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Simplifying 4x2 + -32 + 2x2 + x = 180 Reorder the terms: -32 + x + 4x2 + 2x2 = 180 Combine like terms: 4x2 + 2x2 = 6x2 -32 + x + 6x2 = 180 Solving -32 + x + 6x2 = 180 Solving for variable 'x'. Reorder the terms: -32 + -180 + x + 6x2 = 180 + -180 Combine like terms: -32 + -180 = -212 -212 + x + 6x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -212 + x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -35.33333333 + 0.1666666667x + x2 = 0 Move the constant term to the right: Add '35.33333333' to each side of the equation. -35.33333333 + 0.1666666667x + 35.33333333 + x2 = 0 + 35.33333333 Reorder the terms: -35.33333333 + 35.33333333 + 0.1666666667x + x2 = 0 + 35.33333333 Combine like terms: -35.33333333 + 35.33333333 = 0.00000000 0.00000000 + 0.1666666667x + x2 = 0 + 35.33333333 0.1666666667x + x2 = 0 + 35.33333333 Combine like terms: 0 + 35.33333333 = 35.33333333 0.1666666667x + x2 = 35.33333333 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.1666666667x + 0.25 + x2 = 35.33333333 + 0.25 Reorder the terms: 0.25 + 0.1666666667x + x2 = 35.33333333 + 0.25 Combine like terms: 35.33333333 + 0.25 = 35.58333333 0.25 + 0.1666666667x + x2 = 35.58333333 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 35.58333333 Calculate the square root of the right side: 5.965176722 Break this problem into two subproblems by setting (x + 0.5) equal to 5.965176722 and -5.965176722.Subproblem 1
x + 0.5 = 5.965176722 Simplifying x + 0.5 = 5.965176722 Reorder the terms: 0.5 + x = 5.965176722 Solving 0.5 + x = 5.965176722 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 5.965176722 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 5.965176722 + -0.5 x = 5.965176722 + -0.5 Combine like terms: 5.965176722 + -0.5 = 5.465176722 x = 5.465176722 Simplifying x = 5.465176722Subproblem 2
x + 0.5 = -5.965176722 Simplifying x + 0.5 = -5.965176722 Reorder the terms: 0.5 + x = -5.965176722 Solving 0.5 + x = -5.965176722 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -5.965176722 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -5.965176722 + -0.5 x = -5.965176722 + -0.5 Combine like terms: -5.965176722 + -0.5 = -6.465176722 x = -6.465176722 Simplifying x = -6.465176722Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.465176722, -6.465176722}
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