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Simplifying -4k2 + 37k + -75 = 1 Reorder the terms: -75 + 37k + -4k2 = 1 Solving -75 + 37k + -4k2 = 1 Solving for variable 'k'. Reorder the terms: -75 + -1 + 37k + -4k2 = 1 + -1 Combine like terms: -75 + -1 = -76 -76 + 37k + -4k2 = 1 + -1 Combine like terms: 1 + -1 = 0 -76 + 37k + -4k2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. 19 + -9.25k + k2 = 0 Move the constant term to the right: Add '-19' to each side of the equation. 19 + -9.25k + -19 + k2 = 0 + -19 Reorder the terms: 19 + -19 + -9.25k + k2 = 0 + -19 Combine like terms: 19 + -19 = 0 0 + -9.25k + k2 = 0 + -19 -9.25k + k2 = 0 + -19 Combine like terms: 0 + -19 = -19 -9.25k + k2 = -19 The k term is -9.25k. Take half its coefficient (-4.625). Square it (21.390625) and add it to both sides. Add '21.390625' to each side of the equation. -9.25k + 21.390625 + k2 = -19 + 21.390625 Reorder the terms: 21.390625 + -9.25k + k2 = -19 + 21.390625 Combine like terms: -19 + 21.390625 = 2.390625 21.390625 + -9.25k + k2 = 2.390625 Factor a perfect square on the left side: (k + -4.625)(k + -4.625) = 2.390625 Calculate the square root of the right side: 1.54616461 Break this problem into two subproblems by setting (k + -4.625) equal to 1.54616461 and -1.54616461.Subproblem 1
k + -4.625 = 1.54616461 Simplifying k + -4.625 = 1.54616461 Reorder the terms: -4.625 + k = 1.54616461 Solving -4.625 + k = 1.54616461 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '4.625' to each side of the equation. -4.625 + 4.625 + k = 1.54616461 + 4.625 Combine like terms: -4.625 + 4.625 = 0.000 0.000 + k = 1.54616461 + 4.625 k = 1.54616461 + 4.625 Combine like terms: 1.54616461 + 4.625 = 6.17116461 k = 6.17116461 Simplifying k = 6.17116461Subproblem 2
k + -4.625 = -1.54616461 Simplifying k + -4.625 = -1.54616461 Reorder the terms: -4.625 + k = -1.54616461 Solving -4.625 + k = -1.54616461 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '4.625' to each side of the equation. -4.625 + 4.625 + k = -1.54616461 + 4.625 Combine like terms: -4.625 + 4.625 = 0.000 0.000 + k = -1.54616461 + 4.625 k = -1.54616461 + 4.625 Combine like terms: -1.54616461 + 4.625 = 3.07883539 k = 3.07883539 Simplifying k = 3.07883539Solution
The solution to the problem is based on the solutions from the subproblems. k = {6.17116461, 3.07883539}
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