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Simplifying -5z2 + -4z + 5 = 0 Reorder the terms: 5 + -4z + -5z2 = 0 Solving 5 + -4z + -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 + 0.8z + z2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 0.8z + 1 + z2 = 0 + 1 Reorder the terms: -1 + 1 + 0.8z + z2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 0.8z + z2 = 0 + 1 0.8z + z2 = 0 + 1 Combine like terms: 0 + 1 = 1 0.8z + z2 = 1 The z term is 0.8z. Take half its coefficient (0.4). Square it (0.16) and add it to both sides. Add '0.16' to each side of the equation. 0.8z + 0.16 + z2 = 1 + 0.16 Reorder the terms: 0.16 + 0.8z + z2 = 1 + 0.16 Combine like terms: 1 + 0.16 = 1.16 0.16 + 0.8z + z2 = 1.16 Factor a perfect square on the left side: (z + 0.4)(z + 0.4) = 1.16 Calculate the square root of the right side: 1.077032961 Break this problem into two subproblems by setting (z + 0.4) equal to 1.077032961 and -1.077032961.Subproblem 1
z + 0.4 = 1.077032961 Simplifying z + 0.4 = 1.077032961 Reorder the terms: 0.4 + z = 1.077032961 Solving 0.4 + z = 1.077032961 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.4' to each side of the equation. 0.4 + -0.4 + z = 1.077032961 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + z = 1.077032961 + -0.4 z = 1.077032961 + -0.4 Combine like terms: 1.077032961 + -0.4 = 0.677032961 z = 0.677032961 Simplifying z = 0.677032961Subproblem 2
z + 0.4 = -1.077032961 Simplifying z + 0.4 = -1.077032961 Reorder the terms: 0.4 + z = -1.077032961 Solving 0.4 + z = -1.077032961 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.4' to each side of the equation. 0.4 + -0.4 + z = -1.077032961 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + z = -1.077032961 + -0.4 z = -1.077032961 + -0.4 Combine like terms: -1.077032961 + -0.4 = -1.477032961 z = -1.477032961 Simplifying z = -1.477032961Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.677032961, -1.477032961}
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