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Simplifying -16x2 + 9x + 23 = 0 Reorder the terms: 23 + 9x + -16x2 = 0 Solving 23 + 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'. -1.4375 + -0.5625x + x2 = 0 Move the constant term to the right: Add '1.4375' to each side of the equation. -1.4375 + -0.5625x + 1.4375 + x2 = 0 + 1.4375 Reorder the terms: -1.4375 + 1.4375 + -0.5625x + x2 = 0 + 1.4375 Combine like terms: -1.4375 + 1.4375 = 0.0000 0.0000 + -0.5625x + x2 = 0 + 1.4375 -0.5625x + x2 = 0 + 1.4375 Combine like terms: 0 + 1.4375 = 1.4375 -0.5625x + x2 = 1.4375 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 = 1.4375 + 0.0791015625 Reorder the terms: 0.0791015625 + -0.5625x + x2 = 1.4375 + 0.0791015625 Combine like terms: 1.4375 + 0.0791015625 = 1.5166015625 0.0791015625 + -0.5625x + x2 = 1.5166015625 Factor a perfect square on the left side: (x + -0.28125)(x + -0.28125) = 1.5166015625 Calculate the square root of the right side: 1.231503781 Break this problem into two subproblems by setting (x + -0.28125) equal to 1.231503781 and -1.231503781.Subproblem 1
x + -0.28125 = 1.231503781 Simplifying x + -0.28125 = 1.231503781 Reorder the terms: -0.28125 + x = 1.231503781 Solving -0.28125 + x = 1.231503781 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 = 1.231503781 + 0.28125 Combine like terms: -0.28125 + 0.28125 = 0.00000 0.00000 + x = 1.231503781 + 0.28125 x = 1.231503781 + 0.28125 Combine like terms: 1.231503781 + 0.28125 = 1.512753781 x = 1.512753781 Simplifying x = 1.512753781Subproblem 2
x + -0.28125 = -1.231503781 Simplifying x + -0.28125 = -1.231503781 Reorder the terms: -0.28125 + x = -1.231503781 Solving -0.28125 + x = -1.231503781 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 = -1.231503781 + 0.28125 Combine like terms: -0.28125 + 0.28125 = 0.00000 0.00000 + x = -1.231503781 + 0.28125 x = -1.231503781 + 0.28125 Combine like terms: -1.231503781 + 0.28125 = -0.950253781 x = -0.950253781 Simplifying x = -0.950253781Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.512753781, -0.950253781}
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