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Simplifying 0 = -16x2 + 35x + 6.5 Reorder the terms: 0 = 6.5 + 35x + -16x2 Solving 0 = 6.5 + 35x + -16x2 Solving for variable 'x'. Combine like terms: 0 + -6.5 = -6.5 -6.5 + -35x + 16x2 = 6.5 + 35x + -16x2 + -6.5 + -35x + 16x2 Reorder the terms: -6.5 + -35x + 16x2 = 6.5 + -6.5 + 35x + -35x + -16x2 + 16x2 Combine like terms: 6.5 + -6.5 = 0.0 -6.5 + -35x + 16x2 = 0.0 + 35x + -35x + -16x2 + 16x2 -6.5 + -35x + 16x2 = 35x + -35x + -16x2 + 16x2 Combine like terms: 35x + -35x = 0 -6.5 + -35x + 16x2 = 0 + -16x2 + 16x2 -6.5 + -35x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 -6.5 + -35x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.40625 + -2.1875x + x2 = 0 Move the constant term to the right: Add '0.40625' to each side of the equation. -0.40625 + -2.1875x + 0.40625 + x2 = 0 + 0.40625 Reorder the terms: -0.40625 + 0.40625 + -2.1875x + x2 = 0 + 0.40625 Combine like terms: -0.40625 + 0.40625 = 0.00000 0.00000 + -2.1875x + x2 = 0 + 0.40625 -2.1875x + x2 = 0 + 0.40625 Combine like terms: 0 + 0.40625 = 0.40625 -2.1875x + x2 = 0.40625 The x term is -2.1875x. Take half its coefficient (-1.09375). Square it (1.196289063) and add it to both sides. Add '1.196289063' to each side of the equation. -2.1875x + 1.196289063 + x2 = 0.40625 + 1.196289063 Reorder the terms: 1.196289063 + -2.1875x + x2 = 0.40625 + 1.196289063 Combine like terms: 0.40625 + 1.196289063 = 1.602539063 1.196289063 + -2.1875x + x2 = 1.602539063 Factor a perfect square on the left side: (x + -1.09375)(x + -1.09375) = 1.602539063 Calculate the square root of the right side: 1.265914319 Break this problem into two subproblems by setting (x + -1.09375) equal to 1.265914319 and -1.265914319.Subproblem 1
x + -1.09375 = 1.265914319 Simplifying x + -1.09375 = 1.265914319 Reorder the terms: -1.09375 + x = 1.265914319 Solving -1.09375 + x = 1.265914319 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.09375' to each side of the equation. -1.09375 + 1.09375 + x = 1.265914319 + 1.09375 Combine like terms: -1.09375 + 1.09375 = 0.00000 0.00000 + x = 1.265914319 + 1.09375 x = 1.265914319 + 1.09375 Combine like terms: 1.265914319 + 1.09375 = 2.359664319 x = 2.359664319 Simplifying x = 2.359664319Subproblem 2
x + -1.09375 = -1.265914319 Simplifying x + -1.09375 = -1.265914319 Reorder the terms: -1.09375 + x = -1.265914319 Solving -1.09375 + x = -1.265914319 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.09375' to each side of the equation. -1.09375 + 1.09375 + x = -1.265914319 + 1.09375 Combine like terms: -1.09375 + 1.09375 = 0.00000 0.00000 + x = -1.265914319 + 1.09375 x = -1.265914319 + 1.09375 Combine like terms: -1.265914319 + 1.09375 = -0.172164319 x = -0.172164319 Simplifying x = -0.172164319Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.359664319, -0.172164319}
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