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Simplifying f2 + -10f + 19 = 0 Reorder the terms: 19 + -10f + f2 = 0 Solving 19 + -10f + f2 = 0 Solving for variable 'f'. Begin completing the square. Move the constant term to the right: Add '-19' to each side of the equation. 19 + -10f + -19 + f2 = 0 + -19 Reorder the terms: 19 + -19 + -10f + f2 = 0 + -19 Combine like terms: 19 + -19 = 0 0 + -10f + f2 = 0 + -19 -10f + f2 = 0 + -19 Combine like terms: 0 + -19 = -19 -10f + f2 = -19 The f term is -10f. Take half its coefficient (-5). Square it (25) and add it to both sides. Add '25' to each side of the equation. -10f + 25 + f2 = -19 + 25 Reorder the terms: 25 + -10f + f2 = -19 + 25 Combine like terms: -19 + 25 = 6 25 + -10f + f2 = 6 Factor a perfect square on the left side: (f + -5)(f + -5) = 6 Calculate the square root of the right side: 2.449489743 Break this problem into two subproblems by setting (f + -5) equal to 2.449489743 and -2.449489743.Subproblem 1
f + -5 = 2.449489743 Simplifying f + -5 = 2.449489743 Reorder the terms: -5 + f = 2.449489743 Solving -5 + f = 2.449489743 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + f = 2.449489743 + 5 Combine like terms: -5 + 5 = 0 0 + f = 2.449489743 + 5 f = 2.449489743 + 5 Combine like terms: 2.449489743 + 5 = 7.449489743 f = 7.449489743 Simplifying f = 7.449489743Subproblem 2
f + -5 = -2.449489743 Simplifying f + -5 = -2.449489743 Reorder the terms: -5 + f = -2.449489743 Solving -5 + f = -2.449489743 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + f = -2.449489743 + 5 Combine like terms: -5 + 5 = 0 0 + f = -2.449489743 + 5 f = -2.449489743 + 5 Combine like terms: -2.449489743 + 5 = 2.550510257 f = 2.550510257 Simplifying f = 2.550510257Solution
The solution to the problem is based on the solutions from the subproblems. f = {7.449489743, 2.550510257}
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