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Simplifying 2x2 + 3x + -219 = 0 Reorder the terms: -219 + 3x + 2x2 = 0 Solving -219 + 3x + 2x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -109.5 + 1.5x + x2 = 0 Move the constant term to the right: Add '109.5' to each side of the equation. -109.5 + 1.5x + 109.5 + x2 = 0 + 109.5 Reorder the terms: -109.5 + 109.5 + 1.5x + x2 = 0 + 109.5 Combine like terms: -109.5 + 109.5 = 0.0 0.0 + 1.5x + x2 = 0 + 109.5 1.5x + x2 = 0 + 109.5 Combine like terms: 0 + 109.5 = 109.5 1.5x + x2 = 109.5 The x term is 1.5x. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5x + 0.5625 + x2 = 109.5 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 109.5 + 0.5625 Combine like terms: 109.5 + 0.5625 = 110.0625 0.5625 + 1.5x + x2 = 110.0625 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 110.0625 Calculate the square root of the right side: 10.491067629 Break this problem into two subproblems by setting (x + 0.75) equal to 10.491067629 and -10.491067629.Subproblem 1
x + 0.75 = 10.491067629 Simplifying x + 0.75 = 10.491067629 Reorder the terms: 0.75 + x = 10.491067629 Solving 0.75 + x = 10.491067629 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 10.491067629 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 10.491067629 + -0.75 x = 10.491067629 + -0.75 Combine like terms: 10.491067629 + -0.75 = 9.741067629 x = 9.741067629 Simplifying x = 9.741067629Subproblem 2
x + 0.75 = -10.491067629 Simplifying x + 0.75 = -10.491067629 Reorder the terms: 0.75 + x = -10.491067629 Solving 0.75 + x = -10.491067629 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -10.491067629 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -10.491067629 + -0.75 x = -10.491067629 + -0.75 Combine like terms: -10.491067629 + -0.75 = -11.241067629 x = -11.241067629 Simplifying x = -11.241067629Solution
The solution to the problem is based on the solutions from the subproblems. x = {9.741067629, -11.241067629}
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