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Simplifying 64x2 + -128x + 64 = 28 Reorder the terms: 64 + -128x + 64x2 = 28 Solving 64 + -128x + 64x2 = 28 Solving for variable 'x'. Reorder the terms: 64 + -28 + -128x + 64x2 = 28 + -28 Combine like terms: 64 + -28 = 36 36 + -128x + 64x2 = 28 + -28 Combine like terms: 28 + -28 = 0 36 + -128x + 64x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(9 + -32x + 16x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(9 + -32x + 16x2)' equal to zero and attempt to solve: Simplifying 9 + -32x + 16x2 = 0 Solving 9 + -32x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.5625 + -2x + x2 = 0 Move the constant term to the right: Add '-0.5625' to each side of the equation. 0.5625 + -2x + -0.5625 + x2 = 0 + -0.5625 Reorder the terms: 0.5625 + -0.5625 + -2x + x2 = 0 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + -2x + x2 = 0 + -0.5625 -2x + x2 = 0 + -0.5625 Combine like terms: 0 + -0.5625 = -0.5625 -2x + x2 = -0.5625 The x term is -2x. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2x + 1 + x2 = -0.5625 + 1 Reorder the terms: 1 + -2x + x2 = -0.5625 + 1 Combine like terms: -0.5625 + 1 = 0.4375 1 + -2x + x2 = 0.4375 Factor a perfect square on the left side: (x + -1)(x + -1) = 0.4375 Calculate the square root of the right side: 0.661437828 Break this problem into two subproblems by setting (x + -1) equal to 0.661437828 and -0.661437828.Subproblem 1
x + -1 = 0.661437828 Simplifying x + -1 = 0.661437828 Reorder the terms: -1 + x = 0.661437828 Solving -1 + x = 0.661437828 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = 0.661437828 + 1 Combine like terms: -1 + 1 = 0 0 + x = 0.661437828 + 1 x = 0.661437828 + 1 Combine like terms: 0.661437828 + 1 = 1.661437828 x = 1.661437828 Simplifying x = 1.661437828Subproblem 2
x + -1 = -0.661437828 Simplifying x + -1 = -0.661437828 Reorder the terms: -1 + x = -0.661437828 Solving -1 + x = -0.661437828 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = -0.661437828 + 1 Combine like terms: -1 + 1 = 0 0 + x = -0.661437828 + 1 x = -0.661437828 + 1 Combine like terms: -0.661437828 + 1 = 0.338562172 x = 0.338562172 Simplifying x = 0.338562172Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.661437828, 0.338562172}Solution
x = {1.661437828, 0.338562172}
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