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Simplifying 19f2 + 13f + 1 = 0 Reorder the terms: 1 + 13f + 19f2 = 0 Solving 1 + 13f + 19f2 = 0 Solving for variable 'f'. Begin completing the square. Divide all terms by 19 the coefficient of the squared term: Divide each side by '19'. 0.05263157895 + 0.6842105263f + f2 = 0 Move the constant term to the right: Add '-0.05263157895' to each side of the equation. 0.05263157895 + 0.6842105263f + -0.05263157895 + f2 = 0 + -0.05263157895 Reorder the terms: 0.05263157895 + -0.05263157895 + 0.6842105263f + f2 = 0 + -0.05263157895 Combine like terms: 0.05263157895 + -0.05263157895 = 0.00000000000 0.00000000000 + 0.6842105263f + f2 = 0 + -0.05263157895 0.6842105263f + f2 = 0 + -0.05263157895 Combine like terms: 0 + -0.05263157895 = -0.05263157895 0.6842105263f + f2 = -0.05263157895 The f term is 0.6842105263f. Take half its coefficient (0.3421052632). Square it (0.1170360111) and add it to both sides. Add '0.1170360111' to each side of the equation. 0.6842105263f + 0.1170360111 + f2 = -0.05263157895 + 0.1170360111 Reorder the terms: 0.1170360111 + 0.6842105263f + f2 = -0.05263157895 + 0.1170360111 Combine like terms: -0.05263157895 + 0.1170360111 = 0.06440443215 0.1170360111 + 0.6842105263f + f2 = 0.06440443215 Factor a perfect square on the left side: (f + 0.3421052632)(f + 0.3421052632) = 0.06440443215 Calculate the square root of the right side: 0.253780283 Break this problem into two subproblems by setting (f + 0.3421052632) equal to 0.253780283 and -0.253780283.Subproblem 1
f + 0.3421052632 = 0.253780283 Simplifying f + 0.3421052632 = 0.253780283 Reorder the terms: 0.3421052632 + f = 0.253780283 Solving 0.3421052632 + f = 0.253780283 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.3421052632' to each side of the equation. 0.3421052632 + -0.3421052632 + f = 0.253780283 + -0.3421052632 Combine like terms: 0.3421052632 + -0.3421052632 = 0.0000000000 0.0000000000 + f = 0.253780283 + -0.3421052632 f = 0.253780283 + -0.3421052632 Combine like terms: 0.253780283 + -0.3421052632 = -0.0883249802 f = -0.0883249802 Simplifying f = -0.0883249802Subproblem 2
f + 0.3421052632 = -0.253780283 Simplifying f + 0.3421052632 = -0.253780283 Reorder the terms: 0.3421052632 + f = -0.253780283 Solving 0.3421052632 + f = -0.253780283 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.3421052632' to each side of the equation. 0.3421052632 + -0.3421052632 + f = -0.253780283 + -0.3421052632 Combine like terms: 0.3421052632 + -0.3421052632 = 0.0000000000 0.0000000000 + f = -0.253780283 + -0.3421052632 f = -0.253780283 + -0.3421052632 Combine like terms: -0.253780283 + -0.3421052632 = -0.5958855462 f = -0.5958855462 Simplifying f = -0.5958855462Solution
The solution to the problem is based on the solutions from the subproblems. f = {-0.0883249802, -0.5958855462}
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