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Simplifying 22.5 = -10x2 + 30x Reorder the terms: 22.5 = 30x + -10x2 Solving 22.5 = 30x + -10x2 Solving for variable 'x'. Reorder the terms: 22.5 + -30x + 10x2 = 30x + -30x + -10x2 + 10x2 Combine like terms: 30x + -30x = 0 22.5 + -30x + 10x2 = 0 + -10x2 + 10x2 22.5 + -30x + 10x2 = -10x2 + 10x2 Combine like terms: -10x2 + 10x2 = 0 22.5 + -30x + 10x2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. 2.25 + -3x + x2 = 0 Move the constant term to the right: Add '-2.25' to each side of the equation. 2.25 + -3x + -2.25 + x2 = 0 + -2.25 Reorder the terms: 2.25 + -2.25 + -3x + x2 = 0 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + -3x + x2 = 0 + -2.25 -3x + x2 = 0 + -2.25 Combine like terms: 0 + -2.25 = -2.25 -3x + x2 = -2.25 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 = -2.25 + 2.25 Reorder the terms: 2.25 + -3x + x2 = -2.25 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 2.25 + -3x + x2 = 0.00 Factor a perfect square on the left side: (x + -1.5)(x + -1.5) = 0.00 Calculate the square root of the right side: 0.00 Break this problem into two subproblems by setting (x + -1.5) equal to 0.00 and 0.00.Subproblem 1
x + -1.5 = 0.00 Simplifying x + -1.5 = 0.00 Reorder the terms: -1.5 + x = 0.00 Solving -1.5 + x = 0.00 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 = 0.00 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 0.00 + 1.5 x = 0.00 + 1.5 Combine like terms: 0.00 + 1.5 = 1.5 x = 1.5 Simplifying x = 1.5Subproblem 2
x + -1.5 = 0.00 Simplifying x + -1.5 = 0.00 Reorder the terms: -1.5 + x = 0.00 Solving -1.5 + x = 0.00 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 = 0.00 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 0.00 + 1.5 x = 0.00 + 1.5 Combine like terms: 0.00 + 1.5 = 1.5 x = 1.5 Simplifying x = 1.5Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.5, 1.5}
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