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Simplifying 9x2 + -32x + 25 = 0 Reorder the terms: 25 + -32x + 9x2 = 0 Solving 25 + -32x + 9x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 2.777777778 + -3.555555556x + x2 = 0 Move the constant term to the right: Add '-2.777777778' to each side of the equation. 2.777777778 + -3.555555556x + -2.777777778 + x2 = 0 + -2.777777778 Reorder the terms: 2.777777778 + -2.777777778 + -3.555555556x + x2 = 0 + -2.777777778 Combine like terms: 2.777777778 + -2.777777778 = 0.000000000 0.000000000 + -3.555555556x + x2 = 0 + -2.777777778 -3.555555556x + x2 = 0 + -2.777777778 Combine like terms: 0 + -2.777777778 = -2.777777778 -3.555555556x + x2 = -2.777777778 The x term is -3.555555556x. Take half its coefficient (-1.777777778). Square it (3.160493828) and add it to both sides. Add '3.160493828' to each side of the equation. -3.555555556x + 3.160493828 + x2 = -2.777777778 + 3.160493828 Reorder the terms: 3.160493828 + -3.555555556x + x2 = -2.777777778 + 3.160493828 Combine like terms: -2.777777778 + 3.160493828 = 0.38271605 3.160493828 + -3.555555556x + x2 = 0.38271605 Factor a perfect square on the left side: (x + -1.777777778)(x + -1.777777778) = 0.38271605 Calculate the square root of the right side: 0.618640485 Break this problem into two subproblems by setting (x + -1.777777778) equal to 0.618640485 and -0.618640485.Subproblem 1
x + -1.777777778 = 0.618640485 Simplifying x + -1.777777778 = 0.618640485 Reorder the terms: -1.777777778 + x = 0.618640485 Solving -1.777777778 + x = 0.618640485 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.777777778' to each side of the equation. -1.777777778 + 1.777777778 + x = 0.618640485 + 1.777777778 Combine like terms: -1.777777778 + 1.777777778 = 0.000000000 0.000000000 + x = 0.618640485 + 1.777777778 x = 0.618640485 + 1.777777778 Combine like terms: 0.618640485 + 1.777777778 = 2.396418263 x = 2.396418263 Simplifying x = 2.396418263Subproblem 2
x + -1.777777778 = -0.618640485 Simplifying x + -1.777777778 = -0.618640485 Reorder the terms: -1.777777778 + x = -0.618640485 Solving -1.777777778 + x = -0.618640485 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.777777778' to each side of the equation. -1.777777778 + 1.777777778 + x = -0.618640485 + 1.777777778 Combine like terms: -1.777777778 + 1.777777778 = 0.000000000 0.000000000 + x = -0.618640485 + 1.777777778 x = -0.618640485 + 1.777777778 Combine like terms: -0.618640485 + 1.777777778 = 1.159137293 x = 1.159137293 Simplifying x = 1.159137293Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.396418263, 1.159137293}
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