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Simplifying -4f2 + 28f + -20 = 0 Reorder the terms: -20 + 28f + -4f2 = 0 Solving -20 + 28f + -4f2 = 0 Solving for variable 'f'. Factor out the Greatest Common Factor (GCF), '4'. 4(-5 + 7f + -1f2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-5 + 7f + -1f2)' equal to zero and attempt to solve: Simplifying -5 + 7f + -1f2 = 0 Solving -5 + 7f + -1f2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 5 + -7f + f2 = 0 Move the constant term to the right: Add '-5' to each side of the equation. 5 + -7f + -5 + f2 = 0 + -5 Reorder the terms: 5 + -5 + -7f + f2 = 0 + -5 Combine like terms: 5 + -5 = 0 0 + -7f + f2 = 0 + -5 -7f + f2 = 0 + -5 Combine like terms: 0 + -5 = -5 -7f + f2 = -5 The f term is -7f. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7f + 12.25 + f2 = -5 + 12.25 Reorder the terms: 12.25 + -7f + f2 = -5 + 12.25 Combine like terms: -5 + 12.25 = 7.25 12.25 + -7f + f2 = 7.25 Factor a perfect square on the left side: (f + -3.5)(f + -3.5) = 7.25 Calculate the square root of the right side: 2.692582404 Break this problem into two subproblems by setting (f + -3.5) equal to 2.692582404 and -2.692582404.Subproblem 1
f + -3.5 = 2.692582404 Simplifying f + -3.5 = 2.692582404 Reorder the terms: -3.5 + f = 2.692582404 Solving -3.5 + f = 2.692582404 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + f = 2.692582404 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + f = 2.692582404 + 3.5 f = 2.692582404 + 3.5 Combine like terms: 2.692582404 + 3.5 = 6.192582404 f = 6.192582404 Simplifying f = 6.192582404Subproblem 2
f + -3.5 = -2.692582404 Simplifying f + -3.5 = -2.692582404 Reorder the terms: -3.5 + f = -2.692582404 Solving -3.5 + f = -2.692582404 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + f = -2.692582404 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + f = -2.692582404 + 3.5 f = -2.692582404 + 3.5 Combine like terms: -2.692582404 + 3.5 = 0.807417596 f = 0.807417596 Simplifying f = 0.807417596Solution
The solution to the problem is based on the solutions from the subproblems. f = {6.192582404, 0.807417596}Solution
f = {6.192582404, 0.807417596}
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