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Simplifying x = -16x2 + 11x + 7.5 Reorder the terms: x = 7.5 + 11x + -16x2 Solving x = 7.5 + 11x + -16x2 Solving for variable 'x'. Reorder the terms: -7.5 + x + -11x + 16x2 = 7.5 + 11x + -16x2 + -7.5 + -11x + 16x2 Combine like terms: x + -11x = -10x -7.5 + -10x + 16x2 = 7.5 + 11x + -16x2 + -7.5 + -11x + 16x2 Reorder the terms: -7.5 + -10x + 16x2 = 7.5 + -7.5 + 11x + -11x + -16x2 + 16x2 Combine like terms: 7.5 + -7.5 = 0.0 -7.5 + -10x + 16x2 = 0.0 + 11x + -11x + -16x2 + 16x2 -7.5 + -10x + 16x2 = 11x + -11x + -16x2 + 16x2 Combine like terms: 11x + -11x = 0 -7.5 + -10x + 16x2 = 0 + -16x2 + 16x2 -7.5 + -10x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 -7.5 + -10x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.46875 + -0.625x + x2 = 0 Move the constant term to the right: Add '0.46875' to each side of the equation. -0.46875 + -0.625x + 0.46875 + x2 = 0 + 0.46875 Reorder the terms: -0.46875 + 0.46875 + -0.625x + x2 = 0 + 0.46875 Combine like terms: -0.46875 + 0.46875 = 0.00000 0.00000 + -0.625x + x2 = 0 + 0.46875 -0.625x + x2 = 0 + 0.46875 Combine like terms: 0 + 0.46875 = 0.46875 -0.625x + x2 = 0.46875 The x term is -0.625x. Take half its coefficient (-0.3125). Square it (0.09765625) and add it to both sides. Add '0.09765625' to each side of the equation. -0.625x + 0.09765625 + x2 = 0.46875 + 0.09765625 Reorder the terms: 0.09765625 + -0.625x + x2 = 0.46875 + 0.09765625 Combine like terms: 0.46875 + 0.09765625 = 0.56640625 0.09765625 + -0.625x + x2 = 0.56640625 Factor a perfect square on the left side: (x + -0.3125)(x + -0.3125) = 0.56640625 Calculate the square root of the right side: 0.752599661 Break this problem into two subproblems by setting (x + -0.3125) equal to 0.752599661 and -0.752599661.Subproblem 1
x + -0.3125 = 0.752599661 Simplifying x + -0.3125 = 0.752599661 Reorder the terms: -0.3125 + x = 0.752599661 Solving -0.3125 + x = 0.752599661 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3125' to each side of the equation. -0.3125 + 0.3125 + x = 0.752599661 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + x = 0.752599661 + 0.3125 x = 0.752599661 + 0.3125 Combine like terms: 0.752599661 + 0.3125 = 1.065099661 x = 1.065099661 Simplifying x = 1.065099661Subproblem 2
x + -0.3125 = -0.752599661 Simplifying x + -0.3125 = -0.752599661 Reorder the terms: -0.3125 + x = -0.752599661 Solving -0.3125 + x = -0.752599661 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3125' to each side of the equation. -0.3125 + 0.3125 + x = -0.752599661 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + x = -0.752599661 + 0.3125 x = -0.752599661 + 0.3125 Combine like terms: -0.752599661 + 0.3125 = -0.440099661 x = -0.440099661 Simplifying x = -0.440099661Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.065099661, -0.440099661}
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