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Simplifying 2k(k + 2) + 8k = 84 Reorder the terms: 2k(2 + k) + 8k = 84 (2 * 2k + k * 2k) + 8k = 84 (4k + 2k2) + 8k = 84 Reorder the terms: 4k + 8k + 2k2 = 84 Combine like terms: 4k + 8k = 12k 12k + 2k2 = 84 Solving 12k + 2k2 = 84 Solving for variable 'k'. Reorder the terms: -84 + 12k + 2k2 = 84 + -84 Combine like terms: 84 + -84 = 0 -84 + 12k + 2k2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-42 + 6k + k2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-42 + 6k + k2)' equal to zero and attempt to solve: Simplifying -42 + 6k + k2 = 0 Solving -42 + 6k + k2 = 0 Begin completing the square. Move the constant term to the right: Add '42' to each side of the equation. -42 + 6k + 42 + k2 = 0 + 42 Reorder the terms: -42 + 42 + 6k + k2 = 0 + 42 Combine like terms: -42 + 42 = 0 0 + 6k + k2 = 0 + 42 6k + k2 = 0 + 42 Combine like terms: 0 + 42 = 42 6k + k2 = 42 The k term is 6k. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6k + 9 + k2 = 42 + 9 Reorder the terms: 9 + 6k + k2 = 42 + 9 Combine like terms: 42 + 9 = 51 9 + 6k + k2 = 51 Factor a perfect square on the left side: (k + 3)(k + 3) = 51 Calculate the square root of the right side: 7.141428429 Break this problem into two subproblems by setting (k + 3) equal to 7.141428429 and -7.141428429.Subproblem 1
k + 3 = 7.141428429 Simplifying k + 3 = 7.141428429 Reorder the terms: 3 + k = 7.141428429 Solving 3 + k = 7.141428429 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + k = 7.141428429 + -3 Combine like terms: 3 + -3 = 0 0 + k = 7.141428429 + -3 k = 7.141428429 + -3 Combine like terms: 7.141428429 + -3 = 4.141428429 k = 4.141428429 Simplifying k = 4.141428429Subproblem 2
k + 3 = -7.141428429 Simplifying k + 3 = -7.141428429 Reorder the terms: 3 + k = -7.141428429 Solving 3 + k = -7.141428429 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + k = -7.141428429 + -3 Combine like terms: 3 + -3 = 0 0 + k = -7.141428429 + -3 k = -7.141428429 + -3 Combine like terms: -7.141428429 + -3 = -10.141428429 k = -10.141428429 Simplifying k = -10.141428429Solution
The solution to the problem is based on the solutions from the subproblems. k = {4.141428429, -10.141428429}Solution
k = {4.141428429, -10.141428429}
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