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Simplifying 0 = x2 + -9x + 19 Reorder the terms: 0 = 19 + -9x + x2 Solving 0 = 19 + -9x + x2 Solving for variable 'x'. Combine like terms: 0 + -19 = -19 -19 + 9x + -1x2 = 19 + -9x + x2 + -19 + 9x + -1x2 Reorder the terms: -19 + 9x + -1x2 = 19 + -19 + -9x + 9x + x2 + -1x2 Combine like terms: 19 + -19 = 0 -19 + 9x + -1x2 = 0 + -9x + 9x + x2 + -1x2 -19 + 9x + -1x2 = -9x + 9x + x2 + -1x2 Combine like terms: -9x + 9x = 0 -19 + 9x + -1x2 = 0 + x2 + -1x2 -19 + 9x + -1x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 -19 + 9x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 19 + -9x + x2 = 0 Move the constant term to the right: Add '-19' to each side of the equation. 19 + -9x + -19 + x2 = 0 + -19 Reorder the terms: 19 + -19 + -9x + x2 = 0 + -19 Combine like terms: 19 + -19 = 0 0 + -9x + x2 = 0 + -19 -9x + x2 = 0 + -19 Combine like terms: 0 + -19 = -19 -9x + x2 = -19 The x term is -9x. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9x + 20.25 + x2 = -19 + 20.25 Reorder the terms: 20.25 + -9x + x2 = -19 + 20.25 Combine like terms: -19 + 20.25 = 1.25 20.25 + -9x + x2 = 1.25 Factor a perfect square on the left side: (x + -4.5)(x + -4.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (x + -4.5) equal to 1.118033989 and -1.118033989.Subproblem 1
x + -4.5 = 1.118033989 Simplifying x + -4.5 = 1.118033989 Reorder the terms: -4.5 + x = 1.118033989 Solving -4.5 + x = 1.118033989 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + x = 1.118033989 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x = 1.118033989 + 4.5 x = 1.118033989 + 4.5 Combine like terms: 1.118033989 + 4.5 = 5.618033989 x = 5.618033989 Simplifying x = 5.618033989Subproblem 2
x + -4.5 = -1.118033989 Simplifying x + -4.5 = -1.118033989 Reorder the terms: -4.5 + x = -1.118033989 Solving -4.5 + x = -1.118033989 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + x = -1.118033989 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x = -1.118033989 + 4.5 x = -1.118033989 + 4.5 Combine like terms: -1.118033989 + 4.5 = 3.381966011 x = 3.381966011 Simplifying x = 3.381966011Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.618033989, 3.381966011}
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