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Simplifying -2k2 + -7k + 19 = 0 Reorder the terms: 19 + -7k + -2k2 = 0 Solving 19 + -7k + -2k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -9.5 + 3.5k + k2 = 0 Move the constant term to the right: Add '9.5' to each side of the equation. -9.5 + 3.5k + 9.5 + k2 = 0 + 9.5 Reorder the terms: -9.5 + 9.5 + 3.5k + k2 = 0 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + 3.5k + k2 = 0 + 9.5 3.5k + k2 = 0 + 9.5 Combine like terms: 0 + 9.5 = 9.5 3.5k + k2 = 9.5 The k term is 3.5k. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5k + 3.0625 + k2 = 9.5 + 3.0625 Reorder the terms: 3.0625 + 3.5k + k2 = 9.5 + 3.0625 Combine like terms: 9.5 + 3.0625 = 12.5625 3.0625 + 3.5k + k2 = 12.5625 Factor a perfect square on the left side: (k + 1.75)(k + 1.75) = 12.5625 Calculate the square root of the right side: 3.54436172 Break this problem into two subproblems by setting (k + 1.75) equal to 3.54436172 and -3.54436172.Subproblem 1
k + 1.75 = 3.54436172 Simplifying k + 1.75 = 3.54436172 Reorder the terms: 1.75 + k = 3.54436172 Solving 1.75 + k = 3.54436172 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + k = 3.54436172 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + k = 3.54436172 + -1.75 k = 3.54436172 + -1.75 Combine like terms: 3.54436172 + -1.75 = 1.79436172 k = 1.79436172 Simplifying k = 1.79436172Subproblem 2
k + 1.75 = -3.54436172 Simplifying k + 1.75 = -3.54436172 Reorder the terms: 1.75 + k = -3.54436172 Solving 1.75 + k = -3.54436172 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + k = -3.54436172 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + k = -3.54436172 + -1.75 k = -3.54436172 + -1.75 Combine like terms: -3.54436172 + -1.75 = -5.29436172 k = -5.29436172 Simplifying k = -5.29436172Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.79436172, -5.29436172}
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