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Simplifying 7f2 + -2f + -8 = 0 Reorder the terms: -8 + -2f + 7f2 = 0 Solving -8 + -2f + 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'. -1.142857143 + -0.2857142857f + f2 = 0 Move the constant term to the right: Add '1.142857143' to each side of the equation. -1.142857143 + -0.2857142857f + 1.142857143 + f2 = 0 + 1.142857143 Reorder the terms: -1.142857143 + 1.142857143 + -0.2857142857f + f2 = 0 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + -0.2857142857f + f2 = 0 + 1.142857143 -0.2857142857f + f2 = 0 + 1.142857143 Combine like terms: 0 + 1.142857143 = 1.142857143 -0.2857142857f + f2 = 1.142857143 The f term is -0.2857142857f. Take half its coefficient (-0.1428571429). Square it (0.02040816328) and add it to both sides. Add '0.02040816328' to each side of the equation. -0.2857142857f + 0.02040816328 + f2 = 1.142857143 + 0.02040816328 Reorder the terms: 0.02040816328 + -0.2857142857f + f2 = 1.142857143 + 0.02040816328 Combine like terms: 1.142857143 + 0.02040816328 = 1.16326530628 0.02040816328 + -0.2857142857f + f2 = 1.16326530628 Factor a perfect square on the left side: (f + -0.1428571429)(f + -0.1428571429) = 1.16326530628 Calculate the square root of the right side: 1.078547777 Break this problem into two subproblems by setting (f + -0.1428571429) equal to 1.078547777 and -1.078547777.Subproblem 1
f + -0.1428571429 = 1.078547777 Simplifying f + -0.1428571429 = 1.078547777 Reorder the terms: -0.1428571429 + f = 1.078547777 Solving -0.1428571429 + f = 1.078547777 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.1428571429 + f = 1.078547777 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + f = 1.078547777 + 0.1428571429 f = 1.078547777 + 0.1428571429 Combine like terms: 1.078547777 + 0.1428571429 = 1.2214049199 f = 1.2214049199 Simplifying f = 1.2214049199Subproblem 2
f + -0.1428571429 = -1.078547777 Simplifying f + -0.1428571429 = -1.078547777 Reorder the terms: -0.1428571429 + f = -1.078547777 Solving -0.1428571429 + f = -1.078547777 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.1428571429 + f = -1.078547777 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + f = -1.078547777 + 0.1428571429 f = -1.078547777 + 0.1428571429 Combine like terms: -1.078547777 + 0.1428571429 = -0.9356906341 f = -0.9356906341 Simplifying f = -0.9356906341Solution
The solution to the problem is based on the solutions from the subproblems. f = {1.2214049199, -0.9356906341}
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