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Simplifying 5k2 + 24k + 5 = 0 Reorder the terms: 5 + 24k + 5k2 = 0 Solving 5 + 24k + 5k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 1 + 4.8k + k2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + 4.8k + -1 + k2 = 0 + -1 Reorder the terms: 1 + -1 + 4.8k + k2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 4.8k + k2 = 0 + -1 4.8k + k2 = 0 + -1 Combine like terms: 0 + -1 = -1 4.8k + k2 = -1 The k term is 4.8k. Take half its coefficient (2.4). Square it (5.76) and add it to both sides. Add '5.76' to each side of the equation. 4.8k + 5.76 + k2 = -1 + 5.76 Reorder the terms: 5.76 + 4.8k + k2 = -1 + 5.76 Combine like terms: -1 + 5.76 = 4.76 5.76 + 4.8k + k2 = 4.76 Factor a perfect square on the left side: (k + 2.4)(k + 2.4) = 4.76 Calculate the square root of the right side: 2.181742423 Break this problem into two subproblems by setting (k + 2.4) equal to 2.181742423 and -2.181742423.Subproblem 1
k + 2.4 = 2.181742423 Simplifying k + 2.4 = 2.181742423 Reorder the terms: 2.4 + k = 2.181742423 Solving 2.4 + k = 2.181742423 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.4' to each side of the equation. 2.4 + -2.4 + k = 2.181742423 + -2.4 Combine like terms: 2.4 + -2.4 = 0.0 0.0 + k = 2.181742423 + -2.4 k = 2.181742423 + -2.4 Combine like terms: 2.181742423 + -2.4 = -0.218257577 k = -0.218257577 Simplifying k = -0.218257577Subproblem 2
k + 2.4 = -2.181742423 Simplifying k + 2.4 = -2.181742423 Reorder the terms: 2.4 + k = -2.181742423 Solving 2.4 + k = -2.181742423 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-2.4' to each side of the equation. 2.4 + -2.4 + k = -2.181742423 + -2.4 Combine like terms: 2.4 + -2.4 = 0.0 0.0 + k = -2.181742423 + -2.4 k = -2.181742423 + -2.4 Combine like terms: -2.181742423 + -2.4 = -4.581742423 k = -4.581742423 Simplifying k = -4.581742423Solution
The solution to the problem is based on the solutions from the subproblems. k = {-0.218257577, -4.581742423}
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