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Simplifying x(2x + 9) = 1 Reorder the terms: x(9 + 2x) = 1 (9 * x + 2x * x) = 1 (9x + 2x2) = 1 Solving 9x + 2x2 = 1 Solving for variable 'x'. Reorder the terms: -1 + 9x + 2x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 9x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.5 + 4.5x + x2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 4.5x + 0.5 + x2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 4.5x + x2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 4.5x + x2 = 0 + 0.5 4.5x + x2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 4.5x + x2 = 0.5 The x term is 4.5x. Take half its coefficient (2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. 4.5x + 5.0625 + x2 = 0.5 + 5.0625 Reorder the terms: 5.0625 + 4.5x + x2 = 0.5 + 5.0625 Combine like terms: 0.5 + 5.0625 = 5.5625 5.0625 + 4.5x + x2 = 5.5625 Factor a perfect square on the left side: (x + 2.25)(x + 2.25) = 5.5625 Calculate the square root of the right side: 2.358495283 Break this problem into two subproblems by setting (x + 2.25) equal to 2.358495283 and -2.358495283.Subproblem 1
x + 2.25 = 2.358495283 Simplifying x + 2.25 = 2.358495283 Reorder the terms: 2.25 + x = 2.358495283 Solving 2.25 + x = 2.358495283 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = 2.358495283 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = 2.358495283 + -2.25 x = 2.358495283 + -2.25 Combine like terms: 2.358495283 + -2.25 = 0.108495283 x = 0.108495283 Simplifying x = 0.108495283Subproblem 2
x + 2.25 = -2.358495283 Simplifying x + 2.25 = -2.358495283 Reorder the terms: 2.25 + x = -2.358495283 Solving 2.25 + x = -2.358495283 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = -2.358495283 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = -2.358495283 + -2.25 x = -2.358495283 + -2.25 Combine like terms: -2.358495283 + -2.25 = -4.608495283 x = -4.608495283 Simplifying x = -4.608495283Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.108495283, -4.608495283}
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