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Simplifying 0.5x2 + 1.5x + -2 = 3000 Reorder the terms: -2 + 1.5x + 0.5x2 = 3000 Solving -2 + 1.5x + 0.5x2 = 3000 Solving for variable 'x'. Reorder the terms: -2 + -3000 + 1.5x + 0.5x2 = 3000 + -3000 Combine like terms: -2 + -3000 = -3002 -3002 + 1.5x + 0.5x2 = 3000 + -3000 Combine like terms: 3000 + -3000 = 0 -3002 + 1.5x + 0.5x2 = 0 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -6004 + 3x + x2 = 0 Move the constant term to the right: Add '6004' to each side of the equation. -6004 + 3x + 6004 + x2 = 0 + 6004 Reorder the terms: -6004 + 6004 + 3x + x2 = 0 + 6004 Combine like terms: -6004 + 6004 = 0 0 + 3x + x2 = 0 + 6004 3x + x2 = 0 + 6004 Combine like terms: 0 + 6004 = 6004 3x + x2 = 6004 The x term is 3x. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3x + 2.25 + x2 = 6004 + 2.25 Reorder the terms: 2.25 + 3x + x2 = 6004 + 2.25 Combine like terms: 6004 + 2.25 = 6006.25 2.25 + 3x + x2 = 6006.25 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 6006.25 Calculate the square root of the right side: 77.5 Break this problem into two subproblems by setting (x + 1.5) equal to 77.5 and -77.5.Subproblem 1
x + 1.5 = 77.5 Simplifying x + 1.5 = 77.5 Reorder the terms: 1.5 + x = 77.5 Solving 1.5 + x = 77.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 77.5 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 77.5 + -1.5 x = 77.5 + -1.5 Combine like terms: 77.5 + -1.5 = 76 x = 76 Simplifying x = 76Subproblem 2
x + 1.5 = -77.5 Simplifying x + 1.5 = -77.5 Reorder the terms: 1.5 + x = -77.5 Solving 1.5 + x = -77.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -77.5 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -77.5 + -1.5 x = -77.5 + -1.5 Combine like terms: -77.5 + -1.5 = -79 x = -79 Simplifying x = -79Solution
The solution to the problem is based on the solutions from the subproblems. x = {76, -79}
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