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Simplifying (2z + 6)(z + 6) = 5 Reorder the terms: (6 + 2z)(z + 6) = 5 Reorder the terms: (6 + 2z)(6 + z) = 5 Multiply (6 + 2z) * (6 + z) (6(6 + z) + 2z * (6 + z)) = 5 ((6 * 6 + z * 6) + 2z * (6 + z)) = 5 ((36 + 6z) + 2z * (6 + z)) = 5 (36 + 6z + (6 * 2z + z * 2z)) = 5 (36 + 6z + (12z + 2z2)) = 5 Combine like terms: 6z + 12z = 18z (36 + 18z + 2z2) = 5 Solving 36 + 18z + 2z2 = 5 Solving for variable 'z'. Reorder the terms: 36 + -5 + 18z + 2z2 = 5 + -5 Combine like terms: 36 + -5 = 31 31 + 18z + 2z2 = 5 + -5 Combine like terms: 5 + -5 = 0 31 + 18z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 15.5 + 9z + z2 = 0 Move the constant term to the right: Add '-15.5' to each side of the equation. 15.5 + 9z + -15.5 + z2 = 0 + -15.5 Reorder the terms: 15.5 + -15.5 + 9z + z2 = 0 + -15.5 Combine like terms: 15.5 + -15.5 = 0.0 0.0 + 9z + z2 = 0 + -15.5 9z + z2 = 0 + -15.5 Combine like terms: 0 + -15.5 = -15.5 9z + z2 = -15.5 The z term is 9z. Take half its coefficient (4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. 9z + 20.25 + z2 = -15.5 + 20.25 Reorder the terms: 20.25 + 9z + z2 = -15.5 + 20.25 Combine like terms: -15.5 + 20.25 = 4.75 20.25 + 9z + z2 = 4.75 Factor a perfect square on the left side: (z + 4.5)(z + 4.5) = 4.75 Calculate the square root of the right side: 2.179449472 Break this problem into two subproblems by setting (z + 4.5) equal to 2.179449472 and -2.179449472.Subproblem 1
z + 4.5 = 2.179449472 Simplifying z + 4.5 = 2.179449472 Reorder the terms: 4.5 + z = 2.179449472 Solving 4.5 + z = 2.179449472 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + z = 2.179449472 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + z = 2.179449472 + -4.5 z = 2.179449472 + -4.5 Combine like terms: 2.179449472 + -4.5 = -2.320550528 z = -2.320550528 Simplifying z = -2.320550528Subproblem 2
z + 4.5 = -2.179449472 Simplifying z + 4.5 = -2.179449472 Reorder the terms: 4.5 + z = -2.179449472 Solving 4.5 + z = -2.179449472 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + z = -2.179449472 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + z = -2.179449472 + -4.5 z = -2.179449472 + -4.5 Combine like terms: -2.179449472 + -4.5 = -6.679449472 z = -6.679449472 Simplifying z = -6.679449472Solution
The solution to the problem is based on the solutions from the subproblems. z = {-2.320550528, -6.679449472}
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