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Simplifying 8z2 + 3z + -3 = 0 Reorder the terms: -3 + 3z + 8z2 = 0 Solving -3 + 3z + 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'. -0.375 + 0.375z + z2 = 0 Move the constant term to the right: Add '0.375' to each side of the equation. -0.375 + 0.375z + 0.375 + z2 = 0 + 0.375 Reorder the terms: -0.375 + 0.375 + 0.375z + z2 = 0 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + 0.375z + z2 = 0 + 0.375 0.375z + z2 = 0 + 0.375 Combine like terms: 0 + 0.375 = 0.375 0.375z + z2 = 0.375 The z term is 0.375z. Take half its coefficient (0.1875). Square it (0.03515625) and add it to both sides. Add '0.03515625' to each side of the equation. 0.375z + 0.03515625 + z2 = 0.375 + 0.03515625 Reorder the terms: 0.03515625 + 0.375z + z2 = 0.375 + 0.03515625 Combine like terms: 0.375 + 0.03515625 = 0.41015625 0.03515625 + 0.375z + z2 = 0.41015625 Factor a perfect square on the left side: (z + 0.1875)(z + 0.1875) = 0.41015625 Calculate the square root of the right side: 0.640434423 Break this problem into two subproblems by setting (z + 0.1875) equal to 0.640434423 and -0.640434423.Subproblem 1
z + 0.1875 = 0.640434423 Simplifying z + 0.1875 = 0.640434423 Reorder the terms: 0.1875 + z = 0.640434423 Solving 0.1875 + z = 0.640434423 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + z = 0.640434423 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + z = 0.640434423 + -0.1875 z = 0.640434423 + -0.1875 Combine like terms: 0.640434423 + -0.1875 = 0.452934423 z = 0.452934423 Simplifying z = 0.452934423Subproblem 2
z + 0.1875 = -0.640434423 Simplifying z + 0.1875 = -0.640434423 Reorder the terms: 0.1875 + z = -0.640434423 Solving 0.1875 + z = -0.640434423 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + z = -0.640434423 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + z = -0.640434423 + -0.1875 z = -0.640434423 + -0.1875 Combine like terms: -0.640434423 + -0.1875 = -0.827934423 z = -0.827934423 Simplifying z = -0.827934423Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.452934423, -0.827934423}
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