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Simplifying x2 + 1.5x = 1000 Reorder the terms: 1.5x + x2 = 1000 Solving 1.5x + x2 = 1000 Solving for variable 'x'. Reorder the terms: -1000 + 1.5x + x2 = 1000 + -1000 Combine like terms: 1000 + -1000 = 0 -1000 + 1.5x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '1000' to each side of the equation. -1000 + 1.5x + 1000 + x2 = 0 + 1000 Reorder the terms: -1000 + 1000 + 1.5x + x2 = 0 + 1000 Combine like terms: -1000 + 1000 = 0 0 + 1.5x + x2 = 0 + 1000 1.5x + x2 = 0 + 1000 Combine like terms: 0 + 1000 = 1000 1.5x + x2 = 1000 The x term is 1.5x. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5x + 0.5625 + x2 = 1000 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 1000 + 0.5625 Combine like terms: 1000 + 0.5625 = 1000.5625 0.5625 + 1.5x + x2 = 1000.5625 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 1000.5625 Calculate the square root of the right side: 31.631669257 Break this problem into two subproblems by setting (x + 0.75) equal to 31.631669257 and -31.631669257.Subproblem 1
x + 0.75 = 31.631669257 Simplifying x + 0.75 = 31.631669257 Reorder the terms: 0.75 + x = 31.631669257 Solving 0.75 + x = 31.631669257 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 31.631669257 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 31.631669257 + -0.75 x = 31.631669257 + -0.75 Combine like terms: 31.631669257 + -0.75 = 30.881669257 x = 30.881669257 Simplifying x = 30.881669257Subproblem 2
x + 0.75 = -31.631669257 Simplifying x + 0.75 = -31.631669257 Reorder the terms: 0.75 + x = -31.631669257 Solving 0.75 + x = -31.631669257 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -31.631669257 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -31.631669257 + -0.75 x = -31.631669257 + -0.75 Combine like terms: -31.631669257 + -0.75 = -32.381669257 x = -32.381669257 Simplifying x = -32.381669257Solution
The solution to the problem is based on the solutions from the subproblems. x = {30.881669257, -32.381669257}
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