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Simplifying 9z2 + -36z + 31 = 0 Reorder the terms: 31 + -36z + 9z2 = 0 Solving 31 + -36z + 9z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 3.444444444 + -4z + z2 = 0 Move the constant term to the right: Add '-3.444444444' to each side of the equation. 3.444444444 + -4z + -3.444444444 + z2 = 0 + -3.444444444 Reorder the terms: 3.444444444 + -3.444444444 + -4z + z2 = 0 + -3.444444444 Combine like terms: 3.444444444 + -3.444444444 = 0.000000000 0.000000000 + -4z + z2 = 0 + -3.444444444 -4z + z2 = 0 + -3.444444444 Combine like terms: 0 + -3.444444444 = -3.444444444 -4z + z2 = -3.444444444 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 = -3.444444444 + 4 Reorder the terms: 4 + -4z + z2 = -3.444444444 + 4 Combine like terms: -3.444444444 + 4 = 0.555555556 4 + -4z + z2 = 0.555555556 Factor a perfect square on the left side: (z + -2)(z + -2) = 0.555555556 Calculate the square root of the right side: 0.745355993 Break this problem into two subproblems by setting (z + -2) equal to 0.745355993 and -0.745355993.Subproblem 1
z + -2 = 0.745355993 Simplifying z + -2 = 0.745355993 Reorder the terms: -2 + z = 0.745355993 Solving -2 + z = 0.745355993 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 = 0.745355993 + 2 Combine like terms: -2 + 2 = 0 0 + z = 0.745355993 + 2 z = 0.745355993 + 2 Combine like terms: 0.745355993 + 2 = 2.745355993 z = 2.745355993 Simplifying z = 2.745355993Subproblem 2
z + -2 = -0.745355993 Simplifying z + -2 = -0.745355993 Reorder the terms: -2 + z = -0.745355993 Solving -2 + z = -0.745355993 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 = -0.745355993 + 2 Combine like terms: -2 + 2 = 0 0 + z = -0.745355993 + 2 z = -0.745355993 + 2 Combine like terms: -0.745355993 + 2 = 1.254644007 z = 1.254644007 Simplifying z = 1.254644007Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.745355993, 1.254644007}
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