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Simplifying -4f2 + 8f + 172 = 0 Reorder the terms: 172 + 8f + -4f2 = 0 Solving 172 + 8f + -4f2 = 0 Solving for variable 'f'. Factor out the Greatest Common Factor (GCF), '4'. 4(43 + 2f + -1f2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(43 + 2f + -1f2)' equal to zero and attempt to solve: Simplifying 43 + 2f + -1f2 = 0 Solving 43 + 2f + -1f2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -43 + -2f + f2 = 0 Move the constant term to the right: Add '43' to each side of the equation. -43 + -2f + 43 + f2 = 0 + 43 Reorder the terms: -43 + 43 + -2f + f2 = 0 + 43 Combine like terms: -43 + 43 = 0 0 + -2f + f2 = 0 + 43 -2f + f2 = 0 + 43 Combine like terms: 0 + 43 = 43 -2f + f2 = 43 The f term is -2f. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2f + 1 + f2 = 43 + 1 Reorder the terms: 1 + -2f + f2 = 43 + 1 Combine like terms: 43 + 1 = 44 1 + -2f + f2 = 44 Factor a perfect square on the left side: (f + -1)(f + -1) = 44 Calculate the square root of the right side: 6.633249581 Break this problem into two subproblems by setting (f + -1) equal to 6.633249581 and -6.633249581.Subproblem 1
f + -1 = 6.633249581 Simplifying f + -1 = 6.633249581 Reorder the terms: -1 + f = 6.633249581 Solving -1 + f = 6.633249581 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + f = 6.633249581 + 1 Combine like terms: -1 + 1 = 0 0 + f = 6.633249581 + 1 f = 6.633249581 + 1 Combine like terms: 6.633249581 + 1 = 7.633249581 f = 7.633249581 Simplifying f = 7.633249581Subproblem 2
f + -1 = -6.633249581 Simplifying f + -1 = -6.633249581 Reorder the terms: -1 + f = -6.633249581 Solving -1 + f = -6.633249581 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + f = -6.633249581 + 1 Combine like terms: -1 + 1 = 0 0 + f = -6.633249581 + 1 f = -6.633249581 + 1 Combine like terms: -6.633249581 + 1 = -5.633249581 f = -5.633249581 Simplifying f = -5.633249581Solution
The solution to the problem is based on the solutions from the subproblems. f = {7.633249581, -5.633249581}Solution
f = {7.633249581, -5.633249581}
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