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Simplifying 5z2 + 8z + 1 = 0 Reorder the terms: 1 + 8z + 5z2 = 0 Solving 1 + 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'. 0.2 + 1.6z + z2 = 0 Move the constant term to the right: Add '-0.2' to each side of the equation. 0.2 + 1.6z + -0.2 + z2 = 0 + -0.2 Reorder the terms: 0.2 + -0.2 + 1.6z + z2 = 0 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + 1.6z + z2 = 0 + -0.2 1.6z + z2 = 0 + -0.2 Combine like terms: 0 + -0.2 = -0.2 1.6z + z2 = -0.2 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 = -0.2 + 0.64 Reorder the terms: 0.64 + 1.6z + z2 = -0.2 + 0.64 Combine like terms: -0.2 + 0.64 = 0.44 0.64 + 1.6z + z2 = 0.44 Factor a perfect square on the left side: (z + 0.8)(z + 0.8) = 0.44 Calculate the square root of the right side: 0.663324958 Break this problem into two subproblems by setting (z + 0.8) equal to 0.663324958 and -0.663324958.Subproblem 1
z + 0.8 = 0.663324958 Simplifying z + 0.8 = 0.663324958 Reorder the terms: 0.8 + z = 0.663324958 Solving 0.8 + z = 0.663324958 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 = 0.663324958 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + z = 0.663324958 + -0.8 z = 0.663324958 + -0.8 Combine like terms: 0.663324958 + -0.8 = -0.136675042 z = -0.136675042 Simplifying z = -0.136675042Subproblem 2
z + 0.8 = -0.663324958 Simplifying z + 0.8 = -0.663324958 Reorder the terms: 0.8 + z = -0.663324958 Solving 0.8 + z = -0.663324958 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 = -0.663324958 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + z = -0.663324958 + -0.8 z = -0.663324958 + -0.8 Combine like terms: -0.663324958 + -0.8 = -1.463324958 z = -1.463324958 Simplifying z = -1.463324958Solution
The solution to the problem is based on the solutions from the subproblems. z = {-0.136675042, -1.463324958}
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