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Simplifying -0.5x2 + 176x + -1440 = 12000 Reorder the terms: -1440 + 176x + -0.5x2 = 12000 Solving -1440 + 176x + -0.5x2 = 12000 Solving for variable 'x'. Reorder the terms: -1440 + -12000 + 176x + -0.5x2 = 12000 + -12000 Combine like terms: -1440 + -12000 = -13440 -13440 + 176x + -0.5x2 = 12000 + -12000 Combine like terms: 12000 + -12000 = 0 -13440 + 176x + -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'. 26880 + -352x + x2 = 0 Move the constant term to the right: Add '-26880' to each side of the equation. 26880 + -352x + -26880 + x2 = 0 + -26880 Reorder the terms: 26880 + -26880 + -352x + x2 = 0 + -26880 Combine like terms: 26880 + -26880 = 0 0 + -352x + x2 = 0 + -26880 -352x + x2 = 0 + -26880 Combine like terms: 0 + -26880 = -26880 -352x + x2 = -26880 The x term is -352x. Take half its coefficient (-176). Square it (30976) and add it to both sides. Add '30976' to each side of the equation. -352x + 30976 + x2 = -26880 + 30976 Reorder the terms: 30976 + -352x + x2 = -26880 + 30976 Combine like terms: -26880 + 30976 = 4096 30976 + -352x + x2 = 4096 Factor a perfect square on the left side: (x + -176)(x + -176) = 4096 Calculate the square root of the right side: 64 Break this problem into two subproblems by setting (x + -176) equal to 64 and -64.Subproblem 1
x + -176 = 64 Simplifying x + -176 = 64 Reorder the terms: -176 + x = 64 Solving -176 + x = 64 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '176' to each side of the equation. -176 + 176 + x = 64 + 176 Combine like terms: -176 + 176 = 0 0 + x = 64 + 176 x = 64 + 176 Combine like terms: 64 + 176 = 240 x = 240 Simplifying x = 240Subproblem 2
x + -176 = -64 Simplifying x + -176 = -64 Reorder the terms: -176 + x = -64 Solving -176 + x = -64 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '176' to each side of the equation. -176 + 176 + x = -64 + 176 Combine like terms: -176 + 176 = 0 0 + x = -64 + 176 x = -64 + 176 Combine like terms: -64 + 176 = 112 x = 112 Simplifying x = 112Solution
The solution to the problem is based on the solutions from the subproblems. x = {240, 112}
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