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Simplifying 10k2 + 3k + -0.4 = 0 Reorder the terms: -0.4 + 3k + 10k2 = 0 Solving -0.4 + 3k + 10k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -0.04 + 0.3k + k2 = 0 Move the constant term to the right: Add '0.04' to each side of the equation. -0.04 + 0.3k + 0.04 + k2 = 0 + 0.04 Reorder the terms: -0.04 + 0.04 + 0.3k + k2 = 0 + 0.04 Combine like terms: -0.04 + 0.04 = 0.00 0.00 + 0.3k + k2 = 0 + 0.04 0.3k + k2 = 0 + 0.04 Combine like terms: 0 + 0.04 = 0.04 0.3k + k2 = 0.04 The k term is 0.3k. Take half its coefficient (0.15). Square it (0.0225) and add it to both sides. Add '0.0225' to each side of the equation. 0.3k + 0.0225 + k2 = 0.04 + 0.0225 Reorder the terms: 0.0225 + 0.3k + k2 = 0.04 + 0.0225 Combine like terms: 0.04 + 0.0225 = 0.0625 0.0225 + 0.3k + k2 = 0.0625 Factor a perfect square on the left side: (k + 0.15)(k + 0.15) = 0.0625 Calculate the square root of the right side: 0.25 Break this problem into two subproblems by setting (k + 0.15) equal to 0.25 and -0.25.Subproblem 1
k + 0.15 = 0.25 Simplifying k + 0.15 = 0.25 Reorder the terms: 0.15 + k = 0.25 Solving 0.15 + k = 0.25 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.15' to each side of the equation. 0.15 + -0.15 + k = 0.25 + -0.15 Combine like terms: 0.15 + -0.15 = 0.00 0.00 + k = 0.25 + -0.15 k = 0.25 + -0.15 Combine like terms: 0.25 + -0.15 = 0.1 k = 0.1 Simplifying k = 0.1Subproblem 2
k + 0.15 = -0.25 Simplifying k + 0.15 = -0.25 Reorder the terms: 0.15 + k = -0.25 Solving 0.15 + k = -0.25 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-0.15' to each side of the equation. 0.15 + -0.15 + k = -0.25 + -0.15 Combine like terms: 0.15 + -0.15 = 0.00 0.00 + k = -0.25 + -0.15 k = -0.25 + -0.15 Combine like terms: -0.25 + -0.15 = -0.4 k = -0.4 Simplifying k = -0.4Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.1, -0.4}
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