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Simplifying 5z2 + 8z + -5 = 0 Reorder the terms: -5 + 8z + 5z2 = 0 Solving -5 + 8z + 5z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -1 + 1.6z + z2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 1.6z + 1 + z2 = 0 + 1 Reorder the terms: -1 + 1 + 1.6z + z2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 1.6z + z2 = 0 + 1 1.6z + z2 = 0 + 1 Combine like terms: 0 + 1 = 1 1.6z + z2 = 1 The z term is 1.6z. Take half its coefficient (0.8). Square it (0.64) and add it to both sides. Add '0.64' to each side of the equation. 1.6z + 0.64 + z2 = 1 + 0.64 Reorder the terms: 0.64 + 1.6z + z2 = 1 + 0.64 Combine like terms: 1 + 0.64 = 1.64 0.64 + 1.6z + z2 = 1.64 Factor a perfect square on the left side: (z + 0.8)(z + 0.8) = 1.64 Calculate the square root of the right side: 1.280624847 Break this problem into two subproblems by setting (z + 0.8) equal to 1.280624847 and -1.280624847.Subproblem 1
z + 0.8 = 1.280624847 Simplifying z + 0.8 = 1.280624847 Reorder the terms: 0.8 + z = 1.280624847 Solving 0.8 + z = 1.280624847 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + z = 1.280624847 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + z = 1.280624847 + -0.8 z = 1.280624847 + -0.8 Combine like terms: 1.280624847 + -0.8 = 0.480624847 z = 0.480624847 Simplifying z = 0.480624847Subproblem 2
z + 0.8 = -1.280624847 Simplifying z + 0.8 = -1.280624847 Reorder the terms: 0.8 + z = -1.280624847 Solving 0.8 + z = -1.280624847 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + z = -1.280624847 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + z = -1.280624847 + -0.8 z = -1.280624847 + -0.8 Combine like terms: -1.280624847 + -0.8 = -2.080624847 z = -2.080624847 Simplifying z = -2.080624847Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.480624847, -2.080624847}
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