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Simplifying (2c + -3)(c + 5) + 3(c + -3)(-3c + 1) = 0 Reorder the terms: (-3 + 2c)(c + 5) + 3(c + -3)(-3c + 1) = 0 Reorder the terms: (-3 + 2c)(5 + c) + 3(c + -3)(-3c + 1) = 0 Multiply (-3 + 2c) * (5 + c) (-3(5 + c) + 2c * (5 + c)) + 3(c + -3)(-3c + 1) = 0 ((5 * -3 + c * -3) + 2c * (5 + c)) + 3(c + -3)(-3c + 1) = 0 ((-15 + -3c) + 2c * (5 + c)) + 3(c + -3)(-3c + 1) = 0 (-15 + -3c + (5 * 2c + c * 2c)) + 3(c + -3)(-3c + 1) = 0 (-15 + -3c + (10c + 2c2)) + 3(c + -3)(-3c + 1) = 0 Combine like terms: -3c + 10c = 7c (-15 + 7c + 2c2) + 3(c + -3)(-3c + 1) = 0 Reorder the terms: -15 + 7c + 2c2 + 3(-3 + c)(-3c + 1) = 0 Reorder the terms: -15 + 7c + 2c2 + 3(-3 + c)(1 + -3c) = 0 Multiply (-3 + c) * (1 + -3c) -15 + 7c + 2c2 + 3(-3(1 + -3c) + c(1 + -3c)) = 0 -15 + 7c + 2c2 + 3((1 * -3 + -3c * -3) + c(1 + -3c)) = 0 -15 + 7c + 2c2 + 3((-3 + 9c) + c(1 + -3c)) = 0 -15 + 7c + 2c2 + 3(-3 + 9c + (1 * c + -3c * c)) = 0 -15 + 7c + 2c2 + 3(-3 + 9c + (1c + -3c2)) = 0 Combine like terms: 9c + 1c = 10c -15 + 7c + 2c2 + 3(-3 + 10c + -3c2) = 0 -15 + 7c + 2c2 + (-3 * 3 + 10c * 3 + -3c2 * 3) = 0 -15 + 7c + 2c2 + (-9 + 30c + -9c2) = 0 Reorder the terms: -15 + -9 + 7c + 30c + 2c2 + -9c2 = 0 Combine like terms: -15 + -9 = -24 -24 + 7c + 30c + 2c2 + -9c2 = 0 Combine like terms: 7c + 30c = 37c -24 + 37c + 2c2 + -9c2 = 0 Combine like terms: 2c2 + -9c2 = -7c2 -24 + 37c + -7c2 = 0 Solving -24 + 37c + -7c2 = 0 Solving for variable 'c'. Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. 3.428571429 + -5.285714286c + c2 = 0 Move the constant term to the right: Add '-3.428571429' to each side of the equation. 3.428571429 + -5.285714286c + -3.428571429 + c2 = 0 + -3.428571429 Reorder the terms: 3.428571429 + -3.428571429 + -5.285714286c + c2 = 0 + -3.428571429 Combine like terms: 3.428571429 + -3.428571429 = 0.000000000 0.000000000 + -5.285714286c + c2 = 0 + -3.428571429 -5.285714286c + c2 = 0 + -3.428571429 Combine like terms: 0 + -3.428571429 = -3.428571429 -5.285714286c + c2 = -3.428571429 The c term is -5.285714286c. Take half its coefficient (-2.642857143). Square it (6.984693878) and add it to both sides. Add '6.984693878' to each side of the equation. -5.285714286c + 6.984693878 + c2 = -3.428571429 + 6.984693878 Reorder the terms: 6.984693878 + -5.285714286c + c2 = -3.428571429 + 6.984693878 Combine like terms: -3.428571429 + 6.984693878 = 3.556122449 6.984693878 + -5.285714286c + c2 = 3.556122449 Factor a perfect square on the left side: (c + -2.642857143)(c + -2.642857143) = 3.556122449 Calculate the square root of the right side: 1.885768397 Break this problem into two subproblems by setting (c + -2.642857143) equal to 1.885768397 and -1.885768397.Subproblem 1
c + -2.642857143 = 1.885768397 Simplifying c + -2.642857143 = 1.885768397 Reorder the terms: -2.642857143 + c = 1.885768397 Solving -2.642857143 + c = 1.885768397 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '2.642857143' to each side of the equation. -2.642857143 + 2.642857143 + c = 1.885768397 + 2.642857143 Combine like terms: -2.642857143 + 2.642857143 = 0.000000000 0.000000000 + c = 1.885768397 + 2.642857143 c = 1.885768397 + 2.642857143 Combine like terms: 1.885768397 + 2.642857143 = 4.52862554 c = 4.52862554 Simplifying c = 4.52862554Subproblem 2
c + -2.642857143 = -1.885768397 Simplifying c + -2.642857143 = -1.885768397 Reorder the terms: -2.642857143 + c = -1.885768397 Solving -2.642857143 + c = -1.885768397 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '2.642857143' to each side of the equation. -2.642857143 + 2.642857143 + c = -1.885768397 + 2.642857143 Combine like terms: -2.642857143 + 2.642857143 = 0.000000000 0.000000000 + c = -1.885768397 + 2.642857143 c = -1.885768397 + 2.642857143 Combine like terms: -1.885768397 + 2.642857143 = 0.757088746 c = 0.757088746 Simplifying c = 0.757088746Solution
The solution to the problem is based on the solutions from the subproblems. c = {4.52862554, 0.757088746}
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