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Simplifying 0.25(x + 8) = 0.5(x2 + 1) Reorder the terms: 0.25(8 + x) = 0.5(x2 + 1) (8 * 0.25 + x * 0.25) = 0.5(x2 + 1) (2 + 0.25x) = 0.5(x2 + 1) Reorder the terms: 2 + 0.25x = 0.5(1 + x2) 2 + 0.25x = (1 * 0.5 + x2 * 0.5) 2 + 0.25x = (0.5 + 0.5x2) Solving 2 + 0.25x = 0.5 + 0.5x2 Solving for variable 'x'. Reorder the terms: 2 + -0.5 + 0.25x + -0.5x2 = 0.5 + 0.5x2 + -0.5 + -0.5x2 Combine like terms: 2 + -0.5 = 1.5 1.5 + 0.25x + -0.5x2 = 0.5 + 0.5x2 + -0.5 + -0.5x2 Reorder the terms: 1.5 + 0.25x + -0.5x2 = 0.5 + -0.5 + 0.5x2 + -0.5x2 Combine like terms: 0.5 + -0.5 = 0.0 1.5 + 0.25x + -0.5x2 = 0.0 + 0.5x2 + -0.5x2 1.5 + 0.25x + -0.5x2 = 0.5x2 + -0.5x2 Combine like terms: 0.5x2 + -0.5x2 = 0.0 1.5 + 0.25x + -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'. -3 + -0.5x + x2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + -0.5x + 3 + x2 = 0 + 3 Reorder the terms: -3 + 3 + -0.5x + x2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + -0.5x + x2 = 0 + 3 -0.5x + x2 = 0 + 3 Combine like terms: 0 + 3 = 3 -0.5x + x2 = 3 The x term is -0.5x. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5x + 0.0625 + x2 = 3 + 0.0625 Reorder the terms: 0.0625 + -0.5x + x2 = 3 + 0.0625 Combine like terms: 3 + 0.0625 = 3.0625 0.0625 + -0.5x + x2 = 3.0625 Factor a perfect square on the left side: (x + -0.25)(x + -0.25) = 3.0625 Calculate the square root of the right side: 1.75 Break this problem into two subproblems by setting (x + -0.25) equal to 1.75 and -1.75.Subproblem 1
x + -0.25 = 1.75 Simplifying x + -0.25 = 1.75 Reorder the terms: -0.25 + x = 1.75 Solving -0.25 + x = 1.75 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = 1.75 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = 1.75 + 0.25 x = 1.75 + 0.25 Combine like terms: 1.75 + 0.25 = 2 x = 2 Simplifying x = 2Subproblem 2
x + -0.25 = -1.75 Simplifying x + -0.25 = -1.75 Reorder the terms: -0.25 + x = -1.75 Solving -0.25 + x = -1.75 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = -1.75 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = -1.75 + 0.25 x = -1.75 + 0.25 Combine like terms: -1.75 + 0.25 = -1.5 x = -1.5 Simplifying x = -1.5Solution
The solution to the problem is based on the solutions from the subproblems. x = {2, -1.5}
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