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Simplifying 12k2 + 15k = 16k + 21 Reorder the terms: 15k + 12k2 = 16k + 21 Reorder the terms: 15k + 12k2 = 21 + 16k Solving 15k + 12k2 = 21 + 16k Solving for variable 'k'. Reorder the terms: -21 + 15k + -16k + 12k2 = 21 + 16k + -21 + -16k Combine like terms: 15k + -16k = -1k -21 + -1k + 12k2 = 21 + 16k + -21 + -16k Reorder the terms: -21 + -1k + 12k2 = 21 + -21 + 16k + -16k Combine like terms: 21 + -21 = 0 -21 + -1k + 12k2 = 0 + 16k + -16k -21 + -1k + 12k2 = 16k + -16k Combine like terms: 16k + -16k = 0 -21 + -1k + 12k2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -1.75 + -0.08333333333k + k2 = 0 Move the constant term to the right: Add '1.75' to each side of the equation. -1.75 + -0.08333333333k + 1.75 + k2 = 0 + 1.75 Reorder the terms: -1.75 + 1.75 + -0.08333333333k + k2 = 0 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + -0.08333333333k + k2 = 0 + 1.75 -0.08333333333k + k2 = 0 + 1.75 Combine like terms: 0 + 1.75 = 1.75 -0.08333333333k + k2 = 1.75 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 = 1.75 + 0.001736111111 Reorder the terms: 0.001736111111 + -0.08333333333k + k2 = 1.75 + 0.001736111111 Combine like terms: 1.75 + 0.001736111111 = 1.751736111111 0.001736111111 + -0.08333333333k + k2 = 1.751736111111 Factor a perfect square on the left side: (k + -0.04166666667)(k + -0.04166666667) = 1.751736111111 Calculate the square root of the right side: 1.323531681 Break this problem into two subproblems by setting (k + -0.04166666667) equal to 1.323531681 and -1.323531681.Subproblem 1
k + -0.04166666667 = 1.323531681 Simplifying k + -0.04166666667 = 1.323531681 Reorder the terms: -0.04166666667 + k = 1.323531681 Solving -0.04166666667 + k = 1.323531681 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 = 1.323531681 + 0.04166666667 Combine like terms: -0.04166666667 + 0.04166666667 = 0.00000000000 0.00000000000 + k = 1.323531681 + 0.04166666667 k = 1.323531681 + 0.04166666667 Combine like terms: 1.323531681 + 0.04166666667 = 1.36519834767 k = 1.36519834767 Simplifying k = 1.36519834767Subproblem 2
k + -0.04166666667 = -1.323531681 Simplifying k + -0.04166666667 = -1.323531681 Reorder the terms: -0.04166666667 + k = -1.323531681 Solving -0.04166666667 + k = -1.323531681 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 = -1.323531681 + 0.04166666667 Combine like terms: -0.04166666667 + 0.04166666667 = 0.00000000000 0.00000000000 + k = -1.323531681 + 0.04166666667 k = -1.323531681 + 0.04166666667 Combine like terms: -1.323531681 + 0.04166666667 = -1.28186501433 k = -1.28186501433 Simplifying k = -1.28186501433Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.36519834767, -1.28186501433}
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