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Simplifying 15f2 + 37f = 18 Reorder the terms: 37f + 15f2 = 18 Solving 37f + 15f2 = 18 Solving for variable 'f'. Reorder the terms: -18 + 37f + 15f2 = 18 + -18 Combine like terms: 18 + -18 = 0 -18 + 37f + 15f2 = 0 Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -1.2 + 2.466666667f + f2 = 0 Move the constant term to the right: Add '1.2' to each side of the equation. -1.2 + 2.466666667f + 1.2 + f2 = 0 + 1.2 Reorder the terms: -1.2 + 1.2 + 2.466666667f + f2 = 0 + 1.2 Combine like terms: -1.2 + 1.2 = 0.0 0.0 + 2.466666667f + f2 = 0 + 1.2 2.466666667f + f2 = 0 + 1.2 Combine like terms: 0 + 1.2 = 1.2 2.466666667f + f2 = 1.2 The f term is 2.466666667f. Take half its coefficient (1.233333334). Square it (1.521111113) and add it to both sides. Add '1.521111113' to each side of the equation. 2.466666667f + 1.521111113 + f2 = 1.2 + 1.521111113 Reorder the terms: 1.521111113 + 2.466666667f + f2 = 1.2 + 1.521111113 Combine like terms: 1.2 + 1.521111113 = 2.721111113 1.521111113 + 2.466666667f + f2 = 2.721111113 Factor a perfect square on the left side: (f + 1.233333334)(f + 1.233333334) = 2.721111113 Calculate the square root of the right side: 1.649579071 Break this problem into two subproblems by setting (f + 1.233333334) equal to 1.649579071 and -1.649579071.Subproblem 1
f + 1.233333334 = 1.649579071 Simplifying f + 1.233333334 = 1.649579071 Reorder the terms: 1.233333334 + f = 1.649579071 Solving 1.233333334 + f = 1.649579071 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-1.233333334' to each side of the equation. 1.233333334 + -1.233333334 + f = 1.649579071 + -1.233333334 Combine like terms: 1.233333334 + -1.233333334 = 0.000000000 0.000000000 + f = 1.649579071 + -1.233333334 f = 1.649579071 + -1.233333334 Combine like terms: 1.649579071 + -1.233333334 = 0.416245737 f = 0.416245737 Simplifying f = 0.416245737Subproblem 2
f + 1.233333334 = -1.649579071 Simplifying f + 1.233333334 = -1.649579071 Reorder the terms: 1.233333334 + f = -1.649579071 Solving 1.233333334 + f = -1.649579071 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-1.233333334' to each side of the equation. 1.233333334 + -1.233333334 + f = -1.649579071 + -1.233333334 Combine like terms: 1.233333334 + -1.233333334 = 0.000000000 0.000000000 + f = -1.649579071 + -1.233333334 f = -1.649579071 + -1.233333334 Combine like terms: -1.649579071 + -1.233333334 = -2.882912405 f = -2.882912405 Simplifying f = -2.882912405Solution
The solution to the problem is based on the solutions from the subproblems. f = {0.416245737, -2.882912405}
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