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Simplifying 1 + 0.5x = x2 + -6x + 9 Reorder the terms: 1 + 0.5x = 9 + -6x + x2 Solving 1 + 0.5x = 9 + -6x + x2 Solving for variable 'x'. Reorder the terms: 1 + -9 + 0.5x + 6x + -1x2 = 9 + -6x + x2 + -9 + 6x + -1x2 Combine like terms: 1 + -9 = -8 -8 + 0.5x + 6x + -1x2 = 9 + -6x + x2 + -9 + 6x + -1x2 Combine like terms: 0.5x + 6x = 6.5x -8 + 6.5x + -1x2 = 9 + -6x + x2 + -9 + 6x + -1x2 Reorder the terms: -8 + 6.5x + -1x2 = 9 + -9 + -6x + 6x + x2 + -1x2 Combine like terms: 9 + -9 = 0 -8 + 6.5x + -1x2 = 0 + -6x + 6x + x2 + -1x2 -8 + 6.5x + -1x2 = -6x + 6x + x2 + -1x2 Combine like terms: -6x + 6x = 0 -8 + 6.5x + -1x2 = 0 + x2 + -1x2 -8 + 6.5x + -1x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 -8 + 6.5x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 8 + -6.5x + x2 = 0 Move the constant term to the right: Add '-8' to each side of the equation. 8 + -6.5x + -8 + x2 = 0 + -8 Reorder the terms: 8 + -8 + -6.5x + x2 = 0 + -8 Combine like terms: 8 + -8 = 0 0 + -6.5x + x2 = 0 + -8 -6.5x + x2 = 0 + -8 Combine like terms: 0 + -8 = -8 -6.5x + x2 = -8 The x term is -6.5x. Take half its coefficient (-3.25). Square it (10.5625) and add it to both sides. Add '10.5625' to each side of the equation. -6.5x + 10.5625 + x2 = -8 + 10.5625 Reorder the terms: 10.5625 + -6.5x + x2 = -8 + 10.5625 Combine like terms: -8 + 10.5625 = 2.5625 10.5625 + -6.5x + x2 = 2.5625 Factor a perfect square on the left side: (x + -3.25)(x + -3.25) = 2.5625 Calculate the square root of the right side: 1.600781059 Break this problem into two subproblems by setting (x + -3.25) equal to 1.600781059 and -1.600781059.Subproblem 1
x + -3.25 = 1.600781059 Simplifying x + -3.25 = 1.600781059 Reorder the terms: -3.25 + x = 1.600781059 Solving -3.25 + x = 1.600781059 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + x = 1.600781059 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + x = 1.600781059 + 3.25 x = 1.600781059 + 3.25 Combine like terms: 1.600781059 + 3.25 = 4.850781059 x = 4.850781059 Simplifying x = 4.850781059Subproblem 2
x + -3.25 = -1.600781059 Simplifying x + -3.25 = -1.600781059 Reorder the terms: -3.25 + x = -1.600781059 Solving -3.25 + x = -1.600781059 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + x = -1.600781059 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + x = -1.600781059 + 3.25 x = -1.600781059 + 3.25 Combine like terms: -1.600781059 + 3.25 = 1.649218941 x = 1.649218941 Simplifying x = 1.649218941Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.850781059, 1.649218941}
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