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Simplifying 6x + -2 = 40x2 + 10x + -75 Reorder the terms: -2 + 6x = 40x2 + 10x + -75 Reorder the terms: -2 + 6x = -75 + 10x + 40x2 Solving -2 + 6x = -75 + 10x + 40x2 Solving for variable 'x'. Reorder the terms: -2 + 75 + 6x + -10x + -40x2 = -75 + 10x + 40x2 + 75 + -10x + -40x2 Combine like terms: -2 + 75 = 73 73 + 6x + -10x + -40x2 = -75 + 10x + 40x2 + 75 + -10x + -40x2 Combine like terms: 6x + -10x = -4x 73 + -4x + -40x2 = -75 + 10x + 40x2 + 75 + -10x + -40x2 Reorder the terms: 73 + -4x + -40x2 = -75 + 75 + 10x + -10x + 40x2 + -40x2 Combine like terms: -75 + 75 = 0 73 + -4x + -40x2 = 0 + 10x + -10x + 40x2 + -40x2 73 + -4x + -40x2 = 10x + -10x + 40x2 + -40x2 Combine like terms: 10x + -10x = 0 73 + -4x + -40x2 = 0 + 40x2 + -40x2 73 + -4x + -40x2 = 40x2 + -40x2 Combine like terms: 40x2 + -40x2 = 0 73 + -4x + -40x2 = 0 Begin completing the square. Divide all terms by -40 the coefficient of the squared term: Divide each side by '-40'. -1.825 + 0.1x + x2 = 0 Move the constant term to the right: Add '1.825' to each side of the equation. -1.825 + 0.1x + 1.825 + x2 = 0 + 1.825 Reorder the terms: -1.825 + 1.825 + 0.1x + x2 = 0 + 1.825 Combine like terms: -1.825 + 1.825 = 0.000 0.000 + 0.1x + x2 = 0 + 1.825 0.1x + x2 = 0 + 1.825 Combine like terms: 0 + 1.825 = 1.825 0.1x + x2 = 1.825 The x term is 0.1x. Take half its coefficient (0.05). Square it (0.0025) and add it to both sides. Add '0.0025' to each side of the equation. 0.1x + 0.0025 + x2 = 1.825 + 0.0025 Reorder the terms: 0.0025 + 0.1x + x2 = 1.825 + 0.0025 Combine like terms: 1.825 + 0.0025 = 1.8275 0.0025 + 0.1x + x2 = 1.8275 Factor a perfect square on the left side: (x + 0.05)(x + 0.05) = 1.8275 Calculate the square root of the right side: 1.351850583 Break this problem into two subproblems by setting (x + 0.05) equal to 1.351850583 and -1.351850583.Subproblem 1
x + 0.05 = 1.351850583 Simplifying x + 0.05 = 1.351850583 Reorder the terms: 0.05 + x = 1.351850583 Solving 0.05 + x = 1.351850583 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.05' to each side of the equation. 0.05 + -0.05 + x = 1.351850583 + -0.05 Combine like terms: 0.05 + -0.05 = 0.00 0.00 + x = 1.351850583 + -0.05 x = 1.351850583 + -0.05 Combine like terms: 1.351850583 + -0.05 = 1.301850583 x = 1.301850583 Simplifying x = 1.301850583Subproblem 2
x + 0.05 = -1.351850583 Simplifying x + 0.05 = -1.351850583 Reorder the terms: 0.05 + x = -1.351850583 Solving 0.05 + x = -1.351850583 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.05' to each side of the equation. 0.05 + -0.05 + x = -1.351850583 + -0.05 Combine like terms: 0.05 + -0.05 = 0.00 0.00 + x = -1.351850583 + -0.05 x = -1.351850583 + -0.05 Combine like terms: -1.351850583 + -0.05 = -1.401850583 x = -1.401850583 Simplifying x = -1.401850583Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.301850583, -1.401850583}
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