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Simplifying 11x2 + -6x + -27 = 180 Reorder the terms: -27 + -6x + 11x2 = 180 Solving -27 + -6x + 11x2 = 180 Solving for variable 'x'. Reorder the terms: -27 + -180 + -6x + 11x2 = 180 + -180 Combine like terms: -27 + -180 = -207 -207 + -6x + 11x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -207 + -6x + 11x2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -18.81818182 + -0.5454545455x + x2 = 0 Move the constant term to the right: Add '18.81818182' to each side of the equation. -18.81818182 + -0.5454545455x + 18.81818182 + x2 = 0 + 18.81818182 Reorder the terms: -18.81818182 + 18.81818182 + -0.5454545455x + x2 = 0 + 18.81818182 Combine like terms: -18.81818182 + 18.81818182 = 0.00000000 0.00000000 + -0.5454545455x + x2 = 0 + 18.81818182 -0.5454545455x + x2 = 0 + 18.81818182 Combine like terms: 0 + 18.81818182 = 18.81818182 -0.5454545455x + x2 = 18.81818182 The x term is -0.5454545455x. Take half its coefficient (-0.2727272728). Square it (0.07438016533) and add it to both sides. Add '0.07438016533' to each side of the equation. -0.5454545455x + 0.07438016533 + x2 = 18.81818182 + 0.07438016533 Reorder the terms: 0.07438016533 + -0.5454545455x + x2 = 18.81818182 + 0.07438016533 Combine like terms: 18.81818182 + 0.07438016533 = 18.89256198533 0.07438016533 + -0.5454545455x + x2 = 18.89256198533 Factor a perfect square on the left side: (x + -0.2727272728)(x + -0.2727272728) = 18.89256198533 Calculate the square root of the right side: 4.346557487 Break this problem into two subproblems by setting (x + -0.2727272728) equal to 4.346557487 and -4.346557487.Subproblem 1
x + -0.2727272728 = 4.346557487 Simplifying x + -0.2727272728 = 4.346557487 Reorder the terms: -0.2727272728 + x = 4.346557487 Solving -0.2727272728 + x = 4.346557487 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2727272728' to each side of the equation. -0.2727272728 + 0.2727272728 + x = 4.346557487 + 0.2727272728 Combine like terms: -0.2727272728 + 0.2727272728 = 0.0000000000 0.0000000000 + x = 4.346557487 + 0.2727272728 x = 4.346557487 + 0.2727272728 Combine like terms: 4.346557487 + 0.2727272728 = 4.6192847598 x = 4.6192847598 Simplifying x = 4.6192847598Subproblem 2
x + -0.2727272728 = -4.346557487 Simplifying x + -0.2727272728 = -4.346557487 Reorder the terms: -0.2727272728 + x = -4.346557487 Solving -0.2727272728 + x = -4.346557487 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2727272728' to each side of the equation. -0.2727272728 + 0.2727272728 + x = -4.346557487 + 0.2727272728 Combine like terms: -0.2727272728 + 0.2727272728 = 0.0000000000 0.0000000000 + x = -4.346557487 + 0.2727272728 x = -4.346557487 + 0.2727272728 Combine like terms: -4.346557487 + 0.2727272728 = -4.0738302142 x = -4.0738302142 Simplifying x = -4.0738302142Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.6192847598, -4.0738302142}
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