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Simplifying 2f2 + 20f + 5 = 0 Reorder the terms: 5 + 20f + 2f2 = 0 Solving 5 + 20f + 2f2 = 0 Solving for variable 'f'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 2.5 + 10f + f2 = 0 Move the constant term to the right: Add '-2.5' to each side of the equation. 2.5 + 10f + -2.5 + f2 = 0 + -2.5 Reorder the terms: 2.5 + -2.5 + 10f + f2 = 0 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + 10f + f2 = 0 + -2.5 10f + f2 = 0 + -2.5 Combine like terms: 0 + -2.5 = -2.5 10f + f2 = -2.5 The f term is 10f. Take half its coefficient (5). Square it (25) and add it to both sides. Add '25' to each side of the equation. 10f + 25 + f2 = -2.5 + 25 Reorder the terms: 25 + 10f + f2 = -2.5 + 25 Combine like terms: -2.5 + 25 = 22.5 25 + 10f + f2 = 22.5 Factor a perfect square on the left side: (f + 5)(f + 5) = 22.5 Calculate the square root of the right side: 4.74341649 Break this problem into two subproblems by setting (f + 5) equal to 4.74341649 and -4.74341649.Subproblem 1
f + 5 = 4.74341649 Simplifying f + 5 = 4.74341649 Reorder the terms: 5 + f = 4.74341649 Solving 5 + f = 4.74341649 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + f = 4.74341649 + -5 Combine like terms: 5 + -5 = 0 0 + f = 4.74341649 + -5 f = 4.74341649 + -5 Combine like terms: 4.74341649 + -5 = -0.25658351 f = -0.25658351 Simplifying f = -0.25658351Subproblem 2
f + 5 = -4.74341649 Simplifying f + 5 = -4.74341649 Reorder the terms: 5 + f = -4.74341649 Solving 5 + f = -4.74341649 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + f = -4.74341649 + -5 Combine like terms: 5 + -5 = 0 0 + f = -4.74341649 + -5 f = -4.74341649 + -5 Combine like terms: -4.74341649 + -5 = -9.74341649 f = -9.74341649 Simplifying f = -9.74341649Solution
The solution to the problem is based on the solutions from the subproblems. f = {-0.25658351, -9.74341649}
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