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Simplifying (-0.001x2) + 10x = 7000 + 2x Reorder the terms: 10x + (-0.001x2) = 7000 + 2x Solving 10x + (-0.001x2) = 7000 + 2x Solving for variable 'x'. Reorder the terms: -7000 + 10x + -2x + (-0.001x2) = 7000 + 2x + -7000 + -2x Combine like terms: 10x + -2x = 8x -7000 + 8x + (-0.001x2) = 7000 + 2x + -7000 + -2x Reorder the terms: -7000 + 8x + (-0.001x2) = 7000 + -7000 + 2x + -2x Combine like terms: 7000 + -7000 = 0 -7000 + 8x + (-0.001x2) = 0 + 2x + -2x -7000 + 8x + (-0.001x2) = 2x + -2x Combine like terms: 2x + -2x = 0 -7000 + 8x + (-0.001x2) = 0 Begin completing the square. Divide all terms by -0.001 the coefficient of the squared term: Divide each side by '-0.001'. 7000000 + -8000x + x2 = 0 Move the constant term to the right: Add '-7000000' to each side of the equation. 7000000 + -8000x + -7000000 + x2 = 0 + -7000000 Reorder the terms: 7000000 + -7000000 + -8000x + x2 = 0 + -7000000 Combine like terms: 7000000 + -7000000 = 0 0 + -8000x + x2 = 0 + -7000000 -8000x + x2 = 0 + -7000000 Combine like terms: 0 + -7000000 = -7000000 -8000x + x2 = -7000000 The x term is -8000x. Take half its coefficient (-4000). Square it (16000000) and add it to both sides. Add '16000000' to each side of the equation. -8000x + 16000000 + x2 = -7000000 + 16000000 Reorder the terms: 16000000 + -8000x + x2 = -7000000 + 16000000 Combine like terms: -7000000 + 16000000 = 9000000 16000000 + -8000x + x2 = 9000000 Factor a perfect square on the left side: ((x) + -4000)((x) + -4000) = 9000000 Calculate the square root of the right side: 3000 Break this problem into two subproblems by setting ((x) + -4000) equal to 3000 and -3000.Subproblem 1
(x) + -4000 = 3000 Simplifying (x) + -4000 = 3000 x + -4000 = 3000 Reorder the terms: -4000 + x = 3000 Solving -4000 + x = 3000 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4000' to each side of the equation. -4000 + 4000 + x = 3000 + 4000 Combine like terms: -4000 + 4000 = 0 0 + x = 3000 + 4000 x = 3000 + 4000 Combine like terms: 3000 + 4000 = 7000 x = 7000 Simplifying x = 7000Subproblem 2
(x) + -4000 = -3000 Simplifying (x) + -4000 = -3000 x + -4000 = -3000 Reorder the terms: -4000 + x = -3000 Solving -4000 + x = -3000 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4000' to each side of the equation. -4000 + 4000 + x = -3000 + 4000 Combine like terms: -4000 + 4000 = 0 0 + x = -3000 + 4000 x = -3000 + 4000 Combine like terms: -3000 + 4000 = 1000 x = 1000 Simplifying x = 1000Solution
The solution to the problem is based on the solutions from the subproblems. x = {7000, 1000}
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