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Simplifying 12k2 + 15k = 16k + 203 Reorder the terms: 15k + 12k2 = 16k + 203 Reorder the terms: 15k + 12k2 = 203 + 16k Solving 15k + 12k2 = 203 + 16k Solving for variable 'k'. Reorder the terms: -203 + 15k + -16k + 12k2 = 203 + 16k + -203 + -16k Combine like terms: 15k + -16k = -1k -203 + -1k + 12k2 = 203 + 16k + -203 + -16k Reorder the terms: -203 + -1k + 12k2 = 203 + -203 + 16k + -16k Combine like terms: 203 + -203 = 0 -203 + -1k + 12k2 = 0 + 16k + -16k -203 + -1k + 12k2 = 16k + -16k Combine like terms: 16k + -16k = 0 -203 + -1k + 12k2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -16.91666667 + -0.08333333333k + k2 = 0 Move the constant term to the right: Add '16.91666667' to each side of the equation. -16.91666667 + -0.08333333333k + 16.91666667 + k2 = 0 + 16.91666667 Reorder the terms: -16.91666667 + 16.91666667 + -0.08333333333k + k2 = 0 + 16.91666667 Combine like terms: -16.91666667 + 16.91666667 = 0.00000000 0.00000000 + -0.08333333333k + k2 = 0 + 16.91666667 -0.08333333333k + k2 = 0 + 16.91666667 Combine like terms: 0 + 16.91666667 = 16.91666667 -0.08333333333k + k2 = 16.91666667 The k term is -0.08333333333k. Take half its coefficient (-0.04166666667). Square it (0.001736111111) and add it to both sides. Add '0.001736111111' to each side of the equation. -0.08333333333k + 0.001736111111 + k2 = 16.91666667 + 0.001736111111 Reorder the terms: 0.001736111111 + -0.08333333333k + k2 = 16.91666667 + 0.001736111111 Combine like terms: 16.91666667 + 0.001736111111 = 16.918402781111 0.001736111111 + -0.08333333333k + k2 = 16.918402781111 Factor a perfect square on the left side: (k + -0.04166666667)(k + -0.04166666667) = 16.918402781111 Calculate the square root of the right side: 4.113198607 Break this problem into two subproblems by setting (k + -0.04166666667) equal to 4.113198607 and -4.113198607.Subproblem 1
k + -0.04166666667 = 4.113198607 Simplifying k + -0.04166666667 = 4.113198607 Reorder the terms: -0.04166666667 + k = 4.113198607 Solving -0.04166666667 + k = 4.113198607 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.04166666667' to each side of the equation. -0.04166666667 + 0.04166666667 + k = 4.113198607 + 0.04166666667 Combine like terms: -0.04166666667 + 0.04166666667 = 0.00000000000 0.00000000000 + k = 4.113198607 + 0.04166666667 k = 4.113198607 + 0.04166666667 Combine like terms: 4.113198607 + 0.04166666667 = 4.15486527367 k = 4.15486527367 Simplifying k = 4.15486527367Subproblem 2
k + -0.04166666667 = -4.113198607 Simplifying k + -0.04166666667 = -4.113198607 Reorder the terms: -0.04166666667 + k = -4.113198607 Solving -0.04166666667 + k = -4.113198607 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.04166666667' to each side of the equation. -0.04166666667 + 0.04166666667 + k = -4.113198607 + 0.04166666667 Combine like terms: -0.04166666667 + 0.04166666667 = 0.00000000000 0.00000000000 + k = -4.113198607 + 0.04166666667 k = -4.113198607 + 0.04166666667 Combine like terms: -4.113198607 + 0.04166666667 = -4.07153194033 k = -4.07153194033 Simplifying k = -4.07153194033Solution
The solution to the problem is based on the solutions from the subproblems. k = {4.15486527367, -4.07153194033}
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