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Simplifying 16x2 + 8x + -11 = 0 Reorder the terms: -11 + 8x + 16x2 = 0 Solving -11 + 8x + 16x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.6875 + 0.5x + x2 = 0 Move the constant term to the right: Add '0.6875' to each side of the equation. -0.6875 + 0.5x + 0.6875 + x2 = 0 + 0.6875 Reorder the terms: -0.6875 + 0.6875 + 0.5x + x2 = 0 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + 0.5x + x2 = 0 + 0.6875 0.5x + x2 = 0 + 0.6875 Combine like terms: 0 + 0.6875 = 0.6875 0.5x + x2 = 0.6875 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 = 0.6875 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 0.6875 + 0.0625 Combine like terms: 0.6875 + 0.0625 = 0.75 0.0625 + 0.5x + x2 = 0.75 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (x + 0.25) equal to 0.866025404 and -0.866025404.Subproblem 1
x + 0.25 = 0.866025404 Simplifying x + 0.25 = 0.866025404 Reorder the terms: 0.25 + x = 0.866025404 Solving 0.25 + x = 0.866025404 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 = 0.866025404 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 0.866025404 + -0.25 x = 0.866025404 + -0.25 Combine like terms: 0.866025404 + -0.25 = 0.616025404 x = 0.616025404 Simplifying x = 0.616025404Subproblem 2
x + 0.25 = -0.866025404 Simplifying x + 0.25 = -0.866025404 Reorder the terms: 0.25 + x = -0.866025404 Solving 0.25 + x = -0.866025404 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 = -0.866025404 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -0.866025404 + -0.25 x = -0.866025404 + -0.25 Combine like terms: -0.866025404 + -0.25 = -1.116025404 x = -1.116025404 Simplifying x = -1.116025404Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.616025404, -1.116025404}
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