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Simplifying 2.5k2 + -7.5k + 1.25 = 0 Reorder the terms: 1.25 + -7.5k + 2.5k2 = 0 Solving 1.25 + -7.5k + 2.5k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 2.5 the coefficient of the squared term: Divide each side by '2.5'. 0.5 + -3k + k2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -3k + -0.5 + k2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -3k + k2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -3k + k2 = 0 + -0.5 -3k + k2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -3k + k2 = -0.5 The k term is -3k. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3k + 2.25 + k2 = -0.5 + 2.25 Reorder the terms: 2.25 + -3k + k2 = -0.5 + 2.25 Combine like terms: -0.5 + 2.25 = 1.75 2.25 + -3k + k2 = 1.75 Factor a perfect square on the left side: (k + -1.5)(k + -1.5) = 1.75 Calculate the square root of the right side: 1.322875656 Break this problem into two subproblems by setting (k + -1.5) equal to 1.322875656 and -1.322875656.Subproblem 1
k + -1.5 = 1.322875656 Simplifying k + -1.5 = 1.322875656 Reorder the terms: -1.5 + k = 1.322875656 Solving -1.5 + k = 1.322875656 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + k = 1.322875656 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + k = 1.322875656 + 1.5 k = 1.322875656 + 1.5 Combine like terms: 1.322875656 + 1.5 = 2.822875656 k = 2.822875656 Simplifying k = 2.822875656Subproblem 2
k + -1.5 = -1.322875656 Simplifying k + -1.5 = -1.322875656 Reorder the terms: -1.5 + k = -1.322875656 Solving -1.5 + k = -1.322875656 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + k = -1.322875656 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + k = -1.322875656 + 1.5 k = -1.322875656 + 1.5 Combine like terms: -1.322875656 + 1.5 = 0.177124344 k = 0.177124344 Simplifying k = 0.177124344Solution
The solution to the problem is based on the solutions from the subproblems. k = {2.822875656, 0.177124344}
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