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Simplifying 16z2 + 8z + -5 = 0 Reorder the terms: -5 + 8z + 16z2 = 0 Solving -5 + 8z + 16z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.3125 + 0.5z + z2 = 0 Move the constant term to the right: Add '0.3125' to each side of the equation. -0.3125 + 0.5z + 0.3125 + z2 = 0 + 0.3125 Reorder the terms: -0.3125 + 0.3125 + 0.5z + z2 = 0 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + 0.5z + z2 = 0 + 0.3125 0.5z + z2 = 0 + 0.3125 Combine like terms: 0 + 0.3125 = 0.3125 0.5z + z2 = 0.3125 The z term is 0.5z. 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. 0.5z + 0.0625 + z2 = 0.3125 + 0.0625 Reorder the terms: 0.0625 + 0.5z + z2 = 0.3125 + 0.0625 Combine like terms: 0.3125 + 0.0625 = 0.375 0.0625 + 0.5z + z2 = 0.375 Factor a perfect square on the left side: (z + 0.25)(z + 0.25) = 0.375 Calculate the square root of the right side: 0.612372436 Break this problem into two subproblems by setting (z + 0.25) equal to 0.612372436 and -0.612372436.Subproblem 1
z + 0.25 = 0.612372436 Simplifying z + 0.25 = 0.612372436 Reorder the terms: 0.25 + z = 0.612372436 Solving 0.25 + z = 0.612372436 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + z = 0.612372436 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + z = 0.612372436 + -0.25 z = 0.612372436 + -0.25 Combine like terms: 0.612372436 + -0.25 = 0.362372436 z = 0.362372436 Simplifying z = 0.362372436Subproblem 2
z + 0.25 = -0.612372436 Simplifying z + 0.25 = -0.612372436 Reorder the terms: 0.25 + z = -0.612372436 Solving 0.25 + z = -0.612372436 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + z = -0.612372436 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + z = -0.612372436 + -0.25 z = -0.612372436 + -0.25 Combine like terms: -0.612372436 + -0.25 = -0.862372436 z = -0.862372436 Simplifying z = -0.862372436Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.362372436, -0.862372436}
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