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Simplifying 2z2 + 4z + 4 = 9 Reorder the terms: 4 + 4z + 2z2 = 9 Solving 4 + 4z + 2z2 = 9 Solving for variable 'z'. Reorder the terms: 4 + -9 + 4z + 2z2 = 9 + -9 Combine like terms: 4 + -9 = -5 -5 + 4z + 2z2 = 9 + -9 Combine like terms: 9 + -9 = 0 -5 + 4z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2.5 + 2z + z2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + 2z + 2.5 + z2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + 2z + z2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + 2z + z2 = 0 + 2.5 2z + z2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 2z + z2 = 2.5 The z term is 2z. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2z + 1 + z2 = 2.5 + 1 Reorder the terms: 1 + 2z + z2 = 2.5 + 1 Combine like terms: 2.5 + 1 = 3.5 1 + 2z + z2 = 3.5 Factor a perfect square on the left side: (z + 1)(z + 1) = 3.5 Calculate the square root of the right side: 1.870828693 Break this problem into two subproblems by setting (z + 1) equal to 1.870828693 and -1.870828693.Subproblem 1
z + 1 = 1.870828693 Simplifying z + 1 = 1.870828693 Reorder the terms: 1 + z = 1.870828693 Solving 1 + z = 1.870828693 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + z = 1.870828693 + -1 Combine like terms: 1 + -1 = 0 0 + z = 1.870828693 + -1 z = 1.870828693 + -1 Combine like terms: 1.870828693 + -1 = 0.870828693 z = 0.870828693 Simplifying z = 0.870828693Subproblem 2
z + 1 = -1.870828693 Simplifying z + 1 = -1.870828693 Reorder the terms: 1 + z = -1.870828693 Solving 1 + z = -1.870828693 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + z = -1.870828693 + -1 Combine like terms: 1 + -1 = 0 0 + z = -1.870828693 + -1 z = -1.870828693 + -1 Combine like terms: -1.870828693 + -1 = -2.870828693 z = -2.870828693 Simplifying z = -2.870828693Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.870828693, -2.870828693}
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