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Simplifying 20f(8f + -15) = 3(4f + -5) Reorder the terms: 20f(-15 + 8f) = 3(4f + -5) (-15 * 20f + 8f * 20f) = 3(4f + -5) (-300f + 160f2) = 3(4f + -5) Reorder the terms: -300f + 160f2 = 3(-5 + 4f) -300f + 160f2 = (-5 * 3 + 4f * 3) -300f + 160f2 = (-15 + 12f) Solving -300f + 160f2 = -15 + 12f Solving for variable 'f'. Reorder the terms: 15 + -300f + -12f + 160f2 = -15 + 12f + 15 + -12f Combine like terms: -300f + -12f = -312f 15 + -312f + 160f2 = -15 + 12f + 15 + -12f Reorder the terms: 15 + -312f + 160f2 = -15 + 15 + 12f + -12f Combine like terms: -15 + 15 = 0 15 + -312f + 160f2 = 0 + 12f + -12f 15 + -312f + 160f2 = 12f + -12f Combine like terms: 12f + -12f = 0 15 + -312f + 160f2 = 0 Begin completing the square. Divide all terms by 160 the coefficient of the squared term: Divide each side by '160'. 0.09375 + -1.95f + f2 = 0.0 Move the constant term to the right: Add '-0.09375' to each side of the equation. 0.09375 + -1.95f + -0.09375 + f2 = 0.0 + -0.09375 Reorder the terms: 0.09375 + -0.09375 + -1.95f + f2 = 0.0 + -0.09375 Combine like terms: 0.09375 + -0.09375 = 0.00000 0.00000 + -1.95f + f2 = 0.0 + -0.09375 -1.95f + f2 = 0.0 + -0.09375 Combine like terms: 0.0 + -0.09375 = -0.09375 -1.95f + f2 = -0.09375 The f term is -1.95f. Take half its coefficient (-0.975). Square it (0.950625) and add it to both sides. Add '0.950625' to each side of the equation. -1.95f + 0.950625 + f2 = -0.09375 + 0.950625 Reorder the terms: 0.950625 + -1.95f + f2 = -0.09375 + 0.950625 Combine like terms: -0.09375 + 0.950625 = 0.856875 0.950625 + -1.95f + f2 = 0.856875 Factor a perfect square on the left side: (f + -0.975)(f + -0.975) = 0.856875 Calculate the square root of the right side: 0.925675429 Break this problem into two subproblems by setting (f + -0.975) equal to 0.925675429 and -0.925675429.Subproblem 1
f + -0.975 = 0.925675429 Simplifying f + -0.975 = 0.925675429 Reorder the terms: -0.975 + f = 0.925675429 Solving -0.975 + f = 0.925675429 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '0.975' to each side of the equation. -0.975 + 0.975 + f = 0.925675429 + 0.975 Combine like terms: -0.975 + 0.975 = 0.000 0.000 + f = 0.925675429 + 0.975 f = 0.925675429 + 0.975 Combine like terms: 0.925675429 + 0.975 = 1.900675429 f = 1.900675429 Simplifying f = 1.900675429Subproblem 2
f + -0.975 = -0.925675429 Simplifying f + -0.975 = -0.925675429 Reorder the terms: -0.975 + f = -0.925675429 Solving -0.975 + f = -0.925675429 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '0.975' to each side of the equation. -0.975 + 0.975 + f = -0.925675429 + 0.975 Combine like terms: -0.975 + 0.975 = 0.000 0.000 + f = -0.925675429 + 0.975 f = -0.925675429 + 0.975 Combine like terms: -0.925675429 + 0.975 = 0.049324571 f = 0.049324571 Simplifying f = 0.049324571Solution
The solution to the problem is based on the solutions from the subproblems. f = {1.900675429, 0.049324571}
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