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Simplifying 4z2 + 9z + 3 = 0 Reorder the terms: 3 + 9z + 4z2 = 0 Solving 3 + 9z + 4z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.75 + 2.25z + z2 = 0 Move the constant term to the right: Add '-0.75' to each side of the equation. 0.75 + 2.25z + -0.75 + z2 = 0 + -0.75 Reorder the terms: 0.75 + -0.75 + 2.25z + z2 = 0 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + 2.25z + z2 = 0 + -0.75 2.25z + z2 = 0 + -0.75 Combine like terms: 0 + -0.75 = -0.75 2.25z + z2 = -0.75 The z term is 2.25z. Take half its coefficient (1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. 2.25z + 1.265625 + z2 = -0.75 + 1.265625 Reorder the terms: 1.265625 + 2.25z + z2 = -0.75 + 1.265625 Combine like terms: -0.75 + 1.265625 = 0.515625 1.265625 + 2.25z + z2 = 0.515625 Factor a perfect square on the left side: (z + 1.125)(z + 1.125) = 0.515625 Calculate the square root of the right side: 0.718070331 Break this problem into two subproblems by setting (z + 1.125) equal to 0.718070331 and -0.718070331.Subproblem 1
z + 1.125 = 0.718070331 Simplifying z + 1.125 = 0.718070331 Reorder the terms: 1.125 + z = 0.718070331 Solving 1.125 + z = 0.718070331 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + z = 0.718070331 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + z = 0.718070331 + -1.125 z = 0.718070331 + -1.125 Combine like terms: 0.718070331 + -1.125 = -0.406929669 z = -0.406929669 Simplifying z = -0.406929669Subproblem 2
z + 1.125 = -0.718070331 Simplifying z + 1.125 = -0.718070331 Reorder the terms: 1.125 + z = -0.718070331 Solving 1.125 + z = -0.718070331 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + z = -0.718070331 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + z = -0.718070331 + -1.125 z = -0.718070331 + -1.125 Combine like terms: -0.718070331 + -1.125 = -1.843070331 z = -1.843070331 Simplifying z = -1.843070331Solution
The solution to the problem is based on the solutions from the subproblems. z = {-0.406929669, -1.843070331}
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