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Simplifying 0.5x + 6 = x2 + 12x Reorder the terms: 6 + 0.5x = x2 + 12x Reorder the terms: 6 + 0.5x = 12x + x2 Solving 6 + 0.5x = 12x + x2 Solving for variable 'x'. Combine like terms: 0.5x + -12x = -11.5x 6 + -11.5x + -1x2 = 12x + x2 + -12x + -1x2 Reorder the terms: 6 + -11.5x + -1x2 = 12x + -12x + x2 + -1x2 Combine like terms: 12x + -12x = 0 6 + -11.5x + -1x2 = 0 + x2 + -1x2 6 + -11.5x + -1x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 6 + -11.5x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -6 + 11.5x + x2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + 11.5x + 6 + x2 = 0 + 6 Reorder the terms: -6 + 6 + 11.5x + x2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + 11.5x + x2 = 0 + 6 11.5x + x2 = 0 + 6 Combine like terms: 0 + 6 = 6 11.5x + x2 = 6 The x term is 11.5x. Take half its coefficient (5.75). Square it (33.0625) and add it to both sides. Add '33.0625' to each side of the equation. 11.5x + 33.0625 + x2 = 6 + 33.0625 Reorder the terms: 33.0625 + 11.5x + x2 = 6 + 33.0625 Combine like terms: 6 + 33.0625 = 39.0625 33.0625 + 11.5x + x2 = 39.0625 Factor a perfect square on the left side: (x + 5.75)(x + 5.75) = 39.0625 Calculate the square root of the right side: 6.25 Break this problem into two subproblems by setting (x + 5.75) equal to 6.25 and -6.25.Subproblem 1
x + 5.75 = 6.25 Simplifying x + 5.75 = 6.25 Reorder the terms: 5.75 + x = 6.25 Solving 5.75 + x = 6.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.75' to each side of the equation. 5.75 + -5.75 + x = 6.25 + -5.75 Combine like terms: 5.75 + -5.75 = 0.00 0.00 + x = 6.25 + -5.75 x = 6.25 + -5.75 Combine like terms: 6.25 + -5.75 = 0.5 x = 0.5 Simplifying x = 0.5Subproblem 2
x + 5.75 = -6.25 Simplifying x + 5.75 = -6.25 Reorder the terms: 5.75 + x = -6.25 Solving 5.75 + x = -6.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.75' to each side of the equation. 5.75 + -5.75 + x = -6.25 + -5.75 Combine like terms: 5.75 + -5.75 = 0.00 0.00 + x = -6.25 + -5.75 x = -6.25 + -5.75 Combine like terms: -6.25 + -5.75 = -12 x = -12 Simplifying x = -12Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.5, -12}
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