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Simplifying 16x2 + 80x + 92 = 0 Reorder the terms: 92 + 80x + 16x2 = 0 Solving 92 + 80x + 16x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '4'. 4(23 + 20x + 4x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(23 + 20x + 4x2)' equal to zero and attempt to solve: Simplifying 23 + 20x + 4x2 = 0 Solving 23 + 20x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 5.75 + 5x + x2 = 0 Move the constant term to the right: Add '-5.75' to each side of the equation. 5.75 + 5x + -5.75 + x2 = 0 + -5.75 Reorder the terms: 5.75 + -5.75 + 5x + x2 = 0 + -5.75 Combine like terms: 5.75 + -5.75 = 0.00 0.00 + 5x + x2 = 0 + -5.75 5x + x2 = 0 + -5.75 Combine like terms: 0 + -5.75 = -5.75 5x + x2 = -5.75 The x term is 5x. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5x + 6.25 + x2 = -5.75 + 6.25 Reorder the terms: 6.25 + 5x + x2 = -5.75 + 6.25 Combine like terms: -5.75 + 6.25 = 0.5 6.25 + 5x + x2 = 0.5 Factor a perfect square on the left side: (x + 2.5)(x + 2.5) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (x + 2.5) equal to 0.707106781 and -0.707106781.Subproblem 1
x + 2.5 = 0.707106781 Simplifying x + 2.5 = 0.707106781 Reorder the terms: 2.5 + x = 0.707106781 Solving 2.5 + x = 0.707106781 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = 0.707106781 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 0.707106781 + -2.5 x = 0.707106781 + -2.5 Combine like terms: 0.707106781 + -2.5 = -1.792893219 x = -1.792893219 Simplifying x = -1.792893219Subproblem 2
x + 2.5 = -0.707106781 Simplifying x + 2.5 = -0.707106781 Reorder the terms: 2.5 + x = -0.707106781 Solving 2.5 + x = -0.707106781 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = -0.707106781 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -0.707106781 + -2.5 x = -0.707106781 + -2.5 Combine like terms: -0.707106781 + -2.5 = -3.207106781 x = -3.207106781 Simplifying x = -3.207106781Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.792893219, -3.207106781}Solution
x = {-1.792893219, -3.207106781}
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