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Simplifying 8z2 + 29z + -48 = 0 Reorder the terms: -48 + 29z + 8z2 = 0 Solving -48 + 29z + 8z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -6 + 3.625z + z2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + 3.625z + 6 + z2 = 0 + 6 Reorder the terms: -6 + 6 + 3.625z + z2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + 3.625z + z2 = 0 + 6 3.625z + z2 = 0 + 6 Combine like terms: 0 + 6 = 6 3.625z + z2 = 6 The z term is 3.625z. Take half its coefficient (1.8125). Square it (3.28515625) and add it to both sides. Add '3.28515625' to each side of the equation. 3.625z + 3.28515625 + z2 = 6 + 3.28515625 Reorder the terms: 3.28515625 + 3.625z + z2 = 6 + 3.28515625 Combine like terms: 6 + 3.28515625 = 9.28515625 3.28515625 + 3.625z + z2 = 9.28515625 Factor a perfect square on the left side: (z + 1.8125)(z + 1.8125) = 9.28515625 Calculate the square root of the right side: 3.047155436 Break this problem into two subproblems by setting (z + 1.8125) equal to 3.047155436 and -3.047155436.Subproblem 1
z + 1.8125 = 3.047155436 Simplifying z + 1.8125 = 3.047155436 Reorder the terms: 1.8125 + z = 3.047155436 Solving 1.8125 + z = 3.047155436 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.8125' to each side of the equation. 1.8125 + -1.8125 + z = 3.047155436 + -1.8125 Combine like terms: 1.8125 + -1.8125 = 0.0000 0.0000 + z = 3.047155436 + -1.8125 z = 3.047155436 + -1.8125 Combine like terms: 3.047155436 + -1.8125 = 1.234655436 z = 1.234655436 Simplifying z = 1.234655436Subproblem 2
z + 1.8125 = -3.047155436 Simplifying z + 1.8125 = -3.047155436 Reorder the terms: 1.8125 + z = -3.047155436 Solving 1.8125 + z = -3.047155436 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.8125' to each side of the equation. 1.8125 + -1.8125 + z = -3.047155436 + -1.8125 Combine like terms: 1.8125 + -1.8125 = 0.0000 0.0000 + z = -3.047155436 + -1.8125 z = -3.047155436 + -1.8125 Combine like terms: -3.047155436 + -1.8125 = -4.859655436 z = -4.859655436 Simplifying z = -4.859655436Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.234655436, -4.859655436}
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