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Simplifying (-9c2 + 3c + 9) + -1(9c2 + 7c + -4) = 0 Reorder the terms: (9 + 3c + -9c2) + -1(9c2 + 7c + -4) = 0 Remove parenthesis around (9 + 3c + -9c2) 9 + 3c + -9c2 + -1(9c2 + 7c + -4) = 0 Reorder the terms: 9 + 3c + -9c2 + -1(-4 + 7c + 9c2) = 0 9 + 3c + -9c2 + (-4 * -1 + 7c * -1 + 9c2 * -1) = 0 9 + 3c + -9c2 + (4 + -7c + -9c2) = 0 Reorder the terms: 9 + 4 + 3c + -7c + -9c2 + -9c2 = 0 Combine like terms: 9 + 4 = 13 13 + 3c + -7c + -9c2 + -9c2 = 0 Combine like terms: 3c + -7c = -4c 13 + -4c + -9c2 + -9c2 = 0 Combine like terms: -9c2 + -9c2 = -18c2 13 + -4c + -18c2 = 0 Solving 13 + -4c + -18c2 = 0 Solving for variable 'c'. Begin completing the square. Divide all terms by -18 the coefficient of the squared term: Divide each side by '-18'. -0.7222222222 + 0.2222222222c + c2 = 0 Move the constant term to the right: Add '0.7222222222' to each side of the equation. -0.7222222222 + 0.2222222222c + 0.7222222222 + c2 = 0 + 0.7222222222 Reorder the terms: -0.7222222222 + 0.7222222222 + 0.2222222222c + c2 = 0 + 0.7222222222 Combine like terms: -0.7222222222 + 0.7222222222 = 0.0000000000 0.0000000000 + 0.2222222222c + c2 = 0 + 0.7222222222 0.2222222222c + c2 = 0 + 0.7222222222 Combine like terms: 0 + 0.7222222222 = 0.7222222222 0.2222222222c + c2 = 0.7222222222 The c term is 0.2222222222c. Take half its coefficient (0.1111111111). Square it (0.01234567901) and add it to both sides. Add '0.01234567901' to each side of the equation. 0.2222222222c + 0.01234567901 + c2 = 0.7222222222 + 0.01234567901 Reorder the terms: 0.01234567901 + 0.2222222222c + c2 = 0.7222222222 + 0.01234567901 Combine like terms: 0.7222222222 + 0.01234567901 = 0.73456790121 0.01234567901 + 0.2222222222c + c2 = 0.73456790121 Factor a perfect square on the left side: (c + 0.1111111111)(c + 0.1111111111) = 0.73456790121 Calculate the square root of the right side: 0.857069368 Break this problem into two subproblems by setting (c + 0.1111111111) equal to 0.857069368 and -0.857069368.Subproblem 1
c + 0.1111111111 = 0.857069368 Simplifying c + 0.1111111111 = 0.857069368 Reorder the terms: 0.1111111111 + c = 0.857069368 Solving 0.1111111111 + c = 0.857069368 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-0.1111111111' to each side of the equation. 0.1111111111 + -0.1111111111 + c = 0.857069368 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + c = 0.857069368 + -0.1111111111 c = 0.857069368 + -0.1111111111 Combine like terms: 0.857069368 + -0.1111111111 = 0.7459582569 c = 0.7459582569 Simplifying c = 0.7459582569Subproblem 2
c + 0.1111111111 = -0.857069368 Simplifying c + 0.1111111111 = -0.857069368 Reorder the terms: 0.1111111111 + c = -0.857069368 Solving 0.1111111111 + c = -0.857069368 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-0.1111111111' to each side of the equation. 0.1111111111 + -0.1111111111 + c = -0.857069368 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + c = -0.857069368 + -0.1111111111 c = -0.857069368 + -0.1111111111 Combine like terms: -0.857069368 + -0.1111111111 = -0.9681804791 c = -0.9681804791 Simplifying c = -0.9681804791Solution
The solution to the problem is based on the solutions from the subproblems. c = {0.7459582569, -0.9681804791}
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