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Simplifying f2 + -22f = 3 Reorder the terms: -22f + f2 = 3 Solving -22f + f2 = 3 Solving for variable 'f'. Reorder the terms: -3 + -22f + f2 = 3 + -3 Combine like terms: 3 + -3 = 0 -3 + -22f + f2 = 0 Begin completing the square. Move the constant term to the right: Add '3' to each side of the equation. -3 + -22f + 3 + f2 = 0 + 3 Reorder the terms: -3 + 3 + -22f + f2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + -22f + f2 = 0 + 3 -22f + f2 = 0 + 3 Combine like terms: 0 + 3 = 3 -22f + f2 = 3 The f term is -22f. Take half its coefficient (-11). Square it (121) and add it to both sides. Add '121' to each side of the equation. -22f + 121 + f2 = 3 + 121 Reorder the terms: 121 + -22f + f2 = 3 + 121 Combine like terms: 3 + 121 = 124 121 + -22f + f2 = 124 Factor a perfect square on the left side: (f + -11)(f + -11) = 124 Calculate the square root of the right side: 11.135528726 Break this problem into two subproblems by setting (f + -11) equal to 11.135528726 and -11.135528726.Subproblem 1
f + -11 = 11.135528726 Simplifying f + -11 = 11.135528726 Reorder the terms: -11 + f = 11.135528726 Solving -11 + f = 11.135528726 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + f = 11.135528726 + 11 Combine like terms: -11 + 11 = 0 0 + f = 11.135528726 + 11 f = 11.135528726 + 11 Combine like terms: 11.135528726 + 11 = 22.135528726 f = 22.135528726 Simplifying f = 22.135528726Subproblem 2
f + -11 = -11.135528726 Simplifying f + -11 = -11.135528726 Reorder the terms: -11 + f = -11.135528726 Solving -11 + f = -11.135528726 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + f = -11.135528726 + 11 Combine like terms: -11 + 11 = 0 0 + f = -11.135528726 + 11 f = -11.135528726 + 11 Combine like terms: -11.135528726 + 11 = -0.135528726 f = -0.135528726 Simplifying f = -0.135528726Solution
The solution to the problem is based on the solutions from the subproblems. f = {22.135528726, -0.135528726}
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