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Simplifying -4f(4f + -5) = -19 Reorder the terms: -4f(-5 + 4f) = -19 (-5 * -4f + 4f * -4f) = -19 (20f + -16f2) = -19 Solving 20f + -16f2 = -19 Solving for variable 'f'. Reorder the terms: 19 + 20f + -16f2 = -19 + 19 Combine like terms: -19 + 19 = 0 19 + 20f + -16f2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -1.1875 + -1.25f + f2 = 0 Move the constant term to the right: Add '1.1875' to each side of the equation. -1.1875 + -1.25f + 1.1875 + f2 = 0 + 1.1875 Reorder the terms: -1.1875 + 1.1875 + -1.25f + f2 = 0 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + -1.25f + f2 = 0 + 1.1875 -1.25f + f2 = 0 + 1.1875 Combine like terms: 0 + 1.1875 = 1.1875 -1.25f + f2 = 1.1875 The f term is -1.25f. Take half its coefficient (-0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. -1.25f + 0.390625 + f2 = 1.1875 + 0.390625 Reorder the terms: 0.390625 + -1.25f + f2 = 1.1875 + 0.390625 Combine like terms: 1.1875 + 0.390625 = 1.578125 0.390625 + -1.25f + f2 = 1.578125 Factor a perfect square on the left side: (f + -0.625)(f + -0.625) = 1.578125 Calculate the square root of the right side: 1.256234453 Break this problem into two subproblems by setting (f + -0.625) equal to 1.256234453 and -1.256234453.Subproblem 1
f + -0.625 = 1.256234453 Simplifying f + -0.625 = 1.256234453 Reorder the terms: -0.625 + f = 1.256234453 Solving -0.625 + f = 1.256234453 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + f = 1.256234453 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + f = 1.256234453 + 0.625 f = 1.256234453 + 0.625 Combine like terms: 1.256234453 + 0.625 = 1.881234453 f = 1.881234453 Simplifying f = 1.881234453Subproblem 2
f + -0.625 = -1.256234453 Simplifying f + -0.625 = -1.256234453 Reorder the terms: -0.625 + f = -1.256234453 Solving -0.625 + f = -1.256234453 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + f = -1.256234453 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + f = -1.256234453 + 0.625 f = -1.256234453 + 0.625 Combine like terms: -1.256234453 + 0.625 = -0.631234453 f = -0.631234453 Simplifying f = -0.631234453Solution
The solution to the problem is based on the solutions from the subproblems. f = {1.881234453, -0.631234453}
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