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Simplifying 2k2 + -14 = 9k Reorder the terms: -14 + 2k2 = 9k Solving -14 + 2k2 = 9k Solving for variable 'k'. Reorder the terms: -14 + -9k + 2k2 = 9k + -9k Combine like terms: 9k + -9k = 0 -14 + -9k + 2k2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -7 + -4.5k + k2 = 0 Move the constant term to the right: Add '7' to each side of the equation. -7 + -4.5k + 7 + k2 = 0 + 7 Reorder the terms: -7 + 7 + -4.5k + k2 = 0 + 7 Combine like terms: -7 + 7 = 0 0 + -4.5k + k2 = 0 + 7 -4.5k + k2 = 0 + 7 Combine like terms: 0 + 7 = 7 -4.5k + k2 = 7 The k term is -4.5k. Take half its coefficient (-2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. -4.5k + 5.0625 + k2 = 7 + 5.0625 Reorder the terms: 5.0625 + -4.5k + k2 = 7 + 5.0625 Combine like terms: 7 + 5.0625 = 12.0625 5.0625 + -4.5k + k2 = 12.0625 Factor a perfect square on the left side: (k + -2.25)(k + -2.25) = 12.0625 Calculate the square root of the right side: 3.473110997 Break this problem into two subproblems by setting (k + -2.25) equal to 3.473110997 and -3.473110997.Subproblem 1
k + -2.25 = 3.473110997 Simplifying k + -2.25 = 3.473110997 Reorder the terms: -2.25 + k = 3.473110997 Solving -2.25 + k = 3.473110997 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + k = 3.473110997 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + k = 3.473110997 + 2.25 k = 3.473110997 + 2.25 Combine like terms: 3.473110997 + 2.25 = 5.723110997 k = 5.723110997 Simplifying k = 5.723110997Subproblem 2
k + -2.25 = -3.473110997 Simplifying k + -2.25 = -3.473110997 Reorder the terms: -2.25 + k = -3.473110997 Solving -2.25 + k = -3.473110997 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + k = -3.473110997 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + k = -3.473110997 + 2.25 k = -3.473110997 + 2.25 Combine like terms: -3.473110997 + 2.25 = -1.223110997 k = -1.223110997 Simplifying k = -1.223110997Solution
The solution to the problem is based on the solutions from the subproblems. k = {5.723110997, -1.223110997}
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