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Simplifying 0 = 16x2 + -67.5x + 48 Reorder the terms: 0 = 48 + -67.5x + 16x2 Solving 0 = 48 + -67.5x + 16x2 Solving for variable 'x'. Combine like terms: 0 + -48 = -48 -48 + 67.5x + -16x2 = 48 + -67.5x + 16x2 + -48 + 67.5x + -16x2 Reorder the terms: -48 + 67.5x + -16x2 = 48 + -48 + -67.5x + 67.5x + 16x2 + -16x2 Combine like terms: 48 + -48 = 0 -48 + 67.5x + -16x2 = 0 + -67.5x + 67.5x + 16x2 + -16x2 -48 + 67.5x + -16x2 = -67.5x + 67.5x + 16x2 + -16x2 Combine like terms: -67.5x + 67.5x = 0.0 -48 + 67.5x + -16x2 = 0.0 + 16x2 + -16x2 -48 + 67.5x + -16x2 = 16x2 + -16x2 Combine like terms: 16x2 + -16x2 = 0 -48 + 67.5x + -16x2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. 3 + -4.21875x + x2 = 0 Move the constant term to the right: Add '-3' to each side of the equation. 3 + -4.21875x + -3 + x2 = 0 + -3 Reorder the terms: 3 + -3 + -4.21875x + x2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + -4.21875x + x2 = 0 + -3 -4.21875x + x2 = 0 + -3 Combine like terms: 0 + -3 = -3 -4.21875x + x2 = -3 The x term is -4.21875x. Take half its coefficient (-2.109375). Square it (4.449462891) and add it to both sides. Add '4.449462891' to each side of the equation. -4.21875x + 4.449462891 + x2 = -3 + 4.449462891 Reorder the terms: 4.449462891 + -4.21875x + x2 = -3 + 4.449462891 Combine like terms: -3 + 4.449462891 = 1.449462891 4.449462891 + -4.21875x + x2 = 1.449462891 Factor a perfect square on the left side: (x + -2.109375)(x + -2.109375) = 1.449462891 Calculate the square root of the right side: 1.203936415 Break this problem into two subproblems by setting (x + -2.109375) equal to 1.203936415 and -1.203936415.Subproblem 1
x + -2.109375 = 1.203936415 Simplifying x + -2.109375 = 1.203936415 Reorder the terms: -2.109375 + x = 1.203936415 Solving -2.109375 + x = 1.203936415 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.109375' to each side of the equation. -2.109375 + 2.109375 + x = 1.203936415 + 2.109375 Combine like terms: -2.109375 + 2.109375 = 0.000000 0.000000 + x = 1.203936415 + 2.109375 x = 1.203936415 + 2.109375 Combine like terms: 1.203936415 + 2.109375 = 3.313311415 x = 3.313311415 Simplifying x = 3.313311415Subproblem 2
x + -2.109375 = -1.203936415 Simplifying x + -2.109375 = -1.203936415 Reorder the terms: -2.109375 + x = -1.203936415 Solving -2.109375 + x = -1.203936415 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.109375' to each side of the equation. -2.109375 + 2.109375 + x = -1.203936415 + 2.109375 Combine like terms: -2.109375 + 2.109375 = 0.000000 0.000000 + x = -1.203936415 + 2.109375 x = -1.203936415 + 2.109375 Combine like terms: -1.203936415 + 2.109375 = 0.905438585 x = 0.905438585 Simplifying x = 0.905438585Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.313311415, 0.905438585}
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