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Simplifying (50 + 4X)(120 + -5X) = 6500 Multiply (50 + 4X) * (120 + -5X) (50(120 + -5X) + 4X * (120 + -5X)) = 6500 ((120 * 50 + -5X * 50) + 4X * (120 + -5X)) = 6500 ((6000 + -250X) + 4X * (120 + -5X)) = 6500 (6000 + -250X + (120 * 4X + -5X * 4X)) = 6500 (6000 + -250X + (480X + -20X2)) = 6500 Combine like terms: -250X + 480X = 230X (6000 + 230X + -20X2) = 6500 Solving 6000 + 230X + -20X2 = 6500 Solving for variable 'X'. Reorder the terms: 6000 + -6500 + 230X + -20X2 = 6500 + -6500 Combine like terms: 6000 + -6500 = -500 -500 + 230X + -20X2 = 6500 + -6500 Combine like terms: 6500 + -6500 = 0 -500 + 230X + -20X2 = 0 Factor out the Greatest Common Factor (GCF), '10'. 10(-50 + 23X + -2X2) = 0 Ignore the factor 10.Subproblem 1
Set the factor '(-50 + 23X + -2X2)' equal to zero and attempt to solve: Simplifying -50 + 23X + -2X2 = 0 Solving -50 + 23X + -2X2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 25 + -11.5X + X2 = 0 Move the constant term to the right: Add '-25' to each side of the equation. 25 + -11.5X + -25 + X2 = 0 + -25 Reorder the terms: 25 + -25 + -11.5X + X2 = 0 + -25 Combine like terms: 25 + -25 = 0 0 + -11.5X + X2 = 0 + -25 -11.5X + X2 = 0 + -25 Combine like terms: 0 + -25 = -25 -11.5X + X2 = -25 The X term is -11.5X. Take half its coefficient (-5.75). Square it (33.0625) and add it to both sides. Add '33.0625' to each side of the equation. -11.5X + 33.0625 + X2 = -25 + 33.0625 Reorder the terms: 33.0625 + -11.5X + X2 = -25 + 33.0625 Combine like terms: -25 + 33.0625 = 8.0625 33.0625 + -11.5X + X2 = 8.0625 Factor a perfect square on the left side: (X + -5.75)(X + -5.75) = 8.0625 Calculate the square root of the right side: 2.839454173 Break this problem into two subproblems by setting (X + -5.75) equal to 2.839454173 and -2.839454173.Subproblem 1
X + -5.75 = 2.839454173 Simplifying X + -5.75 = 2.839454173 Reorder the terms: -5.75 + X = 2.839454173 Solving -5.75 + X = 2.839454173 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '5.75' to each side of the equation. -5.75 + 5.75 + X = 2.839454173 + 5.75 Combine like terms: -5.75 + 5.75 = 0.00 0.00 + X = 2.839454173 + 5.75 X = 2.839454173 + 5.75 Combine like terms: 2.839454173 + 5.75 = 8.589454173 X = 8.589454173 Simplifying X = 8.589454173Subproblem 2
X + -5.75 = -2.839454173 Simplifying X + -5.75 = -2.839454173 Reorder the terms: -5.75 + X = -2.839454173 Solving -5.75 + X = -2.839454173 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '5.75' to each side of the equation. -5.75 + 5.75 + X = -2.839454173 + 5.75 Combine like terms: -5.75 + 5.75 = 0.00 0.00 + X = -2.839454173 + 5.75 X = -2.839454173 + 5.75 Combine like terms: -2.839454173 + 5.75 = 2.910545827 X = 2.910545827 Simplifying X = 2.910545827Solution
The solution to the problem is based on the solutions from the subproblems. X = {8.589454173, 2.910545827}Solution
X = {8.589454173, 2.910545827}
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