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Simplifying 9z2 + 76 = 54z Reorder the terms: 76 + 9z2 = 54z Solving 76 + 9z2 = 54z Solving for variable 'z'. Reorder the terms: 76 + -54z + 9z2 = 54z + -54z Combine like terms: 54z + -54z = 0 76 + -54z + 9z2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 8.444444444 + -6z + z2 = 0 Move the constant term to the right: Add '-8.444444444' to each side of the equation. 8.444444444 + -6z + -8.444444444 + z2 = 0 + -8.444444444 Reorder the terms: 8.444444444 + -8.444444444 + -6z + z2 = 0 + -8.444444444 Combine like terms: 8.444444444 + -8.444444444 = 0.000000000 0.000000000 + -6z + z2 = 0 + -8.444444444 -6z + z2 = 0 + -8.444444444 Combine like terms: 0 + -8.444444444 = -8.444444444 -6z + z2 = -8.444444444 The z term is -6z. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6z + 9 + z2 = -8.444444444 + 9 Reorder the terms: 9 + -6z + z2 = -8.444444444 + 9 Combine like terms: -8.444444444 + 9 = 0.555555556 9 + -6z + z2 = 0.555555556 Factor a perfect square on the left side: (z + -3)(z + -3) = 0.555555556 Calculate the square root of the right side: 0.745355993 Break this problem into two subproblems by setting (z + -3) equal to 0.745355993 and -0.745355993.Subproblem 1
z + -3 = 0.745355993 Simplifying z + -3 = 0.745355993 Reorder the terms: -3 + z = 0.745355993 Solving -3 + z = 0.745355993 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + z = 0.745355993 + 3 Combine like terms: -3 + 3 = 0 0 + z = 0.745355993 + 3 z = 0.745355993 + 3 Combine like terms: 0.745355993 + 3 = 3.745355993 z = 3.745355993 Simplifying z = 3.745355993Subproblem 2
z + -3 = -0.745355993 Simplifying z + -3 = -0.745355993 Reorder the terms: -3 + z = -0.745355993 Solving -3 + z = -0.745355993 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + z = -0.745355993 + 3 Combine like terms: -3 + 3 = 0 0 + z = -0.745355993 + 3 z = -0.745355993 + 3 Combine like terms: -0.745355993 + 3 = 2.254644007 z = 2.254644007 Simplifying z = 2.254644007Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.745355993, 2.254644007}
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