 0.1048526077 + -0.6476190476c + c2 = 0.4858049887 Factor a perfect square on the left side: (c + -0.3238095238)(c + -0.3238095238) = 0.4858049887 Calculate the square root of the right side: 0.696997122 Break this problem into two subproblems by setting (c + -0.3238095238) equal to 0.696997122 and -0.696997122.Subproblem 1
c + -0.3238095238 = 0.696997122 Simplifying c + -0.3238095238 = 0.696997122 Reorder the terms: -0.3238095238 + c = 0.696997122 Solving -0.3238095238 + c = 0.696997122 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '0.3238095238' to each side of the equation. -0.3238095238 + 0.3238095238 + c = 0.696997122 + 0.3238095238 Combine like terms: -0.3238095238 + 0.3238095238 = 0.0000000000 0.0000000000 + c = 0.696997122 + 0.3238095238 c = 0.696997122 + 0.3238095238 Combine like terms: 0.696997122 + 0.3238095238 = 1.0208066458 c = 1.0208066458 Simplifying c = 1.0208066458Subproblem 2
c + -0.3238095238 = -0.696997122 Simplifying c + -0.3238095238 = -0.696997122 Reorder the terms: -0.3238095238 + c = -0.696997122 Solving -0.3238095238 + c = -0.696997122 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '0.3238095238' to each side of the equation. -0.3238095238 + 0.3238095238 + c = -0.696997122 + 0.3238095238 Combine like terms: -0.3238095238 + 0.3238095238 = 0.0000000000 0.0000000000 + c = -0.696997122 + 0.3238095238 c = -0.696997122 + 0.3238095238 Combine like terms: -0.696997122 + 0.3238095238 = -0.3731875982 c = -0.3731875982 Simplifying c = -0.3731875982Solution
The solution to the problem is based on the solutions from the subproblems. c = {1.0208066458, -0.3731875982}
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