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Simplifying 28k2 + 12k + 1 = 0 Reorder the terms: 1 + 12k + 28k2 = 0 Solving 1 + 12k + 28k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 28 the coefficient of the squared term: Divide each side by '28'. 0.03571428571 + 0.4285714286k + k2 = 0 Move the constant term to the right: Add '-0.03571428571' to each side of the equation. 0.03571428571 + 0.4285714286k + -0.03571428571 + k2 = 0 + -0.03571428571 Reorder the terms: 0.03571428571 + -0.03571428571 + 0.4285714286k + k2 = 0 + -0.03571428571 Combine like terms: 0.03571428571 + -0.03571428571 = 0.00000000000 0.00000000000 + 0.4285714286k + k2 = 0 + -0.03571428571 0.4285714286k + k2 = 0 + -0.03571428571 Combine like terms: 0 + -0.03571428571 = -0.03571428571 0.4285714286k + k2 = -0.03571428571 The k term is 0.4285714286k. Take half its coefficient (0.2142857143). Square it (0.04591836735) and add it to both sides. Add '0.04591836735' to each side of the equation. 0.4285714286k + 0.04591836735 + k2 = -0.03571428571 + 0.04591836735 Reorder the terms: 0.04591836735 + 0.4285714286k + k2 = -0.03571428571 + 0.04591836735 Combine like terms: -0.03571428571 + 0.04591836735 = 0.01020408164 0.04591836735 + 0.4285714286k + k2 = 0.01020408164 Factor a perfect square on the left side: (k + 0.2142857143)(k + 0.2142857143) = 0.01020408164 Calculate the square root of the right side: 0.101015254 Break this problem into two subproblems by setting (k + 0.2142857143) equal to 0.101015254 and -0.101015254.Subproblem 1
k + 0.2142857143 = 0.101015254 Simplifying k + 0.2142857143 = 0.101015254 Reorder the terms: 0.2142857143 + k = 0.101015254 Solving 0.2142857143 + k = 0.101015254 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.2142857143' to each side of the equation. 0.2142857143 + -0.2142857143 + k = 0.101015254 + -0.2142857143 Combine like terms: 0.2142857143 + -0.2142857143 = 0.0000000000 0.0000000000 + k = 0.101015254 + -0.2142857143 k = 0.101015254 + -0.2142857143 Combine like terms: 0.101015254 + -0.2142857143 = -0.1132704603 k = -0.1132704603 Simplifying k = -0.1132704603Subproblem 2
k + 0.2142857143 = -0.101015254 Simplifying k + 0.2142857143 = -0.101015254 Reorder the terms: 0.2142857143 + k = -0.101015254 Solving 0.2142857143 + k = -0.101015254 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.2142857143' to each side of the equation. 0.2142857143 + -0.2142857143 + k = -0.101015254 + -0.2142857143 Combine like terms: 0.2142857143 + -0.2142857143 = 0.0000000000 0.0000000000 + k = -0.101015254 + -0.2142857143 k = -0.101015254 + -0.2142857143 Combine like terms: -0.101015254 + -0.2142857143 = -0.3153009683 k = -0.3153009683 Simplifying k = -0.3153009683Solution
The solution to the problem is based on the solutions from the subproblems. k = {-0.1132704603, -0.3153009683}
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