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Simplifying (4x + 1)(4x) = 256 Reorder the terms: (1 + 4x)(4x) = 256 Remove parenthesis around (4x) (1 + 4x) * 4x = 256 Reorder the terms for easier multiplication: 4x(1 + 4x) = 256 (1 * 4x + 4x * 4x) = 256 (4x + 16x2) = 256 Solving 4x + 16x2 = 256 Solving for variable 'x'. Reorder the terms: -256 + 4x + 16x2 = 256 + -256 Combine like terms: 256 + -256 = 0 -256 + 4x + 16x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-64 + x + 4x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-64 + x + 4x2)' equal to zero and attempt to solve: Simplifying -64 + x + 4x2 = 0 Solving -64 + x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -16 + 0.25x + x2 = 0 Move the constant term to the right: Add '16' to each side of the equation. -16 + 0.25x + 16 + x2 = 0 + 16 Reorder the terms: -16 + 16 + 0.25x + x2 = 0 + 16 Combine like terms: -16 + 16 = 0 0 + 0.25x + x2 = 0 + 16 0.25x + x2 = 0 + 16 Combine like terms: 0 + 16 = 16 0.25x + x2 = 16 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. 0.25x + 0.25 + x2 = 16 + 0.25 Reorder the terms: 0.25 + 0.25x + x2 = 16 + 0.25 Combine like terms: 16 + 0.25 = 16.25 0.25 + 0.25x + x2 = 16.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 16.25 Calculate the square root of the right side: 4.031128874 Break this problem into two subproblems by setting (x + 0.5) equal to 4.031128874 and -4.031128874.Subproblem 1
x + 0.5 = 4.031128874 Simplifying x + 0.5 = 4.031128874 Reorder the terms: 0.5 + x = 4.031128874 Solving 0.5 + x = 4.031128874 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 = 4.031128874 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 4.031128874 + -0.5 x = 4.031128874 + -0.5 Combine like terms: 4.031128874 + -0.5 = 3.531128874 x = 3.531128874 Simplifying x = 3.531128874Subproblem 2
x + 0.5 = -4.031128874 Simplifying x + 0.5 = -4.031128874 Reorder the terms: 0.5 + x = -4.031128874 Solving 0.5 + x = -4.031128874 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 = -4.031128874 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -4.031128874 + -0.5 x = -4.031128874 + -0.5 Combine like terms: -4.031128874 + -0.5 = -4.531128874 x = -4.531128874 Simplifying x = -4.531128874Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.531128874, -4.531128874}Solution
x = {3.531128874, -4.531128874}
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