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Simplifying k2 + 10k + 1 = 0 Reorder the terms: 1 + 10k + k2 = 0 Solving 1 + 10k + k2 = 0 Solving for variable 'k'. Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + 10k + -1 + k2 = 0 + -1 Reorder the terms: 1 + -1 + 10k + k2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 10k + k2 = 0 + -1 10k + k2 = 0 + -1 Combine like terms: 0 + -1 = -1 10k + k2 = -1 The k term is 10k. Take half its coefficient (5). Square it (25) and add it to both sides. Add '25' to each side of the equation. 10k + 25 + k2 = -1 + 25 Reorder the terms: 25 + 10k + k2 = -1 + 25 Combine like terms: -1 + 25 = 24 25 + 10k + k2 = 24 Factor a perfect square on the left side: (k + 5)(k + 5) = 24 Calculate the square root of the right side: 4.898979486 Break this problem into two subproblems by setting (k + 5) equal to 4.898979486 and -4.898979486.Subproblem 1
k + 5 = 4.898979486 Simplifying k + 5 = 4.898979486 Reorder the terms: 5 + k = 4.898979486 Solving 5 + k = 4.898979486 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + k = 4.898979486 + -5 Combine like terms: 5 + -5 = 0 0 + k = 4.898979486 + -5 k = 4.898979486 + -5 Combine like terms: 4.898979486 + -5 = -0.101020514 k = -0.101020514 Simplifying k = -0.101020514Subproblem 2
k + 5 = -4.898979486 Simplifying k + 5 = -4.898979486 Reorder the terms: 5 + k = -4.898979486 Solving 5 + k = -4.898979486 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + k = -4.898979486 + -5 Combine like terms: 5 + -5 = 0 0 + k = -4.898979486 + -5 k = -4.898979486 + -5 Combine like terms: -4.898979486 + -5 = -9.898979486 k = -9.898979486 Simplifying k = -9.898979486Solution
The solution to the problem is based on the solutions from the subproblems. k = {-0.101020514, -9.898979486}
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