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Simplifying 16x2 + 16x + -78 = 0 Reorder the terms: -78 + 16x + 16x2 = 0 Solving -78 + 16x + 16x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(-39 + 8x + 8x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-39 + 8x + 8x2)' equal to zero and attempt to solve: Simplifying -39 + 8x + 8x2 = 0 Solving -39 + 8x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -4.875 + x + x2 = 0 Move the constant term to the right: Add '4.875' to each side of the equation. -4.875 + x + 4.875 + x2 = 0 + 4.875 Reorder the terms: -4.875 + 4.875 + x + x2 = 0 + 4.875 Combine like terms: -4.875 + 4.875 = 0.000 0.000 + x + x2 = 0 + 4.875 x + x2 = 0 + 4.875 Combine like terms: 0 + 4.875 = 4.875 x + x2 = 4.875 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. + 0.25 + x2 = 4.875 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 4.875 + 0.25 Combine like terms: 4.875 + 0.25 = 5.125 1.25 + x2 = 5.125 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 5.125 Calculate the square root of the right side: 2.263846285 Break this problem into two subproblems by setting (x + 0.5) equal to 2.263846285 and -2.263846285.Subproblem 1
x + 0.5 = 2.263846285 Simplifying x + 0.5 = 2.263846285 Reorder the terms: 0.5 + x = 2.263846285 Solving 0.5 + x = 2.263846285 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 = 2.263846285 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 2.263846285 + -0.5 x = 2.263846285 + -0.5 Combine like terms: 2.263846285 + -0.5 = 1.763846285 x = 1.763846285 Simplifying x = 1.763846285Subproblem 2
x + 0.5 = -2.263846285 Simplifying x + 0.5 = -2.263846285 Reorder the terms: 0.5 + x = -2.263846285 Solving 0.5 + x = -2.263846285 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 = -2.263846285 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -2.263846285 + -0.5 x = -2.263846285 + -0.5 Combine like terms: -2.263846285 + -0.5 = -2.763846285 x = -2.763846285 Simplifying x = -2.763846285Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.763846285, -2.763846285}Solution
x = {1.763846285, -2.763846285}
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