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Simplifying 0 = -16x2 + 40x + 1.5 Reorder the terms: 0 = 1.5 + 40x + -16x2 Solving 0 = 1.5 + 40x + -16x2 Solving for variable 'x'. Combine like terms: 0 + -1.5 = -1.5 -1.5 + -40x + 16x2 = 1.5 + 40x + -16x2 + -1.5 + -40x + 16x2 Reorder the terms: -1.5 + -40x + 16x2 = 1.5 + -1.5 + 40x + -40x + -16x2 + 16x2 Combine like terms: 1.5 + -1.5 = 0.0 -1.5 + -40x + 16x2 = 0.0 + 40x + -40x + -16x2 + 16x2 -1.5 + -40x + 16x2 = 40x + -40x + -16x2 + 16x2 Combine like terms: 40x + -40x = 0 -1.5 + -40x + 16x2 = 0 + -16x2 + 16x2 -1.5 + -40x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 -1.5 + -40x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.09375 + -2.5x + x2 = 0 Move the constant term to the right: Add '0.09375' to each side of the equation. -0.09375 + -2.5x + 0.09375 + x2 = 0 + 0.09375 Reorder the terms: -0.09375 + 0.09375 + -2.5x + x2 = 0 + 0.09375 Combine like terms: -0.09375 + 0.09375 = 0.00000 0.00000 + -2.5x + x2 = 0 + 0.09375 -2.5x + x2 = 0 + 0.09375 Combine like terms: 0 + 0.09375 = 0.09375 -2.5x + x2 = 0.09375 The x term is -2.5x. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5x + 1.5625 + x2 = 0.09375 + 1.5625 Reorder the terms: 1.5625 + -2.5x + x2 = 0.09375 + 1.5625 Combine like terms: 0.09375 + 1.5625 = 1.65625 1.5625 + -2.5x + x2 = 1.65625 Factor a perfect square on the left side: (x + -1.25)(x + -1.25) = 1.65625 Calculate the square root of the right side: 1.286953768 Break this problem into two subproblems by setting (x + -1.25) equal to 1.286953768 and -1.286953768.Subproblem 1
x + -1.25 = 1.286953768 Simplifying x + -1.25 = 1.286953768 Reorder the terms: -1.25 + x = 1.286953768 Solving -1.25 + x = 1.286953768 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = 1.286953768 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = 1.286953768 + 1.25 x = 1.286953768 + 1.25 Combine like terms: 1.286953768 + 1.25 = 2.536953768 x = 2.536953768 Simplifying x = 2.536953768Subproblem 2
x + -1.25 = -1.286953768 Simplifying x + -1.25 = -1.286953768 Reorder the terms: -1.25 + x = -1.286953768 Solving -1.25 + x = -1.286953768 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = -1.286953768 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = -1.286953768 + 1.25 x = -1.286953768 + 1.25 Combine like terms: -1.286953768 + 1.25 = -0.036953768 x = -0.036953768 Simplifying x = -0.036953768Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.536953768, -0.036953768}
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