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Simplifying -16x2 + -27x + 180 = 0 Reorder the terms: 180 + -27x + -16x2 = 0 Solving 180 + -27x + -16x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -11.25 + 1.6875x + x2 = 0 Move the constant term to the right: Add '11.25' to each side of the equation. -11.25 + 1.6875x + 11.25 + x2 = 0 + 11.25 Reorder the terms: -11.25 + 11.25 + 1.6875x + x2 = 0 + 11.25 Combine like terms: -11.25 + 11.25 = 0.00 0.00 + 1.6875x + x2 = 0 + 11.25 1.6875x + x2 = 0 + 11.25 Combine like terms: 0 + 11.25 = 11.25 1.6875x + x2 = 11.25 The x term is 1.6875x. Take half its coefficient (0.84375). Square it (0.7119140625) and add it to both sides. Add '0.7119140625' to each side of the equation. 1.6875x + 0.7119140625 + x2 = 11.25 + 0.7119140625 Reorder the terms: 0.7119140625 + 1.6875x + x2 = 11.25 + 0.7119140625 Combine like terms: 11.25 + 0.7119140625 = 11.9619140625 0.7119140625 + 1.6875x + x2 = 11.9619140625 Factor a perfect square on the left side: (x + 0.84375)(x + 0.84375) = 11.9619140625 Calculate the square root of the right side: 3.458600015 Break this problem into two subproblems by setting (x + 0.84375) equal to 3.458600015 and -3.458600015.Subproblem 1
x + 0.84375 = 3.458600015 Simplifying x + 0.84375 = 3.458600015 Reorder the terms: 0.84375 + x = 3.458600015 Solving 0.84375 + x = 3.458600015 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.84375' to each side of the equation. 0.84375 + -0.84375 + x = 3.458600015 + -0.84375 Combine like terms: 0.84375 + -0.84375 = 0.00000 0.00000 + x = 3.458600015 + -0.84375 x = 3.458600015 + -0.84375 Combine like terms: 3.458600015 + -0.84375 = 2.614850015 x = 2.614850015 Simplifying x = 2.614850015Subproblem 2
x + 0.84375 = -3.458600015 Simplifying x + 0.84375 = -3.458600015 Reorder the terms: 0.84375 + x = -3.458600015 Solving 0.84375 + x = -3.458600015 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.84375' to each side of the equation. 0.84375 + -0.84375 + x = -3.458600015 + -0.84375 Combine like terms: 0.84375 + -0.84375 = 0.00000 0.00000 + x = -3.458600015 + -0.84375 x = -3.458600015 + -0.84375 Combine like terms: -3.458600015 + -0.84375 = -4.302350015 x = -4.302350015 Simplifying x = -4.302350015Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.614850015, -4.302350015}
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