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Simplifying 2z2 + 8 = -9z Reorder the terms: 8 + 2z2 = -9z Solving 8 + 2z2 = -9z Solving for variable 'z'. Reorder the terms: 8 + 9z + 2z2 = -9z + 9z Combine like terms: -9z + 9z = 0 8 + 9z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 4 + 4.5z + z2 = 0 Move the constant term to the right: Add '-4' to each side of the equation. 4 + 4.5z + -4 + z2 = 0 + -4 Reorder the terms: 4 + -4 + 4.5z + z2 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + 4.5z + z2 = 0 + -4 4.5z + z2 = 0 + -4 Combine like terms: 0 + -4 = -4 4.5z + z2 = -4 The z term is 4.5z. Take half its coefficient (2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. 4.5z + 5.0625 + z2 = -4 + 5.0625 Reorder the terms: 5.0625 + 4.5z + z2 = -4 + 5.0625 Combine like terms: -4 + 5.0625 = 1.0625 5.0625 + 4.5z + z2 = 1.0625 Factor a perfect square on the left side: (z + 2.25)(z + 2.25) = 1.0625 Calculate the square root of the right side: 1.030776406 Break this problem into two subproblems by setting (z + 2.25) equal to 1.030776406 and -1.030776406.Subproblem 1
z + 2.25 = 1.030776406 Simplifying z + 2.25 = 1.030776406 Reorder the terms: 2.25 + z = 1.030776406 Solving 2.25 + z = 1.030776406 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + z = 1.030776406 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + z = 1.030776406 + -2.25 z = 1.030776406 + -2.25 Combine like terms: 1.030776406 + -2.25 = -1.219223594 z = -1.219223594 Simplifying z = -1.219223594Subproblem 2
z + 2.25 = -1.030776406 Simplifying z + 2.25 = -1.030776406 Reorder the terms: 2.25 + z = -1.030776406 Solving 2.25 + z = -1.030776406 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + z = -1.030776406 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + z = -1.030776406 + -2.25 z = -1.030776406 + -2.25 Combine like terms: -1.030776406 + -2.25 = -3.280776406 z = -3.280776406 Simplifying z = -3.280776406Solution
The solution to the problem is based on the solutions from the subproblems. z = {-1.219223594, -3.280776406}
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