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Simplifying k + -2 = 7 + -5k * 4k Reorder the terms: -2 + k = 7 + -5k * 4k Reorder the terms for easier multiplication: -2 + k = 7 + -5 * 4k * k Multiply -5 * 4 -2 + k = 7 + -20k * k Multiply k * k -2 + k = 7 + -20k2 Solving -2 + k = 7 + -20k2 Solving for variable 'k'. Reorder the terms: -2 + -7 + k + 20k2 = 7 + -20k2 + -7 + 20k2 Combine like terms: -2 + -7 = -9 -9 + k + 20k2 = 7 + -20k2 + -7 + 20k2 Reorder the terms: -9 + k + 20k2 = 7 + -7 + -20k2 + 20k2 Combine like terms: 7 + -7 = 0 -9 + k + 20k2 = 0 + -20k2 + 20k2 -9 + k + 20k2 = -20k2 + 20k2 Combine like terms: -20k2 + 20k2 = 0 -9 + k + 20k2 = 0 Begin completing the square. Divide all terms by 20 the coefficient of the squared term: Divide each side by '20'. -0.45 + 0.05k + k2 = 0 Move the constant term to the right: Add '0.45' to each side of the equation. -0.45 + 0.05k + 0.45 + k2 = 0 + 0.45 Reorder the terms: -0.45 + 0.45 + 0.05k + k2 = 0 + 0.45 Combine like terms: -0.45 + 0.45 = 0.00 0.00 + 0.05k + k2 = 0 + 0.45 0.05k + k2 = 0 + 0.45 Combine like terms: 0 + 0.45 = 0.45 0.05k + k2 = 0.45 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. 0.05k + 0.25 + k2 = 0.45 + 0.25 Reorder the terms: 0.25 + 0.05k + k2 = 0.45 + 0.25 Combine like terms: 0.45 + 0.25 = 0.7 0.25 + 0.05k + k2 = 0.7 Factor a perfect square on the left side: (k + 0.5)(k + 0.5) = 0.7 Calculate the square root of the right side: 0.836660027 Break this problem into two subproblems by setting (k + 0.5) equal to 0.836660027 and -0.836660027.Subproblem 1
k + 0.5 = 0.836660027 Simplifying k + 0.5 = 0.836660027 Reorder the terms: 0.5 + k = 0.836660027 Solving 0.5 + k = 0.836660027 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 = 0.836660027 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = 0.836660027 + -0.5 k = 0.836660027 + -0.5 Combine like terms: 0.836660027 + -0.5 = 0.336660027 k = 0.336660027 Simplifying k = 0.336660027Subproblem 2
k + 0.5 = -0.836660027 Simplifying k + 0.5 = -0.836660027 Reorder the terms: 0.5 + k = -0.836660027 Solving 0.5 + k = -0.836660027 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 = -0.836660027 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + k = -0.836660027 + -0.5 k = -0.836660027 + -0.5 Combine like terms: -0.836660027 + -0.5 = -1.336660027 k = -1.336660027 Simplifying k = -1.336660027Solution
The solution to the problem is based on the solutions from the subproblems. k = {0.336660027, -1.336660027}
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