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Simplifying 0 = -16x2 + 8x + 1.5 Reorder the terms: 0 = 1.5 + 8x + -16x2 Solving 0 = 1.5 + 8x + -16x2 Solving for variable 'x'. Combine like terms: 0 + -1.5 = -1.5 -1.5 + -8x + 16x2 = 1.5 + 8x + -16x2 + -1.5 + -8x + 16x2 Reorder the terms: -1.5 + -8x + 16x2 = 1.5 + -1.5 + 8x + -8x + -16x2 + 16x2 Combine like terms: 1.5 + -1.5 = 0.0 -1.5 + -8x + 16x2 = 0.0 + 8x + -8x + -16x2 + 16x2 -1.5 + -8x + 16x2 = 8x + -8x + -16x2 + 16x2 Combine like terms: 8x + -8x = 0 -1.5 + -8x + 16x2 = 0 + -16x2 + 16x2 -1.5 + -8x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 -1.5 + -8x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.09375 + -0.5x + x2 = 0 Move the constant term to the right: Add '0.09375' to each side of the equation. -0.09375 + -0.5x + 0.09375 + x2 = 0 + 0.09375 Reorder the terms: -0.09375 + 0.09375 + -0.5x + x2 = 0 + 0.09375 Combine like terms: -0.09375 + 0.09375 = 0.00000 0.00000 + -0.5x + x2 = 0 + 0.09375 -0.5x + x2 = 0 + 0.09375 Combine like terms: 0 + 0.09375 = 0.09375 -0.5x + x2 = 0.09375 The x term is -0.5x. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5x + 0.0625 + x2 = 0.09375 + 0.0625 Reorder the terms: 0.0625 + -0.5x + x2 = 0.09375 + 0.0625 Combine like terms: 0.09375 + 0.0625 = 0.15625 0.0625 + -0.5x + x2 = 0.15625 Factor a perfect square on the left side: (x + -0.25)(x + -0.25) = 0.15625 Calculate the square root of the right side: 0.395284708 Break this problem into two subproblems by setting (x + -0.25) equal to 0.395284708 and -0.395284708.Subproblem 1
x + -0.25 = 0.395284708 Simplifying x + -0.25 = 0.395284708 Reorder the terms: -0.25 + x = 0.395284708 Solving -0.25 + x = 0.395284708 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = 0.395284708 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = 0.395284708 + 0.25 x = 0.395284708 + 0.25 Combine like terms: 0.395284708 + 0.25 = 0.645284708 x = 0.645284708 Simplifying x = 0.645284708Subproblem 2
x + -0.25 = -0.395284708 Simplifying x + -0.25 = -0.395284708 Reorder the terms: -0.25 + x = -0.395284708 Solving -0.25 + x = -0.395284708 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = -0.395284708 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = -0.395284708 + 0.25 x = -0.395284708 + 0.25 Combine like terms: -0.395284708 + 0.25 = -0.145284708 x = -0.145284708 Simplifying x = -0.145284708Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.645284708, -0.145284708}
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