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Simplifying 5k2 + -3k + 1 = 6 Reorder the terms: 1 + -3k + 5k2 = 6 Solving 1 + -3k + 5k2 = 6 Solving for variable 'k'. Reorder the terms: 1 + -6 + -3k + 5k2 = 6 + -6 Combine like terms: 1 + -6 = -5 -5 + -3k + 5k2 = 6 + -6 Combine like terms: 6 + -6 = 0 -5 + -3k + 5k2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -1 + -0.6k + k2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -0.6k + 1 + k2 = 0 + 1 Reorder the terms: -1 + 1 + -0.6k + k2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -0.6k + k2 = 0 + 1 -0.6k + k2 = 0 + 1 Combine like terms: 0 + 1 = 1 -0.6k + k2 = 1 The k term is -0.6k. Take half its coefficient (-0.3). Square it (0.09) and add it to both sides. Add '0.09' to each side of the equation. -0.6k + 0.09 + k2 = 1 + 0.09 Reorder the terms: 0.09 + -0.6k + k2 = 1 + 0.09 Combine like terms: 1 + 0.09 = 1.09 0.09 + -0.6k + k2 = 1.09 Factor a perfect square on the left side: (k + -0.3)(k + -0.3) = 1.09 Calculate the square root of the right side: 1.044030651 Break this problem into two subproblems by setting (k + -0.3) equal to 1.044030651 and -1.044030651.Subproblem 1
k + -0.3 = 1.044030651 Simplifying k + -0.3 = 1.044030651 Reorder the terms: -0.3 + k = 1.044030651 Solving -0.3 + k = 1.044030651 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.3' to each side of the equation. -0.3 + 0.3 + k = 1.044030651 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + k = 1.044030651 + 0.3 k = 1.044030651 + 0.3 Combine like terms: 1.044030651 + 0.3 = 1.344030651 k = 1.344030651 Simplifying k = 1.344030651Subproblem 2
k + -0.3 = -1.044030651 Simplifying k + -0.3 = -1.044030651 Reorder the terms: -0.3 + k = -1.044030651 Solving -0.3 + k = -1.044030651 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.3' to each side of the equation. -0.3 + 0.3 + k = -1.044030651 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + k = -1.044030651 + 0.3 k = -1.044030651 + 0.3 Combine like terms: -1.044030651 + 0.3 = -0.744030651 k = -0.744030651 Simplifying k = -0.744030651Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.344030651, -0.744030651}
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