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Simplifying k + 20 = 9k2 Reorder the terms: 20 + k = 9k2 Solving 20 + k = 9k2 Solving for variable 'k'. Combine like terms: 9k2 + -9k2 = 0 20 + k + -9k2 = 0 Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -2.222222222 + -0.1111111111k + k2 = 0 Move the constant term to the right: Add '2.222222222' to each side of the equation. -2.222222222 + -0.1111111111k + 2.222222222 + k2 = 0 + 2.222222222 Reorder the terms: -2.222222222 + 2.222222222 + -0.1111111111k + k2 = 0 + 2.222222222 Combine like terms: -2.222222222 + 2.222222222 = 0.000000000 0.000000000 + -0.1111111111k + k2 = 0 + 2.222222222 -0.1111111111k + k2 = 0 + 2.222222222 Combine like terms: 0 + 2.222222222 = 2.222222222 -0.1111111111k + k2 = 2.222222222 The k term is k. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -0.1111111111k + 0.25 + k2 = 2.222222222 + 0.25 Reorder the terms: 0.25 + -0.1111111111k + k2 = 2.222222222 + 0.25 Combine like terms: 2.222222222 + 0.25 = 2.472222222 0.25 + -0.1111111111k + k2 = 2.472222222 Factor a perfect square on the left side: (k + 0.5)(k + 0.5) = 2.472222222 Calculate the square root of the right side: 1.572330189 Break this problem into two subproblems by setting (k + 0.5) equal to 1.572330189 and -1.572330189.Subproblem 1
k + 0.5 = 1.572330189 Simplifying k + 0.5 = 1.572330189 Reorder the terms: 0.5 + k = 1.572330189 Solving 0.5 + k = 1.572330189 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + k = 1.572330189 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = 1.572330189 + -0.5 k = 1.572330189 + -0.5 Combine like terms: 1.572330189 + -0.5 = 1.072330189 k = 1.072330189 Simplifying k = 1.072330189Subproblem 2
k + 0.5 = -1.572330189 Simplifying k + 0.5 = -1.572330189 Reorder the terms: 0.5 + k = -1.572330189 Solving 0.5 + k = -1.572330189 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + k = -1.572330189 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = -1.572330189 + -0.5 k = -1.572330189 + -0.5 Combine like terms: -1.572330189 + -0.5 = -2.072330189 k = -2.072330189 Simplifying k = -2.072330189Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.072330189, -2.072330189}
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