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Simplifying 3c(2c + -5) + -5(4c + 3) = 0 Reorder the terms: 3c(-5 + 2c) + -5(4c + 3) = 0 (-5 * 3c + 2c * 3c) + -5(4c + 3) = 0 (-15c + 6c2) + -5(4c + 3) = 0 Reorder the terms: -15c + 6c2 + -5(3 + 4c) = 0 -15c + 6c2 + (3 * -5 + 4c * -5) = 0 -15c + 6c2 + (-15 + -20c) = 0 Reorder the terms: -15 + -15c + -20c + 6c2 = 0 Combine like terms: -15c + -20c = -35c -15 + -35c + 6c2 = 0 Solving -15 + -35c + 6c2 = 0 Solving for variable 'c'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -2.5 + -5.833333333c + c2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + -5.833333333c + 2.5 + c2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + -5.833333333c + c2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + -5.833333333c + c2 = 0 + 2.5 -5.833333333c + c2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 -5.833333333c + c2 = 2.5 The c term is -5.833333333c. Take half its coefficient (-2.916666667). Square it (8.506944446) and add it to both sides. Add '8.506944446' to each side of the equation. -5.833333333c + 8.506944446 + c2 = 2.5 + 8.506944446 Reorder the terms: 8.506944446 + -5.833333333c + c2 = 2.5 + 8.506944446 Combine like terms: 2.5 + 8.506944446 = 11.006944446 8.506944446 + -5.833333333c + c2 = 11.006944446 Factor a perfect square on the left side: (c + -2.916666667)(c + -2.916666667) = 11.006944446 Calculate the square root of the right side: 3.31767154 Break this problem into two subproblems by setting (c + -2.916666667) equal to 3.31767154 and -3.31767154.Subproblem 1
c + -2.916666667 = 3.31767154 Simplifying c + -2.916666667 = 3.31767154 Reorder the terms: -2.916666667 + c = 3.31767154 Solving -2.916666667 + c = 3.31767154 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '2.916666667' to each side of the equation. -2.916666667 + 2.916666667 + c = 3.31767154 + 2.916666667 Combine like terms: -2.916666667 + 2.916666667 = 0.000000000 0.000000000 + c = 3.31767154 + 2.916666667 c = 3.31767154 + 2.916666667 Combine like terms: 3.31767154 + 2.916666667 = 6.234338207 c = 6.234338207 Simplifying c = 6.234338207Subproblem 2
c + -2.916666667 = -3.31767154 Simplifying c + -2.916666667 = -3.31767154 Reorder the terms: -2.916666667 + c = -3.31767154 Solving -2.916666667 + c = -3.31767154 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '2.916666667' to each side of the equation. -2.916666667 + 2.916666667 + c = -3.31767154 + 2.916666667 Combine like terms: -2.916666667 + 2.916666667 = 0.000000000 0.000000000 + c = -3.31767154 + 2.916666667 c = -3.31767154 + 2.916666667 Combine like terms: -3.31767154 + 2.916666667 = -0.401004873 c = -0.401004873 Simplifying c = -0.401004873Solution
The solution to the problem is based on the solutions from the subproblems. c = {6.234338207, -0.401004873}
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