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Simplifying f * f + -2f = 0.85 Multiply f * f f2 + -2f = 0.85 Reorder the terms: -2f + f2 = 0.85 Solving -2f + f2 = 0.85 Solving for variable 'f'. Reorder the terms: -0.85 + -2f + f2 = 0.85 + -0.85 Combine like terms: 0.85 + -0.85 = 0.00 -0.85 + -2f + f2 = 0.00 Begin completing the square. Move the constant term to the right: Add '0.85' to each side of the equation. -0.85 + -2f + 0.85 + f2 = 0.00 + 0.85 Reorder the terms: -0.85 + 0.85 + -2f + f2 = 0.00 + 0.85 Combine like terms: -0.85 + 0.85 = 0.00 0.00 + -2f + f2 = 0.00 + 0.85 -2f + f2 = 0.00 + 0.85 Combine like terms: 0.00 + 0.85 = 0.85 -2f + f2 = 0.85 The f term is -2f. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2f + 1 + f2 = 0.85 + 1 Reorder the terms: 1 + -2f + f2 = 0.85 + 1 Combine like terms: 0.85 + 1 = 1.85 1 + -2f + f2 = 1.85 Factor a perfect square on the left side: (f + -1)(f + -1) = 1.85 Calculate the square root of the right side: 1.360147051 Break this problem into two subproblems by setting (f + -1) equal to 1.360147051 and -1.360147051.Subproblem 1
f + -1 = 1.360147051 Simplifying f + -1 = 1.360147051 Reorder the terms: -1 + f = 1.360147051 Solving -1 + f = 1.360147051 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + f = 1.360147051 + 1 Combine like terms: -1 + 1 = 0 0 + f = 1.360147051 + 1 f = 1.360147051 + 1 Combine like terms: 1.360147051 + 1 = 2.360147051 f = 2.360147051 Simplifying f = 2.360147051Subproblem 2
f + -1 = -1.360147051 Simplifying f + -1 = -1.360147051 Reorder the terms: -1 + f = -1.360147051 Solving -1 + f = -1.360147051 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + f = -1.360147051 + 1 Combine like terms: -1 + 1 = 0 0 + f = -1.360147051 + 1 f = -1.360147051 + 1 Combine like terms: -1.360147051 + 1 = -0.360147051 f = -0.360147051 Simplifying f = -0.360147051Solution
The solution to the problem is based on the solutions from the subproblems. f = {2.360147051, -0.360147051}
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