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Simplifying 2z2 + 9z + -4 = 0 Reorder the terms: -4 + 9z + 2z2 = 0 Solving -4 + 9z + 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 + 4.5z + z2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 4.5z + 2 + z2 = 0 + 2 Reorder the terms: -2 + 2 + 4.5z + z2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 4.5z + z2 = 0 + 2 4.5z + z2 = 0 + 2 Combine like terms: 0 + 2 = 2 4.5z + z2 = 2 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 = 2 + 5.0625 Reorder the terms: 5.0625 + 4.5z + z2 = 2 + 5.0625 Combine like terms: 2 + 5.0625 = 7.0625 5.0625 + 4.5z + z2 = 7.0625 Factor a perfect square on the left side: (z + 2.25)(z + 2.25) = 7.0625 Calculate the square root of the right side: 2.657536453 Break this problem into two subproblems by setting (z + 2.25) equal to 2.657536453 and -2.657536453.Subproblem 1
z + 2.25 = 2.657536453 Simplifying z + 2.25 = 2.657536453 Reorder the terms: 2.25 + z = 2.657536453 Solving 2.25 + z = 2.657536453 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 = 2.657536453 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + z = 2.657536453 + -2.25 z = 2.657536453 + -2.25 Combine like terms: 2.657536453 + -2.25 = 0.407536453 z = 0.407536453 Simplifying z = 0.407536453Subproblem 2
z + 2.25 = -2.657536453 Simplifying z + 2.25 = -2.657536453 Reorder the terms: 2.25 + z = -2.657536453 Solving 2.25 + z = -2.657536453 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 = -2.657536453 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + z = -2.657536453 + -2.25 z = -2.657536453 + -2.25 Combine like terms: -2.657536453 + -2.25 = -4.907536453 z = -4.907536453 Simplifying z = -4.907536453Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.407536453, -4.907536453}
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