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Simplifying 3f2 + 24f + 40 = 0 Reorder the terms: 40 + 24f + 3f2 = 0 Solving 40 + 24f + 3f2 = 0 Solving for variable 'f'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 13.33333333 + 8f + f2 = 0 Move the constant term to the right: Add '-13.33333333' to each side of the equation. 13.33333333 + 8f + -13.33333333 + f2 = 0 + -13.33333333 Reorder the terms: 13.33333333 + -13.33333333 + 8f + f2 = 0 + -13.33333333 Combine like terms: 13.33333333 + -13.33333333 = 0.00000000 0.00000000 + 8f + f2 = 0 + -13.33333333 8f + f2 = 0 + -13.33333333 Combine like terms: 0 + -13.33333333 = -13.33333333 8f + f2 = -13.33333333 The f term is 8f. Take half its coefficient (4). Square it (16) and add it to both sides. Add '16' to each side of the equation. 8f + 16 + f2 = -13.33333333 + 16 Reorder the terms: 16 + 8f + f2 = -13.33333333 + 16 Combine like terms: -13.33333333 + 16 = 2.66666667 16 + 8f + f2 = 2.66666667 Factor a perfect square on the left side: (f + 4)(f + 4) = 2.66666667 Calculate the square root of the right side: 1.632993163 Break this problem into two subproblems by setting (f + 4) equal to 1.632993163 and -1.632993163.Subproblem 1
f + 4 = 1.632993163 Simplifying f + 4 = 1.632993163 Reorder the terms: 4 + f = 1.632993163 Solving 4 + f = 1.632993163 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + f = 1.632993163 + -4 Combine like terms: 4 + -4 = 0 0 + f = 1.632993163 + -4 f = 1.632993163 + -4 Combine like terms: 1.632993163 + -4 = -2.367006837 f = -2.367006837 Simplifying f = -2.367006837Subproblem 2
f + 4 = -1.632993163 Simplifying f + 4 = -1.632993163 Reorder the terms: 4 + f = -1.632993163 Solving 4 + f = -1.632993163 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + f = -1.632993163 + -4 Combine like terms: 4 + -4 = 0 0 + f = -1.632993163 + -4 f = -1.632993163 + -4 Combine like terms: -1.632993163 + -4 = -5.632993163 f = -5.632993163 Simplifying f = -5.632993163Solution
The solution to the problem is based on the solutions from the subproblems. f = {-2.367006837, -5.632993163}
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