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Simplifying 18z2 + 79z + 9 = 0 Reorder the terms: 9 + 79z + 18z2 = 0 Solving 9 + 79z + 18z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. 0.5 + 4.388888889z + z2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + 4.388888889z + -0.5 + z2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + 4.388888889z + z2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + 4.388888889z + z2 = 0 + -0.5 4.388888889z + z2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 4.388888889z + z2 = -0.5 The z term is 4.388888889z. Take half its coefficient (2.194444445). Square it (4.815586422) and add it to both sides. Add '4.815586422' to each side of the equation. 4.388888889z + 4.815586422 + z2 = -0.5 + 4.815586422 Reorder the terms: 4.815586422 + 4.388888889z + z2 = -0.5 + 4.815586422 Combine like terms: -0.5 + 4.815586422 = 4.315586422 4.815586422 + 4.388888889z + z2 = 4.315586422 Factor a perfect square on the left side: (z + 2.194444445)(z + 2.194444445) = 4.315586422 Calculate the square root of the right side: 2.077398956 Break this problem into two subproblems by setting (z + 2.194444445) equal to 2.077398956 and -2.077398956.Subproblem 1
z + 2.194444445 = 2.077398956 Simplifying z + 2.194444445 = 2.077398956 Reorder the terms: 2.194444445 + z = 2.077398956 Solving 2.194444445 + z = 2.077398956 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.194444445' to each side of the equation. 2.194444445 + -2.194444445 + z = 2.077398956 + -2.194444445 Combine like terms: 2.194444445 + -2.194444445 = 0.000000000 0.000000000 + z = 2.077398956 + -2.194444445 z = 2.077398956 + -2.194444445 Combine like terms: 2.077398956 + -2.194444445 = -0.117045489 z = -0.117045489 Simplifying z = -0.117045489Subproblem 2
z + 2.194444445 = -2.077398956 Simplifying z + 2.194444445 = -2.077398956 Reorder the terms: 2.194444445 + z = -2.077398956 Solving 2.194444445 + z = -2.077398956 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.194444445' to each side of the equation. 2.194444445 + -2.194444445 + z = -2.077398956 + -2.194444445 Combine like terms: 2.194444445 + -2.194444445 = 0.000000000 0.000000000 + z = -2.077398956 + -2.194444445 z = -2.077398956 + -2.194444445 Combine like terms: -2.077398956 + -2.194444445 = -4.271843401 z = -4.271843401 Simplifying z = -4.271843401Solution
The solution to the problem is based on the solutions from the subproblems. z = {-0.117045489, -4.271843401}
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