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Simplifying (4x + 1)(4x + 1) = 19 Reorder the terms: (1 + 4x)(4x + 1) = 19 Reorder the terms: (1 + 4x)(1 + 4x) = 19 Multiply (1 + 4x) * (1 + 4x) (1(1 + 4x) + 4x * (1 + 4x)) = 19 ((1 * 1 + 4x * 1) + 4x * (1 + 4x)) = 19 ((1 + 4x) + 4x * (1 + 4x)) = 19 (1 + 4x + (1 * 4x + 4x * 4x)) = 19 (1 + 4x + (4x + 16x2)) = 19 Combine like terms: 4x + 4x = 8x (1 + 8x + 16x2) = 19 Solving 1 + 8x + 16x2 = 19 Solving for variable 'x'. Reorder the terms: 1 + -19 + 8x + 16x2 = 19 + -19 Combine like terms: 1 + -19 = -18 -18 + 8x + 16x2 = 19 + -19 Combine like terms: 19 + -19 = 0 -18 + 8x + 16x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-9 + 4x + 8x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-9 + 4x + 8x2)' equal to zero and attempt to solve: Simplifying -9 + 4x + 8x2 = 0 Solving -9 + 4x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -1.125 + 0.5x + x2 = 0 Move the constant term to the right: Add '1.125' to each side of the equation. -1.125 + 0.5x + 1.125 + x2 = 0 + 1.125 Reorder the terms: -1.125 + 1.125 + 0.5x + x2 = 0 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + 0.5x + x2 = 0 + 1.125 0.5x + x2 = 0 + 1.125 Combine like terms: 0 + 1.125 = 1.125 0.5x + x2 = 1.125 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 = 1.125 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 1.125 + 0.0625 Combine like terms: 1.125 + 0.0625 = 1.1875 0.0625 + 0.5x + x2 = 1.1875 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 1.1875 Calculate the square root of the right side: 1.089724736 Break this problem into two subproblems by setting (x + 0.25) equal to 1.089724736 and -1.089724736.Subproblem 1
x + 0.25 = 1.089724736 Simplifying x + 0.25 = 1.089724736 Reorder the terms: 0.25 + x = 1.089724736 Solving 0.25 + x = 1.089724736 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 = 1.089724736 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 1.089724736 + -0.25 x = 1.089724736 + -0.25 Combine like terms: 1.089724736 + -0.25 = 0.839724736 x = 0.839724736 Simplifying x = 0.839724736Subproblem 2
x + 0.25 = -1.089724736 Simplifying x + 0.25 = -1.089724736 Reorder the terms: 0.25 + x = -1.089724736 Solving 0.25 + x = -1.089724736 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 = -1.089724736 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -1.089724736 + -0.25 x = -1.089724736 + -0.25 Combine like terms: -1.089724736 + -0.25 = -1.339724736 x = -1.339724736 Simplifying x = -1.339724736Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.839724736, -1.339724736}Solution
x = {0.839724736, -1.339724736}
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