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Simplifying (4c2 + -12c + 7) + -1(c2 + 2c + -5) = 0 Reorder the terms: (7 + -12c + 4c2) + -1(c2 + 2c + -5) = 0 Remove parenthesis around (7 + -12c + 4c2) 7 + -12c + 4c2 + -1(c2 + 2c + -5) = 0 Reorder the terms: 7 + -12c + 4c2 + -1(-5 + 2c + c2) = 0 7 + -12c + 4c2 + (-5 * -1 + 2c * -1 + c2 * -1) = 0 7 + -12c + 4c2 + (5 + -2c + -1c2) = 0 Reorder the terms: 7 + 5 + -12c + -2c + 4c2 + -1c2 = 0 Combine like terms: 7 + 5 = 12 12 + -12c + -2c + 4c2 + -1c2 = 0 Combine like terms: -12c + -2c = -14c 12 + -14c + 4c2 + -1c2 = 0 Combine like terms: 4c2 + -1c2 = 3c2 12 + -14c + 3c2 = 0 Solving 12 + -14c + 3c2 = 0 Solving for variable 'c'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 4 + -4.666666667c + c2 = 0 Move the constant term to the right: Add '-4' to each side of the equation. 4 + -4.666666667c + -4 + c2 = 0 + -4 Reorder the terms: 4 + -4 + -4.666666667c + c2 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + -4.666666667c + c2 = 0 + -4 -4.666666667c + c2 = 0 + -4 Combine like terms: 0 + -4 = -4 -4.666666667c + c2 = -4 The c term is -4.666666667c. Take half its coefficient (-2.333333334). Square it (5.444444448) and add it to both sides. Add '5.444444448' to each side of the equation. -4.666666667c + 5.444444448 + c2 = -4 + 5.444444448 Reorder the terms: 5.444444448 + -4.666666667c + c2 = -4 + 5.444444448 Combine like terms: -4 + 5.444444448 = 1.444444448 5.444444448 + -4.666666667c + c2 = 1.444444448 Factor a perfect square on the left side: (c + -2.333333334)(c + -2.333333334) = 1.444444448 Calculate the square root of the right side: 1.201850427 Break this problem into two subproblems by setting (c + -2.333333334) equal to 1.201850427 and -1.201850427.Subproblem 1
c + -2.333333334 = 1.201850427 Simplifying c + -2.333333334 = 1.201850427 Reorder the terms: -2.333333334 + c = 1.201850427 Solving -2.333333334 + c = 1.201850427 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '2.333333334' to each side of the equation. -2.333333334 + 2.333333334 + c = 1.201850427 + 2.333333334 Combine like terms: -2.333333334 + 2.333333334 = 0.000000000 0.000000000 + c = 1.201850427 + 2.333333334 c = 1.201850427 + 2.333333334 Combine like terms: 1.201850427 + 2.333333334 = 3.535183761 c = 3.535183761 Simplifying c = 3.535183761Subproblem 2
c + -2.333333334 = -1.201850427 Simplifying c + -2.333333334 = -1.201850427 Reorder the terms: -2.333333334 + c = -1.201850427 Solving -2.333333334 + c = -1.201850427 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '2.333333334' to each side of the equation. -2.333333334 + 2.333333334 + c = -1.201850427 + 2.333333334 Combine like terms: -2.333333334 + 2.333333334 = 0.000000000 0.000000000 + c = -1.201850427 + 2.333333334 c = -1.201850427 + 2.333333334 Combine like terms: -1.201850427 + 2.333333334 = 1.131482907 c = 1.131482907 Simplifying c = 1.131482907Solution
The solution to the problem is based on the solutions from the subproblems. c = {3.535183761, 1.131482907}
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