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Simplifying -2z2 + 8z + 5 = 0 Reorder the terms: 5 + 8z + -2z2 = 0 Solving 5 + 8z + -2z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -2.5 + -4z + z2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + -4z + 2.5 + z2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + -4z + z2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + -4z + z2 = 0 + 2.5 -4z + z2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 -4z + z2 = 2.5 The z term is -4z. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4z + 4 + z2 = 2.5 + 4 Reorder the terms: 4 + -4z + z2 = 2.5 + 4 Combine like terms: 2.5 + 4 = 6.5 4 + -4z + z2 = 6.5 Factor a perfect square on the left side: (z + -2)(z + -2) = 6.5 Calculate the square root of the right side: 2.549509757 Break this problem into two subproblems by setting (z + -2) equal to 2.549509757 and -2.549509757.Subproblem 1
z + -2 = 2.549509757 Simplifying z + -2 = 2.549509757 Reorder the terms: -2 + z = 2.549509757 Solving -2 + z = 2.549509757 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + z = 2.549509757 + 2 Combine like terms: -2 + 2 = 0 0 + z = 2.549509757 + 2 z = 2.549509757 + 2 Combine like terms: 2.549509757 + 2 = 4.549509757 z = 4.549509757 Simplifying z = 4.549509757Subproblem 2
z + -2 = -2.549509757 Simplifying z + -2 = -2.549509757 Reorder the terms: -2 + z = -2.549509757 Solving -2 + z = -2.549509757 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + z = -2.549509757 + 2 Combine like terms: -2 + 2 = 0 0 + z = -2.549509757 + 2 z = -2.549509757 + 2 Combine like terms: -2.549509757 + 2 = -0.549509757 z = -0.549509757 Simplifying z = -0.549509757Solution
The solution to the problem is based on the solutions from the subproblems. z = {4.549509757, -0.549509757}
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