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Simplifying (2c + 7)(3c + -4) = 12 Reorder the terms: (7 + 2c)(3c + -4) = 12 Reorder the terms: (7 + 2c)(-4 + 3c) = 12 Multiply (7 + 2c) * (-4 + 3c) (7(-4 + 3c) + 2c * (-4 + 3c)) = 12 ((-4 * 7 + 3c * 7) + 2c * (-4 + 3c)) = 12 ((-28 + 21c) + 2c * (-4 + 3c)) = 12 (-28 + 21c + (-4 * 2c + 3c * 2c)) = 12 (-28 + 21c + (-8c + 6c2)) = 12 Combine like terms: 21c + -8c = 13c (-28 + 13c + 6c2) = 12 Solving -28 + 13c + 6c2 = 12 Solving for variable 'c'. Reorder the terms: -28 + -12 + 13c + 6c2 = 12 + -12 Combine like terms: -28 + -12 = -40 -40 + 13c + 6c2 = 12 + -12 Combine like terms: 12 + -12 = 0 -40 + 13c + 6c2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -6.666666667 + 2.166666667c + c2 = 0 Move the constant term to the right: Add '6.666666667' to each side of the equation. -6.666666667 + 2.166666667c + 6.666666667 + c2 = 0 + 6.666666667 Reorder the terms: -6.666666667 + 6.666666667 + 2.166666667c + c2 = 0 + 6.666666667 Combine like terms: -6.666666667 + 6.666666667 = 0.000000000 0.000000000 + 2.166666667c + c2 = 0 + 6.666666667 2.166666667c + c2 = 0 + 6.666666667 Combine like terms: 0 + 6.666666667 = 6.666666667 2.166666667c + c2 = 6.666666667 The c term is 2.166666667c. Take half its coefficient (1.083333334). Square it (1.173611113) and add it to both sides. Add '1.173611113' to each side of the equation. 2.166666667c + 1.173611113 + c2 = 6.666666667 + 1.173611113 Reorder the terms: 1.173611113 + 2.166666667c + c2 = 6.666666667 + 1.173611113 Combine like terms: 6.666666667 + 1.173611113 = 7.84027778 1.173611113 + 2.166666667c + c2 = 7.84027778 Factor a perfect square on the left side: (c + 1.083333334)(c + 1.083333334) = 7.84027778 Calculate the square root of the right side: 2.800049603 Break this problem into two subproblems by setting (c + 1.083333334) equal to 2.800049603 and -2.800049603.Subproblem 1
c + 1.083333334 = 2.800049603 Simplifying c + 1.083333334 = 2.800049603 Reorder the terms: 1.083333334 + c = 2.800049603 Solving 1.083333334 + c = 2.800049603 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-1.083333334' to each side of the equation. 1.083333334 + -1.083333334 + c = 2.800049603 + -1.083333334 Combine like terms: 1.083333334 + -1.083333334 = 0.000000000 0.000000000 + c = 2.800049603 + -1.083333334 c = 2.800049603 + -1.083333334 Combine like terms: 2.800049603 + -1.083333334 = 1.716716269 c = 1.716716269 Simplifying c = 1.716716269Subproblem 2
c + 1.083333334 = -2.800049603 Simplifying c + 1.083333334 = -2.800049603 Reorder the terms: 1.083333334 + c = -2.800049603 Solving 1.083333334 + c = -2.800049603 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-1.083333334' to each side of the equation. 1.083333334 + -1.083333334 + c = -2.800049603 + -1.083333334 Combine like terms: 1.083333334 + -1.083333334 = 0.000000000 0.000000000 + c = -2.800049603 + -1.083333334 c = -2.800049603 + -1.083333334 Combine like terms: -2.800049603 + -1.083333334 = -3.883382937 c = -3.883382937 Simplifying c = -3.883382937Solution
The solution to the problem is based on the solutions from the subproblems. c = {1.716716269, -3.883382937}
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