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Simplifying 5c2 + 10c + -9 + c + -1 + 15c2 + -6c + 6 = 0 Reorder the terms: -9 + -1 + 6 + 10c + c + -6c + 5c2 + 15c2 = 0 Combine like terms: -9 + -1 = -10 -10 + 6 + 10c + c + -6c + 5c2 + 15c2 = 0 Combine like terms: -10 + 6 = -4 -4 + 10c + c + -6c + 5c2 + 15c2 = 0 Combine like terms: 10c + c = 11c -4 + 11c + -6c + 5c2 + 15c2 = 0 Combine like terms: 11c + -6c = 5c -4 + 5c + 5c2 + 15c2 = 0 Combine like terms: 5c2 + 15c2 = 20c2 -4 + 5c + 20c2 = 0 Solving -4 + 5c + 20c2 = 0 Solving for variable 'c'. Begin completing the square. Divide all terms by 20 the coefficient of the squared term: Divide each side by '20'. -0.2 + 0.25c + c2 = 0.0 Move the constant term to the right: Add '0.2' to each side of the equation. -0.2 + 0.25c + 0.2 + c2 = 0.0 + 0.2 Reorder the terms: -0.2 + 0.2 + 0.25c + c2 = 0.0 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + 0.25c + c2 = 0.0 + 0.2 0.25c + c2 = 0.0 + 0.2 Combine like terms: 0.0 + 0.2 = 0.2 0.25c + c2 = 0.2 The c term is 0.25c. Take half its coefficient (0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. 0.25c + 0.015625 + c2 = 0.2 + 0.015625 Reorder the terms: 0.015625 + 0.25c + c2 = 0.2 + 0.015625 Combine like terms: 0.2 + 0.015625 = 0.215625 0.015625 + 0.25c + c2 = 0.215625 Factor a perfect square on the left side: (c + 0.125)(c + 0.125) = 0.215625 Calculate the square root of the right side: 0.464354391 Break this problem into two subproblems by setting (c + 0.125) equal to 0.464354391 and -0.464354391.Subproblem 1
c + 0.125 = 0.464354391 Simplifying c + 0.125 = 0.464354391 Reorder the terms: 0.125 + c = 0.464354391 Solving 0.125 + c = 0.464354391 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + c = 0.464354391 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + c = 0.464354391 + -0.125 c = 0.464354391 + -0.125 Combine like terms: 0.464354391 + -0.125 = 0.339354391 c = 0.339354391 Simplifying c = 0.339354391Subproblem 2
c + 0.125 = -0.464354391 Simplifying c + 0.125 = -0.464354391 Reorder the terms: 0.125 + c = -0.464354391 Solving 0.125 + c = -0.464354391 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + c = -0.464354391 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + c = -0.464354391 + -0.125 c = -0.464354391 + -0.125 Combine like terms: -0.464354391 + -0.125 = -0.589354391 c = -0.589354391 Simplifying c = -0.589354391Solution
The solution to the problem is based on the solutions from the subproblems. c = {0.339354391, -0.589354391}
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