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Simplifying 7f2 + 3f + -3 = 0 Reorder the terms: -3 + 3f + 7f2 = 0 Solving -3 + 3f + 7f2 = 0 Solving for variable 'f'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.4285714286 + 0.4285714286f + f2 = 0 Move the constant term to the right: Add '0.4285714286' to each side of the equation. -0.4285714286 + 0.4285714286f + 0.4285714286 + f2 = 0 + 0.4285714286 Reorder the terms: -0.4285714286 + 0.4285714286 + 0.4285714286f + f2 = 0 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + 0.4285714286f + f2 = 0 + 0.4285714286 0.4285714286f + f2 = 0 + 0.4285714286 Combine like terms: 0 + 0.4285714286 = 0.4285714286 0.4285714286f + f2 = 0.4285714286 The f term is 0.4285714286f. Take half its coefficient (0.2142857143). Square it (0.04591836735) and add it to both sides. Add '0.04591836735' to each side of the equation. 0.4285714286f + 0.04591836735 + f2 = 0.4285714286 + 0.04591836735 Reorder the terms: 0.04591836735 + 0.4285714286f + f2 = 0.4285714286 + 0.04591836735 Combine like terms: 0.4285714286 + 0.04591836735 = 0.47448979595 0.04591836735 + 0.4285714286f + f2 = 0.47448979595 Factor a perfect square on the left side: (f + 0.2142857143)(f + 0.2142857143) = 0.47448979595 Calculate the square root of the right side: 0.688832197 Break this problem into two subproblems by setting (f + 0.2142857143) equal to 0.688832197 and -0.688832197.Subproblem 1
f + 0.2142857143 = 0.688832197 Simplifying f + 0.2142857143 = 0.688832197 Reorder the terms: 0.2142857143 + f = 0.688832197 Solving 0.2142857143 + f = 0.688832197 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.2142857143' to each side of the equation. 0.2142857143 + -0.2142857143 + f = 0.688832197 + -0.2142857143 Combine like terms: 0.2142857143 + -0.2142857143 = 0.0000000000 0.0000000000 + f = 0.688832197 + -0.2142857143 f = 0.688832197 + -0.2142857143 Combine like terms: 0.688832197 + -0.2142857143 = 0.4745464827 f = 0.4745464827 Simplifying f = 0.4745464827Subproblem 2
f + 0.2142857143 = -0.688832197 Simplifying f + 0.2142857143 = -0.688832197 Reorder the terms: 0.2142857143 + f = -0.688832197 Solving 0.2142857143 + f = -0.688832197 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.2142857143' to each side of the equation. 0.2142857143 + -0.2142857143 + f = -0.688832197 + -0.2142857143 Combine like terms: 0.2142857143 + -0.2142857143 = 0.0000000000 0.0000000000 + f = -0.688832197 + -0.2142857143 f = -0.688832197 + -0.2142857143 Combine like terms: -0.688832197 + -0.2142857143 = -0.9031179113 f = -0.9031179113 Simplifying f = -0.9031179113Solution
The solution to the problem is based on the solutions from the subproblems. f = {0.4745464827, -0.9031179113}
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