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Simplifying -16x2 + 14x + 13.5 = 5 Reorder the terms: 13.5 + 14x + -16x2 = 5 Solving 13.5 + 14x + -16x2 = 5 Solving for variable 'x'. Reorder the terms: 13.5 + -5 + 14x + -16x2 = 5 + -5 Combine like terms: 13.5 + -5 = 8.5 8.5 + 14x + -16x2 = 5 + -5 Combine like terms: 5 + -5 = 0 8.5 + 14x + -16x2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -0.53125 + -0.875x + x2 = 0 Move the constant term to the right: Add '0.53125' to each side of the equation. -0.53125 + -0.875x + 0.53125 + x2 = 0 + 0.53125 Reorder the terms: -0.53125 + 0.53125 + -0.875x + x2 = 0 + 0.53125 Combine like terms: -0.53125 + 0.53125 = 0.00000 0.00000 + -0.875x + x2 = 0 + 0.53125 -0.875x + x2 = 0 + 0.53125 Combine like terms: 0 + 0.53125 = 0.53125 -0.875x + x2 = 0.53125 The x term is -0.875x. Take half its coefficient (-0.4375). Square it (0.19140625) and add it to both sides. Add '0.19140625' to each side of the equation. -0.875x + 0.19140625 + x2 = 0.53125 + 0.19140625 Reorder the terms: 0.19140625 + -0.875x + x2 = 0.53125 + 0.19140625 Combine like terms: 0.53125 + 0.19140625 = 0.72265625 0.19140625 + -0.875x + x2 = 0.72265625 Factor a perfect square on the left side: (x + -0.4375)(x + -0.4375) = 0.72265625 Calculate the square root of the right side: 0.850091907 Break this problem into two subproblems by setting (x + -0.4375) equal to 0.850091907 and -0.850091907.Subproblem 1
x + -0.4375 = 0.850091907 Simplifying x + -0.4375 = 0.850091907 Reorder the terms: -0.4375 + x = 0.850091907 Solving -0.4375 + x = 0.850091907 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4375' to each side of the equation. -0.4375 + 0.4375 + x = 0.850091907 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + x = 0.850091907 + 0.4375 x = 0.850091907 + 0.4375 Combine like terms: 0.850091907 + 0.4375 = 1.287591907 x = 1.287591907 Simplifying x = 1.287591907Subproblem 2
x + -0.4375 = -0.850091907 Simplifying x + -0.4375 = -0.850091907 Reorder the terms: -0.4375 + x = -0.850091907 Solving -0.4375 + x = -0.850091907 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4375' to each side of the equation. -0.4375 + 0.4375 + x = -0.850091907 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + x = -0.850091907 + 0.4375 x = -0.850091907 + 0.4375 Combine like terms: -0.850091907 + 0.4375 = -0.412591907 x = -0.412591907 Simplifying x = -0.412591907Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.287591907, -0.412591907}
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