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Simplifying 8x = 0.5x2 + 31.5 Reorder the terms: 8x = 31.5 + 0.5x2 Solving 8x = 31.5 + 0.5x2 Solving for variable 'x'. Reorder the terms: -31.5 + 8x + -0.5x2 = 31.5 + 0.5x2 + -31.5 + -0.5x2 Reorder the terms: -31.5 + 8x + -0.5x2 = 31.5 + -31.5 + 0.5x2 + -0.5x2 Combine like terms: 31.5 + -31.5 = 0.0 -31.5 + 8x + -0.5x2 = 0.0 + 0.5x2 + -0.5x2 -31.5 + 8x + -0.5x2 = 0.5x2 + -0.5x2 Combine like terms: 0.5x2 + -0.5x2 = 0.0 -31.5 + 8x + -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'. 63 + -16x + x2 = 0 Move the constant term to the right: Add '-63' to each side of the equation. 63 + -16x + -63 + x2 = 0 + -63 Reorder the terms: 63 + -63 + -16x + x2 = 0 + -63 Combine like terms: 63 + -63 = 0 0 + -16x + x2 = 0 + -63 -16x + x2 = 0 + -63 Combine like terms: 0 + -63 = -63 -16x + x2 = -63 The x term is -16x. Take half its coefficient (-8). Square it (64) and add it to both sides. Add '64' to each side of the equation. -16x + 64 + x2 = -63 + 64 Reorder the terms: 64 + -16x + x2 = -63 + 64 Combine like terms: -63 + 64 = 1 64 + -16x + x2 = 1 Factor a perfect square on the left side: (x + -8)(x + -8) = 1 Calculate the square root of the right side: 1 Break this problem into two subproblems by setting (x + -8) equal to 1 and -1.Subproblem 1
x + -8 = 1 Simplifying x + -8 = 1 Reorder the terms: -8 + x = 1 Solving -8 + x = 1 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + x = 1 + 8 Combine like terms: -8 + 8 = 0 0 + x = 1 + 8 x = 1 + 8 Combine like terms: 1 + 8 = 9 x = 9 Simplifying x = 9Subproblem 2
x + -8 = -1 Simplifying x + -8 = -1 Reorder the terms: -8 + x = -1 Solving -8 + x = -1 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + x = -1 + 8 Combine like terms: -8 + 8 = 0 0 + x = -1 + 8 x = -1 + 8 Combine like terms: -1 + 8 = 7 x = 7 Simplifying x = 7Solution
The solution to the problem is based on the solutions from the subproblems. x = {9, 7}
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