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Simplifying 8k2 + -72 = 5k + -45k Reorder the terms: -72 + 8k2 = 5k + -45k Combine like terms: 5k + -45k = -40k -72 + 8k2 = -40k Solving -72 + 8k2 = -40k Solving for variable 'k'. Reorder the terms: -72 + 40k + 8k2 = -40k + 40k Combine like terms: -40k + 40k = 0 -72 + 40k + 8k2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(-9 + 5k + k2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(-9 + 5k + k2)' equal to zero and attempt to solve: Simplifying -9 + 5k + k2 = 0 Solving -9 + 5k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '9' to each side of the equation. -9 + 5k + 9 + k2 = 0 + 9 Reorder the terms: -9 + 9 + 5k + k2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + 5k + k2 = 0 + 9 5k + k2 = 0 + 9 Combine like terms: 0 + 9 = 9 5k + k2 = 9 The k term is 5k. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5k + 6.25 + k2 = 9 + 6.25 Reorder the terms: 6.25 + 5k + k2 = 9 + 6.25 Combine like terms: 9 + 6.25 = 15.25 6.25 + 5k + k2 = 15.25 Factor a perfect square on the left side: (k + 2.5)(k + 2.5) = 15.25 Calculate the square root of the right side: 3.905124838 Break this problem into two subproblems by setting (k + 2.5) equal to 3.905124838 and -3.905124838.Subproblem 1
k + 2.5 = 3.905124838 Simplifying k + 2.5 = 3.905124838 Reorder the terms: 2.5 + k = 3.905124838 Solving 2.5 + k = 3.905124838 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + k = 3.905124838 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + k = 3.905124838 + -2.5 k = 3.905124838 + -2.5 Combine like terms: 3.905124838 + -2.5 = 1.405124838 k = 1.405124838 Simplifying k = 1.405124838Subproblem 2
k + 2.5 = -3.905124838 Simplifying k + 2.5 = -3.905124838 Reorder the terms: 2.5 + k = -3.905124838 Solving 2.5 + k = -3.905124838 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + k = -3.905124838 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + k = -3.905124838 + -2.5 k = -3.905124838 + -2.5 Combine like terms: -3.905124838 + -2.5 = -6.405124838 k = -6.405124838 Simplifying k = -6.405124838Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.405124838, -6.405124838}Solution
k = {1.405124838, -6.405124838}
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