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Simplifying 105 = 0.5x * x + 0.5x Multiply x * x 105 = 0.5x2 + 0.5x Reorder the terms: 105 = 0.5x + 0.5x2 Solving 105 = 0.5x + 0.5x2 Solving for variable 'x'. Reorder the terms: 105 + -0.5x + -0.5x2 = 0.5x + -0.5x + 0.5x2 + -0.5x2 Combine like terms: 0.5x + -0.5x = 0.0 105 + -0.5x + -0.5x2 = 0.0 + 0.5x2 + -0.5x2 105 + -0.5x + -0.5x2 = 0.5x2 + -0.5x2 Combine like terms: 0.5x2 + -0.5x2 = 0.0 105 + -0.5x + -0.5x2 = 0.0 Begin completing the square. Divide all terms by -0.5 the coefficient of the squared term: Divide each side by '-0.5'. -210 + x + x2 = 0 Move the constant term to the right: Add '210' to each side of the equation. -210 + x + 210 + x2 = 0 + 210 Reorder the terms: -210 + 210 + x + x2 = 0 + 210 Combine like terms: -210 + 210 = 0 0 + x + x2 = 0 + 210 x + x2 = 0 + 210 Combine like terms: 0 + 210 = 210 x + x2 = 210 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + x2 = 210 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 210 + 0.25 Combine like terms: 210 + 0.25 = 210.25 1.25 + x2 = 210.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 210.25 Calculate the square root of the right side: 14.5 Break this problem into two subproblems by setting (x + 0.5) equal to 14.5 and -14.5.Subproblem 1
x + 0.5 = 14.5 Simplifying x + 0.5 = 14.5 Reorder the terms: 0.5 + x = 14.5 Solving 0.5 + x = 14.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 14.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 14.5 + -0.5 x = 14.5 + -0.5 Combine like terms: 14.5 + -0.5 = 14 x = 14 Simplifying x = 14Subproblem 2
x + 0.5 = -14.5 Simplifying x + 0.5 = -14.5 Reorder the terms: 0.5 + x = -14.5 Solving 0.5 + x = -14.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -14.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -14.5 + -0.5 x = -14.5 + -0.5 Combine like terms: -14.5 + -0.5 = -15 x = -15 Simplifying x = -15Solution
The solution to the problem is based on the solutions from the subproblems. x = {14, -15}
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