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Simplifying -16x2 + 23x + 9 = 0 Reorder the terms: 9 + 23x + -16x2 = 0 Solving 9 + 23x + -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'. -0.5625 + -1.4375x + x2 = 0 Move the constant term to the right: Add '0.5625' to each side of the equation. -0.5625 + -1.4375x + 0.5625 + x2 = 0 + 0.5625 Reorder the terms: -0.5625 + 0.5625 + -1.4375x + x2 = 0 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + -1.4375x + x2 = 0 + 0.5625 -1.4375x + x2 = 0 + 0.5625 Combine like terms: 0 + 0.5625 = 0.5625 -1.4375x + x2 = 0.5625 The x term is -1.4375x. Take half its coefficient (-0.71875). Square it (0.5166015625) and add it to both sides. Add '0.5166015625' to each side of the equation. -1.4375x + 0.5166015625 + x2 = 0.5625 + 0.5166015625 Reorder the terms: 0.5166015625 + -1.4375x + x2 = 0.5625 + 0.5166015625 Combine like terms: 0.5625 + 0.5166015625 = 1.0791015625 0.5166015625 + -1.4375x + x2 = 1.0791015625 Factor a perfect square on the left side: (x + -0.71875)(x + -0.71875) = 1.0791015625 Calculate the square root of the right side: 1.038798134 Break this problem into two subproblems by setting (x + -0.71875) equal to 1.038798134 and -1.038798134.Subproblem 1
x + -0.71875 = 1.038798134 Simplifying x + -0.71875 = 1.038798134 Reorder the terms: -0.71875 + x = 1.038798134 Solving -0.71875 + x = 1.038798134 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.71875' to each side of the equation. -0.71875 + 0.71875 + x = 1.038798134 + 0.71875 Combine like terms: -0.71875 + 0.71875 = 0.00000 0.00000 + x = 1.038798134 + 0.71875 x = 1.038798134 + 0.71875 Combine like terms: 1.038798134 + 0.71875 = 1.757548134 x = 1.757548134 Simplifying x = 1.757548134Subproblem 2
x + -0.71875 = -1.038798134 Simplifying x + -0.71875 = -1.038798134 Reorder the terms: -0.71875 + x = -1.038798134 Solving -0.71875 + x = -1.038798134 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.71875' to each side of the equation. -0.71875 + 0.71875 + x = -1.038798134 + 0.71875 Combine like terms: -0.71875 + 0.71875 = 0.00000 0.00000 + x = -1.038798134 + 0.71875 x = -1.038798134 + 0.71875 Combine like terms: -1.038798134 + 0.71875 = -0.320048134 x = -0.320048134 Simplifying x = -0.320048134Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.757548134, -0.320048134}
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