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Simplifying (4x + -3)(2x + 5) = 5000 Reorder the terms: (-3 + 4x)(2x + 5) = 5000 Reorder the terms: (-3 + 4x)(5 + 2x) = 5000 Multiply (-3 + 4x) * (5 + 2x) (-3(5 + 2x) + 4x * (5 + 2x)) = 5000 ((5 * -3 + 2x * -3) + 4x * (5 + 2x)) = 5000 ((-15 + -6x) + 4x * (5 + 2x)) = 5000 (-15 + -6x + (5 * 4x + 2x * 4x)) = 5000 (-15 + -6x + (20x + 8x2)) = 5000 Combine like terms: -6x + 20x = 14x (-15 + 14x + 8x2) = 5000 Solving -15 + 14x + 8x2 = 5000 Solving for variable 'x'. Reorder the terms: -15 + -5000 + 14x + 8x2 = 5000 + -5000 Combine like terms: -15 + -5000 = -5015 -5015 + 14x + 8x2 = 5000 + -5000 Combine like terms: 5000 + -5000 = 0 -5015 + 14x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -626.875 + 1.75x + x2 = 0 Move the constant term to the right: Add '626.875' to each side of the equation. -626.875 + 1.75x + 626.875 + x2 = 0 + 626.875 Reorder the terms: -626.875 + 626.875 + 1.75x + x2 = 0 + 626.875 Combine like terms: -626.875 + 626.875 = 0.000 0.000 + 1.75x + x2 = 0 + 626.875 1.75x + x2 = 0 + 626.875 Combine like terms: 0 + 626.875 = 626.875 1.75x + x2 = 626.875 The x term is 1.75x. Take half its coefficient (0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. 1.75x + 0.765625 + x2 = 626.875 + 0.765625 Reorder the terms: 0.765625 + 1.75x + x2 = 626.875 + 0.765625 Combine like terms: 626.875 + 0.765625 = 627.640625 0.765625 + 1.75x + x2 = 627.640625 Factor a perfect square on the left side: (x + 0.875)(x + 0.875) = 627.640625 Calculate the square root of the right side: 25.052756834 Break this problem into two subproblems by setting (x + 0.875) equal to 25.052756834 and -25.052756834.Subproblem 1
x + 0.875 = 25.052756834 Simplifying x + 0.875 = 25.052756834 Reorder the terms: 0.875 + x = 25.052756834 Solving 0.875 + x = 25.052756834 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + x = 25.052756834 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + x = 25.052756834 + -0.875 x = 25.052756834 + -0.875 Combine like terms: 25.052756834 + -0.875 = 24.177756834 x = 24.177756834 Simplifying x = 24.177756834Subproblem 2
x + 0.875 = -25.052756834 Simplifying x + 0.875 = -25.052756834 Reorder the terms: 0.875 + x = -25.052756834 Solving 0.875 + x = -25.052756834 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + x = -25.052756834 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + x = -25.052756834 + -0.875 x = -25.052756834 + -0.875 Combine like terms: -25.052756834 + -0.875 = -25.927756834 x = -25.927756834 Simplifying x = -25.927756834Solution
The solution to the problem is based on the solutions from the subproblems. x = {24.177756834, -25.927756834}
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