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Simplifying 2f2 + -24f = 98 Reorder the terms: -24f + 2f2 = 98 Solving -24f + 2f2 = 98 Solving for variable 'f'. Reorder the terms: -98 + -24f + 2f2 = 98 + -98 Combine like terms: 98 + -98 = 0 -98 + -24f + 2f2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-49 + -12f + f2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-49 + -12f + f2)' equal to zero and attempt to solve: Simplifying -49 + -12f + f2 = 0 Solving -49 + -12f + f2 = 0 Begin completing the square. Move the constant term to the right: Add '49' to each side of the equation. -49 + -12f + 49 + f2 = 0 + 49 Reorder the terms: -49 + 49 + -12f + f2 = 0 + 49 Combine like terms: -49 + 49 = 0 0 + -12f + f2 = 0 + 49 -12f + f2 = 0 + 49 Combine like terms: 0 + 49 = 49 -12f + f2 = 49 The f term is -12f. Take half its coefficient (-6). Square it (36) and add it to both sides. Add '36' to each side of the equation. -12f + 36 + f2 = 49 + 36 Reorder the terms: 36 + -12f + f2 = 49 + 36 Combine like terms: 49 + 36 = 85 36 + -12f + f2 = 85 Factor a perfect square on the left side: (f + -6)(f + -6) = 85 Calculate the square root of the right side: 9.219544457 Break this problem into two subproblems by setting (f + -6) equal to 9.219544457 and -9.219544457.Subproblem 1
f + -6 = 9.219544457 Simplifying f + -6 = 9.219544457 Reorder the terms: -6 + f = 9.219544457 Solving -6 + f = 9.219544457 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + f = 9.219544457 + 6 Combine like terms: -6 + 6 = 0 0 + f = 9.219544457 + 6 f = 9.219544457 + 6 Combine like terms: 9.219544457 + 6 = 15.219544457 f = 15.219544457 Simplifying f = 15.219544457Subproblem 2
f + -6 = -9.219544457 Simplifying f + -6 = -9.219544457 Reorder the terms: -6 + f = -9.219544457 Solving -6 + f = -9.219544457 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + f = -9.219544457 + 6 Combine like terms: -6 + 6 = 0 0 + f = -9.219544457 + 6 f = -9.219544457 + 6 Combine like terms: -9.219544457 + 6 = -3.219544457 f = -3.219544457 Simplifying f = -3.219544457Solution
The solution to the problem is based on the solutions from the subproblems. f = {15.219544457, -3.219544457}Solution
f = {15.219544457, -3.219544457}
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