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Simplifying 7k + 3 + -5k2 = 0 Reorder the terms: 3 + 7k + -5k2 = 0 Solving 3 + 7k + -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'. -0.6 + -1.4k + k2 = 0 Move the constant term to the right: Add '0.6' to each side of the equation. -0.6 + -1.4k + 0.6 + k2 = 0 + 0.6 Reorder the terms: -0.6 + 0.6 + -1.4k + k2 = 0 + 0.6 Combine like terms: -0.6 + 0.6 = 0.0 0.0 + -1.4k + k2 = 0 + 0.6 -1.4k + k2 = 0 + 0.6 Combine like terms: 0 + 0.6 = 0.6 -1.4k + k2 = 0.6 The k term is -1.4k. Take half its coefficient (-0.7). Square it (0.49) and add it to both sides. Add '0.49' to each side of the equation. -1.4k + 0.49 + k2 = 0.6 + 0.49 Reorder the terms: 0.49 + -1.4k + k2 = 0.6 + 0.49 Combine like terms: 0.6 + 0.49 = 1.09 0.49 + -1.4k + k2 = 1.09 Factor a perfect square on the left side: (k + -0.7)(k + -0.7) = 1.09 Calculate the square root of the right side: 1.044030651 Break this problem into two subproblems by setting (k + -0.7) equal to 1.044030651 and -1.044030651.Subproblem 1
k + -0.7 = 1.044030651 Simplifying k + -0.7 = 1.044030651 Reorder the terms: -0.7 + k = 1.044030651 Solving -0.7 + k = 1.044030651 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.7' to each side of the equation. -0.7 + 0.7 + k = 1.044030651 + 0.7 Combine like terms: -0.7 + 0.7 = 0.0 0.0 + k = 1.044030651 + 0.7 k = 1.044030651 + 0.7 Combine like terms: 1.044030651 + 0.7 = 1.744030651 k = 1.744030651 Simplifying k = 1.744030651Subproblem 2
k + -0.7 = -1.044030651 Simplifying k + -0.7 = -1.044030651 Reorder the terms: -0.7 + k = -1.044030651 Solving -0.7 + k = -1.044030651 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.7' to each side of the equation. -0.7 + 0.7 + k = -1.044030651 + 0.7 Combine like terms: -0.7 + 0.7 = 0.0 0.0 + k = -1.044030651 + 0.7 k = -1.044030651 + 0.7 Combine like terms: -1.044030651 + 0.7 = -0.344030651 k = -0.344030651 Simplifying k = -0.344030651Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.744030651, -0.344030651}
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