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Simplifying 120x = x2 + -60x + 90 Reorder the terms: 120x = 90 + -60x + x2 Solving 120x = 90 + -60x + x2 Solving for variable 'x'. Reorder the terms: -90 + 120x + 60x + -1x2 = 90 + -60x + x2 + -90 + 60x + -1x2 Combine like terms: 120x + 60x = 180x -90 + 180x + -1x2 = 90 + -60x + x2 + -90 + 60x + -1x2 Reorder the terms: -90 + 180x + -1x2 = 90 + -90 + -60x + 60x + x2 + -1x2 Combine like terms: 90 + -90 = 0 -90 + 180x + -1x2 = 0 + -60x + 60x + x2 + -1x2 -90 + 180x + -1x2 = -60x + 60x + x2 + -1x2 Combine like terms: -60x + 60x = 0 -90 + 180x + -1x2 = 0 + x2 + -1x2 -90 + 180x + -1x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 -90 + 180x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 90 + -180x + x2 = 0 Move the constant term to the right: Add '-90' to each side of the equation. 90 + -180x + -90 + x2 = 0 + -90 Reorder the terms: 90 + -90 + -180x + x2 = 0 + -90 Combine like terms: 90 + -90 = 0 0 + -180x + x2 = 0 + -90 -180x + x2 = 0 + -90 Combine like terms: 0 + -90 = -90 -180x + x2 = -90 The x term is -180x. Take half its coefficient (-90). Square it (8100) and add it to both sides. Add '8100' to each side of the equation. -180x + 8100 + x2 = -90 + 8100 Reorder the terms: 8100 + -180x + x2 = -90 + 8100 Combine like terms: -90 + 8100 = 8010 8100 + -180x + x2 = 8010 Factor a perfect square on the left side: (x + -90)(x + -90) = 8010 Calculate the square root of the right side: 89.498603341 Break this problem into two subproblems by setting (x + -90) equal to 89.498603341 and -89.498603341.Subproblem 1
x + -90 = 89.498603341 Simplifying x + -90 = 89.498603341 Reorder the terms: -90 + x = 89.498603341 Solving -90 + x = 89.498603341 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '90' to each side of the equation. -90 + 90 + x = 89.498603341 + 90 Combine like terms: -90 + 90 = 0 0 + x = 89.498603341 + 90 x = 89.498603341 + 90 Combine like terms: 89.498603341 + 90 = 179.498603341 x = 179.498603341 Simplifying x = 179.498603341Subproblem 2
x + -90 = -89.498603341 Simplifying x + -90 = -89.498603341 Reorder the terms: -90 + x = -89.498603341 Solving -90 + x = -89.498603341 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '90' to each side of the equation. -90 + 90 + x = -89.498603341 + 90 Combine like terms: -90 + 90 = 0 0 + x = -89.498603341 + 90 x = -89.498603341 + 90 Combine like terms: -89.498603341 + 90 = 0.501396659 x = 0.501396659 Simplifying x = 0.501396659Solution
The solution to the problem is based on the solutions from the subproblems. x = {179.498603341, 0.501396659}
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