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Simplifying 0 = x2 + 2.25x + 0.75 Reorder the terms: 0 = 0.75 + 2.25x + x2 Solving 0 = 0.75 + 2.25x + x2 Solving for variable 'x'. Combine like terms: 0 + -0.75 = -0.75 -0.75 + -2.25x + -1x2 = 0.75 + 2.25x + x2 + -0.75 + -2.25x + -1x2 Reorder the terms: -0.75 + -2.25x + -1x2 = 0.75 + -0.75 + 2.25x + -2.25x + x2 + -1x2 Combine like terms: 0.75 + -0.75 = 0.00 -0.75 + -2.25x + -1x2 = 0.00 + 2.25x + -2.25x + x2 + -1x2 -0.75 + -2.25x + -1x2 = 2.25x + -2.25x + x2 + -1x2 Combine like terms: 2.25x + -2.25x = 0.00 -0.75 + -2.25x + -1x2 = 0.00 + x2 + -1x2 -0.75 + -2.25x + -1x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 -0.75 + -2.25x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 0.75 + 2.25x + x2 = 0 Move the constant term to the right: Add '-0.75' to each side of the equation. 0.75 + 2.25x + -0.75 + x2 = 0 + -0.75 Reorder the terms: 0.75 + -0.75 + 2.25x + x2 = 0 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + 2.25x + x2 = 0 + -0.75 2.25x + x2 = 0 + -0.75 Combine like terms: 0 + -0.75 = -0.75 2.25x + x2 = -0.75 The x term is 2.25x. Take half its coefficient (1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. 2.25x + 1.265625 + x2 = -0.75 + 1.265625 Reorder the terms: 1.265625 + 2.25x + x2 = -0.75 + 1.265625 Combine like terms: -0.75 + 1.265625 = 0.515625 1.265625 + 2.25x + x2 = 0.515625 Factor a perfect square on the left side: (x + 1.125)(x + 1.125) = 0.515625 Calculate the square root of the right side: 0.718070331 Break this problem into two subproblems by setting (x + 1.125) equal to 0.718070331 and -0.718070331.Subproblem 1
x + 1.125 = 0.718070331 Simplifying x + 1.125 = 0.718070331 Reorder the terms: 1.125 + x = 0.718070331 Solving 1.125 + x = 0.718070331 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + x = 0.718070331 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + x = 0.718070331 + -1.125 x = 0.718070331 + -1.125 Combine like terms: 0.718070331 + -1.125 = -0.406929669 x = -0.406929669 Simplifying x = -0.406929669Subproblem 2
x + 1.125 = -0.718070331 Simplifying x + 1.125 = -0.718070331 Reorder the terms: 1.125 + x = -0.718070331 Solving 1.125 + x = -0.718070331 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + x = -0.718070331 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + x = -0.718070331 + -1.125 x = -0.718070331 + -1.125 Combine like terms: -0.718070331 + -1.125 = -1.843070331 x = -1.843070331 Simplifying x = -1.843070331Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.406929669, -1.843070331}
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