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Simplifying 7k + -8 + 2k(k + 12) = 52 Reorder the terms: 7k + -8 + 2k(12 + k) = 52 7k + -8 + (12 * 2k + k * 2k) = 52 7k + -8 + (24k + 2k2) = 52 Reorder the terms: -8 + 7k + 24k + 2k2 = 52 Combine like terms: 7k + 24k = 31k -8 + 31k + 2k2 = 52 Solving -8 + 31k + 2k2 = 52 Solving for variable 'k'. Reorder the terms: -8 + -52 + 31k + 2k2 = 52 + -52 Combine like terms: -8 + -52 = -60 -60 + 31k + 2k2 = 52 + -52 Combine like terms: 52 + -52 = 0 -60 + 31k + 2k2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -30 + 15.5k + k2 = 0 Move the constant term to the right: Add '30' to each side of the equation. -30 + 15.5k + 30 + k2 = 0 + 30 Reorder the terms: -30 + 30 + 15.5k + k2 = 0 + 30 Combine like terms: -30 + 30 = 0 0 + 15.5k + k2 = 0 + 30 15.5k + k2 = 0 + 30 Combine like terms: 0 + 30 = 30 15.5k + k2 = 30 The k term is 15.5k. Take half its coefficient (7.75). Square it (60.0625) and add it to both sides. Add '60.0625' to each side of the equation. 15.5k + 60.0625 + k2 = 30 + 60.0625 Reorder the terms: 60.0625 + 15.5k + k2 = 30 + 60.0625 Combine like terms: 30 + 60.0625 = 90.0625 60.0625 + 15.5k + k2 = 90.0625 Factor a perfect square on the left side: (k + 7.75)(k + 7.75) = 90.0625 Calculate the square root of the right side: 9.490126448 Break this problem into two subproblems by setting (k + 7.75) equal to 9.490126448 and -9.490126448.Subproblem 1
k + 7.75 = 9.490126448 Simplifying k + 7.75 = 9.490126448 Reorder the terms: 7.75 + k = 9.490126448 Solving 7.75 + k = 9.490126448 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-7.75' to each side of the equation. 7.75 + -7.75 + k = 9.490126448 + -7.75 Combine like terms: 7.75 + -7.75 = 0.00 0.00 + k = 9.490126448 + -7.75 k = 9.490126448 + -7.75 Combine like terms: 9.490126448 + -7.75 = 1.740126448 k = 1.740126448 Simplifying k = 1.740126448Subproblem 2
k + 7.75 = -9.490126448 Simplifying k + 7.75 = -9.490126448 Reorder the terms: 7.75 + k = -9.490126448 Solving 7.75 + k = -9.490126448 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '-7.75' to each side of the equation. 7.75 + -7.75 + k = -9.490126448 + -7.75 Combine like terms: 7.75 + -7.75 = 0.00 0.00 + k = -9.490126448 + -7.75 k = -9.490126448 + -7.75 Combine like terms: -9.490126448 + -7.75 = -17.240126448 k = -17.240126448 Simplifying k = -17.240126448Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.740126448, -17.240126448}
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