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Simplifying 3f2 + 72f + -105 = 0 Reorder the terms: -105 + 72f + 3f2 = 0 Solving -105 + 72f + 3f2 = 0 Solving for variable 'f'. Factor out the Greatest Common Factor (GCF), '3'. 3(-35 + 24f + f2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-35 + 24f + f2)' equal to zero and attempt to solve: Simplifying -35 + 24f + f2 = 0 Solving -35 + 24f + f2 = 0 Begin completing the square. Move the constant term to the right: Add '35' to each side of the equation. -35 + 24f + 35 + f2 = 0 + 35 Reorder the terms: -35 + 35 + 24f + f2 = 0 + 35 Combine like terms: -35 + 35 = 0 0 + 24f + f2 = 0 + 35 24f + f2 = 0 + 35 Combine like terms: 0 + 35 = 35 24f + f2 = 35 The f term is 24f. Take half its coefficient (12). Square it (144) and add it to both sides. Add '144' to each side of the equation. 24f + 144 + f2 = 35 + 144 Reorder the terms: 144 + 24f + f2 = 35 + 144 Combine like terms: 35 + 144 = 179 144 + 24f + f2 = 179 Factor a perfect square on the left side: (f + 12)(f + 12) = 179 Calculate the square root of the right side: 13.37908816 Break this problem into two subproblems by setting (f + 12) equal to 13.37908816 and -13.37908816.Subproblem 1
f + 12 = 13.37908816 Simplifying f + 12 = 13.37908816 Reorder the terms: 12 + f = 13.37908816 Solving 12 + f = 13.37908816 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-12' to each side of the equation. 12 + -12 + f = 13.37908816 + -12 Combine like terms: 12 + -12 = 0 0 + f = 13.37908816 + -12 f = 13.37908816 + -12 Combine like terms: 13.37908816 + -12 = 1.37908816 f = 1.37908816 Simplifying f = 1.37908816Subproblem 2
f + 12 = -13.37908816 Simplifying f + 12 = -13.37908816 Reorder the terms: 12 + f = -13.37908816 Solving 12 + f = -13.37908816 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-12' to each side of the equation. 12 + -12 + f = -13.37908816 + -12 Combine like terms: 12 + -12 = 0 0 + f = -13.37908816 + -12 f = -13.37908816 + -12 Combine like terms: -13.37908816 + -12 = -25.37908816 f = -25.37908816 Simplifying f = -25.37908816Solution
The solution to the problem is based on the solutions from the subproblems. f = {1.37908816, -25.37908816}Solution
f = {1.37908816, -25.37908816}
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