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Simplifying 9z2 + 34 = 36z Reorder the terms: 34 + 9z2 = 36z Solving 34 + 9z2 = 36z Solving for variable 'z'. Reorder the terms: 34 + -36z + 9z2 = 36z + -36z Combine like terms: 36z + -36z = 0 34 + -36z + 9z2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 3.777777778 + -4z + z2 = 0 Move the constant term to the right: Add '-3.777777778' to each side of the equation. 3.777777778 + -4z + -3.777777778 + z2 = 0 + -3.777777778 Reorder the terms: 3.777777778 + -3.777777778 + -4z + z2 = 0 + -3.777777778 Combine like terms: 3.777777778 + -3.777777778 = 0.000000000 0.000000000 + -4z + z2 = 0 + -3.777777778 -4z + z2 = 0 + -3.777777778 Combine like terms: 0 + -3.777777778 = -3.777777778 -4z + z2 = -3.777777778 The z term is -4z. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4z + 4 + z2 = -3.777777778 + 4 Reorder the terms: 4 + -4z + z2 = -3.777777778 + 4 Combine like terms: -3.777777778 + 4 = 0.222222222 4 + -4z + z2 = 0.222222222 Factor a perfect square on the left side: (z + -2)(z + -2) = 0.222222222 Calculate the square root of the right side: 0.471404521 Break this problem into two subproblems by setting (z + -2) equal to 0.471404521 and -0.471404521.Subproblem 1
z + -2 = 0.471404521 Simplifying z + -2 = 0.471404521 Reorder the terms: -2 + z = 0.471404521 Solving -2 + z = 0.471404521 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + z = 0.471404521 + 2 Combine like terms: -2 + 2 = 0 0 + z = 0.471404521 + 2 z = 0.471404521 + 2 Combine like terms: 0.471404521 + 2 = 2.471404521 z = 2.471404521 Simplifying z = 2.471404521Subproblem 2
z + -2 = -0.471404521 Simplifying z + -2 = -0.471404521 Reorder the terms: -2 + z = -0.471404521 Solving -2 + z = -0.471404521 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + z = -0.471404521 + 2 Combine like terms: -2 + 2 = 0 0 + z = -0.471404521 + 2 z = -0.471404521 + 2 Combine like terms: -0.471404521 + 2 = 1.528595479 z = 1.528595479 Simplifying z = 1.528595479Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.471404521, 1.528595479}
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