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Simplifying f2 + 5f + -19 = 0 Reorder the terms: -19 + 5f + f2 = 0 Solving -19 + 5f + 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 + 5f + 19 + f2 = 0 + 19 Reorder the terms: -19 + 19 + 5f + f2 = 0 + 19 Combine like terms: -19 + 19 = 0 0 + 5f + f2 = 0 + 19 5f + f2 = 0 + 19 Combine like terms: 0 + 19 = 19 5f + f2 = 19 The f term is 5f. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5f + 6.25 + f2 = 19 + 6.25 Reorder the terms: 6.25 + 5f + f2 = 19 + 6.25 Combine like terms: 19 + 6.25 = 25.25 6.25 + 5f + f2 = 25.25 Factor a perfect square on the left side: (f + 2.5)(f + 2.5) = 25.25 Calculate the square root of the right side: 5.024937811 Break this problem into two subproblems by setting (f + 2.5) equal to 5.024937811 and -5.024937811.Subproblem 1
f + 2.5 = 5.024937811 Simplifying f + 2.5 = 5.024937811 Reorder the terms: 2.5 + f = 5.024937811 Solving 2.5 + f = 5.024937811 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + f = 5.024937811 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + f = 5.024937811 + -2.5 f = 5.024937811 + -2.5 Combine like terms: 5.024937811 + -2.5 = 2.524937811 f = 2.524937811 Simplifying f = 2.524937811Subproblem 2
f + 2.5 = -5.024937811 Simplifying f + 2.5 = -5.024937811 Reorder the terms: 2.5 + f = -5.024937811 Solving 2.5 + f = -5.024937811 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + f = -5.024937811 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + f = -5.024937811 + -2.5 f = -5.024937811 + -2.5 Combine like terms: -5.024937811 + -2.5 = -7.524937811 f = -7.524937811 Simplifying f = -7.524937811Solution
The solution to the problem is based on the solutions from the subproblems. f = {2.524937811, -7.524937811}
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