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Simplifying 1000 = -1X2 + 90X + -150 Reorder the terms: 1000 = -150 + 90X + -1X2 Solving 1000 = -150 + 90X + -1X2 Solving for variable 'X'. Combine like terms: 1000 + 150 = 1150 1150 + -90X + X2 = -150 + 90X + -1X2 + 150 + -90X + X2 Reorder the terms: 1150 + -90X + X2 = -150 + 150 + 90X + -90X + -1X2 + X2 Combine like terms: -150 + 150 = 0 1150 + -90X + X2 = 0 + 90X + -90X + -1X2 + X2 1150 + -90X + X2 = 90X + -90X + -1X2 + X2 Combine like terms: 90X + -90X = 0 1150 + -90X + X2 = 0 + -1X2 + X2 1150 + -90X + X2 = -1X2 + X2 Combine like terms: -1X2 + X2 = 0 1150 + -90X + X2 = 0 Begin completing the square. Move the constant term to the right: Add '-1150' to each side of the equation. 1150 + -90X + -1150 + X2 = 0 + -1150 Reorder the terms: 1150 + -1150 + -90X + X2 = 0 + -1150 Combine like terms: 1150 + -1150 = 0 0 + -90X + X2 = 0 + -1150 -90X + X2 = 0 + -1150 Combine like terms: 0 + -1150 = -1150 -90X + X2 = -1150 The X term is -90X. Take half its coefficient (-45). Square it (2025) and add it to both sides. Add '2025' to each side of the equation. -90X + 2025 + X2 = -1150 + 2025 Reorder the terms: 2025 + -90X + X2 = -1150 + 2025 Combine like terms: -1150 + 2025 = 875 2025 + -90X + X2 = 875 Factor a perfect square on the left side: (X + -45)(X + -45) = 875 Calculate the square root of the right side: 29.580398915 Break this problem into two subproblems by setting (X + -45) equal to 29.580398915 and -29.580398915.Subproblem 1
X + -45 = 29.580398915 Simplifying X + -45 = 29.580398915 Reorder the terms: -45 + X = 29.580398915 Solving -45 + X = 29.580398915 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '45' to each side of the equation. -45 + 45 + X = 29.580398915 + 45 Combine like terms: -45 + 45 = 0 0 + X = 29.580398915 + 45 X = 29.580398915 + 45 Combine like terms: 29.580398915 + 45 = 74.580398915 X = 74.580398915 Simplifying X = 74.580398915Subproblem 2
X + -45 = -29.580398915 Simplifying X + -45 = -29.580398915 Reorder the terms: -45 + X = -29.580398915 Solving -45 + X = -29.580398915 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '45' to each side of the equation. -45 + 45 + X = -29.580398915 + 45 Combine like terms: -45 + 45 = 0 0 + X = -29.580398915 + 45 X = -29.580398915 + 45 Combine like terms: -29.580398915 + 45 = 15.419601085 X = 15.419601085 Simplifying X = 15.419601085Solution
The solution to the problem is based on the solutions from the subproblems. X = {74.580398915, 15.419601085}
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