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Simplifying 7c(3 + -6c) = 12 + -1(c + 6) * 17c (3 * 7c + -6c * 7c) = 12 + -1(c + 6) * 17c (21c + -42c2) = 12 + -1(c + 6) * 17c Reorder the terms: 21c + -42c2 = 12 + -1(6 + c) * 17c Reorder the terms for easier multiplication: 21c + -42c2 = 12 + -1 * 17c(6 + c) Multiply -1 * 17 21c + -42c2 = 12 + -17c(6 + c) 21c + -42c2 = 12 + (6 * -17c + c * -17c) 21c + -42c2 = 12 + (-102c + -17c2) Solving 21c + -42c2 = 12 + -102c + -17c2 Solving for variable 'c'. Reorder the terms: -12 + 21c + 102c + -42c2 + 17c2 = 12 + -102c + -17c2 + -12 + 102c + 17c2 Combine like terms: 21c + 102c = 123c -12 + 123c + -42c2 + 17c2 = 12 + -102c + -17c2 + -12 + 102c + 17c2 Combine like terms: -42c2 + 17c2 = -25c2 -12 + 123c + -25c2 = 12 + -102c + -17c2 + -12 + 102c + 17c2 Reorder the terms: -12 + 123c + -25c2 = 12 + -12 + -102c + 102c + -17c2 + 17c2 Combine like terms: 12 + -12 = 0 -12 + 123c + -25c2 = 0 + -102c + 102c + -17c2 + 17c2 -12 + 123c + -25c2 = -102c + 102c + -17c2 + 17c2 Combine like terms: -102c + 102c = 0 -12 + 123c + -25c2 = 0 + -17c2 + 17c2 -12 + 123c + -25c2 = -17c2 + 17c2 Combine like terms: -17c2 + 17c2 = 0 -12 + 123c + -25c2 = 0 Begin completing the square. Divide all terms by -25 the coefficient of the squared term: Divide each side by '-25'. 0.48 + -4.92c + c2 = 0 Move the constant term to the right: Add '-0.48' to each side of the equation. 0.48 + -4.92c + -0.48 + c2 = 0 + -0.48 Reorder the terms: 0.48 + -0.48 + -4.92c + c2 = 0 + -0.48 Combine like terms: 0.48 + -0.48 = 0.00 0.00 + -4.92c + c2 = 0 + -0.48 -4.92c + c2 = 0 + -0.48 Combine like terms: 0 + -0.48 = -0.48 -4.92c + c2 = -0.48 The c term is -4.92c. Take half its coefficient (-2.46). Square it (6.0516) and add it to both sides. Add '6.0516' to each side of the equation. -4.92c + 6.0516 + c2 = -0.48 + 6.0516 Reorder the terms: 6.0516 + -4.92c + c2 = -0.48 + 6.0516 Combine like terms: -0.48 + 6.0516 = 5.5716 6.0516 + -4.92c + c2 = 5.5716 Factor a perfect square on the left side: (c + -2.46)(c + -2.46) = 5.5716 Calculate the square root of the right side: 2.360423691 Break this problem into two subproblems by setting (c + -2.46) equal to 2.360423691 and -2.360423691.Subproblem 1
c + -2.46 = 2.360423691 Simplifying c + -2.46 = 2.360423691 Reorder the terms: -2.46 + c = 2.360423691 Solving -2.46 + c = 2.360423691 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '2.46' to each side of the equation. -2.46 + 2.46 + c = 2.360423691 + 2.46 Combine like terms: -2.46 + 2.46 = 0.00 0.00 + c = 2.360423691 + 2.46 c = 2.360423691 + 2.46 Combine like terms: 2.360423691 + 2.46 = 4.820423691 c = 4.820423691 Simplifying c = 4.820423691Subproblem 2
c + -2.46 = -2.360423691 Simplifying c + -2.46 = -2.360423691 Reorder the terms: -2.46 + c = -2.360423691 Solving -2.46 + c = -2.360423691 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '2.46' to each side of the equation. -2.46 + 2.46 + c = -2.360423691 + 2.46 Combine like terms: -2.46 + 2.46 = 0.00 0.00 + c = -2.360423691 + 2.46 c = -2.360423691 + 2.46 Combine like terms: -2.360423691 + 2.46 = 0.099576309 c = 0.099576309 Simplifying c = 0.099576309Solution
The solution to the problem is based on the solutions from the subproblems. c = {4.820423691, 0.099576309}
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