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Simplifying 23c2 + 56c + 10 = 0 Reorder the terms: 10 + 56c + 23c2 = 0 Solving 10 + 56c + 23c2 = 0 Solving for variable 'c'. Begin completing the square. Divide all terms by 23 the coefficient of the squared term: Divide each side by '23'. 0.4347826087 + 2.434782609c + c2 = 0 Move the constant term to the right: Add '-0.4347826087' to each side of the equation. 0.4347826087 + 2.434782609c + -0.4347826087 + c2 = 0 + -0.4347826087 Reorder the terms: 0.4347826087 + -0.4347826087 + 2.434782609c + c2 = 0 + -0.4347826087 Combine like terms: 0.4347826087 + -0.4347826087 = 0.0000000000 0.0000000000 + 2.434782609c + c2 = 0 + -0.4347826087 2.434782609c + c2 = 0 + -0.4347826087 Combine like terms: 0 + -0.4347826087 = -0.4347826087 2.434782609c + c2 = -0.4347826087 The c term is 2.434782609c. Take half its coefficient (1.217391305). Square it (1.482041589) and add it to both sides. Add '1.482041589' to each side of the equation. 2.434782609c + 1.482041589 + c2 = -0.4347826087 + 1.482041589 Reorder the terms: 1.482041589 + 2.434782609c + c2 = -0.4347826087 + 1.482041589 Combine like terms: -0.4347826087 + 1.482041589 = 1.0472589803 1.482041589 + 2.434782609c + c2 = 1.0472589803 Factor a perfect square on the left side: (c + 1.217391305)(c + 1.217391305) = 1.0472589803 Calculate the square root of the right side: 1.023356722 Break this problem into two subproblems by setting (c + 1.217391305) equal to 1.023356722 and -1.023356722.Subproblem 1
c + 1.217391305 = 1.023356722 Simplifying c + 1.217391305 = 1.023356722 Reorder the terms: 1.217391305 + c = 1.023356722 Solving 1.217391305 + c = 1.023356722 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-1.217391305' to each side of the equation. 1.217391305 + -1.217391305 + c = 1.023356722 + -1.217391305 Combine like terms: 1.217391305 + -1.217391305 = 0.000000000 0.000000000 + c = 1.023356722 + -1.217391305 c = 1.023356722 + -1.217391305 Combine like terms: 1.023356722 + -1.217391305 = -0.194034583 c = -0.194034583 Simplifying c = -0.194034583Subproblem 2
c + 1.217391305 = -1.023356722 Simplifying c + 1.217391305 = -1.023356722 Reorder the terms: 1.217391305 + c = -1.023356722 Solving 1.217391305 + c = -1.023356722 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-1.217391305' to each side of the equation. 1.217391305 + -1.217391305 + c = -1.023356722 + -1.217391305 Combine like terms: 1.217391305 + -1.217391305 = 0.000000000 0.000000000 + c = -1.023356722 + -1.217391305 c = -1.023356722 + -1.217391305 Combine like terms: -1.023356722 + -1.217391305 = -2.240748027 c = -2.240748027 Simplifying c = -2.240748027Solution
The solution to the problem is based on the solutions from the subproblems. c = {-0.194034583, -2.240748027}
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