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Simplifying 16x2 + 9x + -65 = 0 Reorder the terms: -65 + 9x + 16x2 = 0 Solving -65 + 9x + 16x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -4.0625 + 0.5625x + x2 = 0 Move the constant term to the right: Add '4.0625' to each side of the equation. -4.0625 + 0.5625x + 4.0625 + x2 = 0 + 4.0625 Reorder the terms: -4.0625 + 4.0625 + 0.5625x + x2 = 0 + 4.0625 Combine like terms: -4.0625 + 4.0625 = 0.0000 0.0000 + 0.5625x + x2 = 0 + 4.0625 0.5625x + x2 = 0 + 4.0625 Combine like terms: 0 + 4.0625 = 4.0625 0.5625x + x2 = 4.0625 The x term is 0.5625x. Take half its coefficient (0.28125). Square it (0.0791015625) and add it to both sides. Add '0.0791015625' to each side of the equation. 0.5625x + 0.0791015625 + x2 = 4.0625 + 0.0791015625 Reorder the terms: 0.0791015625 + 0.5625x + x2 = 4.0625 + 0.0791015625 Combine like terms: 4.0625 + 0.0791015625 = 4.1416015625 0.0791015625 + 0.5625x + x2 = 4.1416015625 Factor a perfect square on the left side: (x + 0.28125)(x + 0.28125) = 4.1416015625 Calculate the square root of the right side: 2.035092519 Break this problem into two subproblems by setting (x + 0.28125) equal to 2.035092519 and -2.035092519.Subproblem 1
x + 0.28125 = 2.035092519 Simplifying x + 0.28125 = 2.035092519 Reorder the terms: 0.28125 + x = 2.035092519 Solving 0.28125 + x = 2.035092519 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.28125' to each side of the equation. 0.28125 + -0.28125 + x = 2.035092519 + -0.28125 Combine like terms: 0.28125 + -0.28125 = 0.00000 0.00000 + x = 2.035092519 + -0.28125 x = 2.035092519 + -0.28125 Combine like terms: 2.035092519 + -0.28125 = 1.753842519 x = 1.753842519 Simplifying x = 1.753842519Subproblem 2
x + 0.28125 = -2.035092519 Simplifying x + 0.28125 = -2.035092519 Reorder the terms: 0.28125 + x = -2.035092519 Solving 0.28125 + x = -2.035092519 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.28125' to each side of the equation. 0.28125 + -0.28125 + x = -2.035092519 + -0.28125 Combine like terms: 0.28125 + -0.28125 = 0.00000 0.00000 + x = -2.035092519 + -0.28125 x = -2.035092519 + -0.28125 Combine like terms: -2.035092519 + -0.28125 = -2.316342519 x = -2.316342519 Simplifying x = -2.316342519Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.753842519, -2.316342519}
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