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Simplifying 7k + -8 + k(k + 12) = 52 Reorder the terms: 7k + -8 + k(12 + k) = 52 7k + -8 + (12 * k + k * k) = 52 7k + -8 + (12k + k2) = 52 Reorder the terms: -8 + 7k + 12k + k2 = 52 Combine like terms: 7k + 12k = 19k -8 + 19k + k2 = 52 Solving -8 + 19k + k2 = 52 Solving for variable 'k'. Reorder the terms: -8 + -52 + 19k + k2 = 52 + -52 Combine like terms: -8 + -52 = -60 -60 + 19k + k2 = 52 + -52 Combine like terms: 52 + -52 = 0 -60 + 19k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '60' to each side of the equation. -60 + 19k + 60 + k2 = 0 + 60 Reorder the terms: -60 + 60 + 19k + k2 = 0 + 60 Combine like terms: -60 + 60 = 0 0 + 19k + k2 = 0 + 60 19k + k2 = 0 + 60 Combine like terms: 0 + 60 = 60 19k + k2 = 60 The k term is 19k. Take half its coefficient (9.5). Square it (90.25) and add it to both sides. Add '90.25' to each side of the equation. 19k + 90.25 + k2 = 60 + 90.25 Reorder the terms: 90.25 + 19k + k2 = 60 + 90.25 Combine like terms: 60 + 90.25 = 150.25 90.25 + 19k + k2 = 150.25 Factor a perfect square on the left side: (k + 9.5)(k + 9.5) = 150.25 Calculate the square root of the right side: 12.257650672 Break this problem into two subproblems by setting (k + 9.5) equal to 12.257650672 and -12.257650672.Subproblem 1
k + 9.5 = 12.257650672 Simplifying k + 9.5 = 12.257650672 Reorder the terms: 9.5 + k = 12.257650672 Solving 9.5 + k = 12.257650672 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-9.5' to each side of the equation. 9.5 + -9.5 + k = 12.257650672 + -9.5 Combine like terms: 9.5 + -9.5 = 0.0 0.0 + k = 12.257650672 + -9.5 k = 12.257650672 + -9.5 Combine like terms: 12.257650672 + -9.5 = 2.757650672 k = 2.757650672 Simplifying k = 2.757650672Subproblem 2
k + 9.5 = -12.257650672 Simplifying k + 9.5 = -12.257650672 Reorder the terms: 9.5 + k = -12.257650672 Solving 9.5 + k = -12.257650672 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-9.5' to each side of the equation. 9.5 + -9.5 + k = -12.257650672 + -9.5 Combine like terms: 9.5 + -9.5 = 0.0 0.0 + k = -12.257650672 + -9.5 k = -12.257650672 + -9.5 Combine like terms: -12.257650672 + -9.5 = -21.757650672 k = -21.757650672 Simplifying k = -21.757650672Solution
The solution to the problem is based on the solutions from the subproblems. k = {2.757650672, -21.757650672}
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