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Simplifying 2z2 + 312.8z = 37223 Reorder the terms: 312.8z + 2z2 = 37223 Solving 312.8z + 2z2 = 37223 Solving for variable 'z'. Reorder the terms: -37223 + 312.8z + 2z2 = 37223 + -37223 Combine like terms: 37223 + -37223 = 0 -37223 + 312.8z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -18611.5 + 156.4z + z2 = 0 Move the constant term to the right: Add '18611.5' to each side of the equation. -18611.5 + 156.4z + 18611.5 + z2 = 0 + 18611.5 Reorder the terms: -18611.5 + 18611.5 + 156.4z + z2 = 0 + 18611.5 Combine like terms: -18611.5 + 18611.5 = 0.0 0.0 + 156.4z + z2 = 0 + 18611.5 156.4z + z2 = 0 + 18611.5 Combine like terms: 0 + 18611.5 = 18611.5 156.4z + z2 = 18611.5 The z term is 156.4z. Take half its coefficient (78.2). Square it (6115.24) and add it to both sides. Add '6115.24' to each side of the equation. 156.4z + 6115.24 + z2 = 18611.5 + 6115.24 Reorder the terms: 6115.24 + 156.4z + z2 = 18611.5 + 6115.24 Combine like terms: 18611.5 + 6115.24 = 24726.74 6115.24 + 156.4z + z2 = 24726.74 Factor a perfect square on the left side: (z + 78.2)(z + 78.2) = 24726.74 Calculate the square root of the right side: 157.247384716 Break this problem into two subproblems by setting (z + 78.2) equal to 157.247384716 and -157.247384716.Subproblem 1
z + 78.2 = 157.247384716 Simplifying z + 78.2 = 157.247384716 Reorder the terms: 78.2 + z = 157.247384716 Solving 78.2 + z = 157.247384716 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-78.2' to each side of the equation. 78.2 + -78.2 + z = 157.247384716 + -78.2 Combine like terms: 78.2 + -78.2 = 0.0 0.0 + z = 157.247384716 + -78.2 z = 157.247384716 + -78.2 Combine like terms: 157.247384716 + -78.2 = 79.047384716 z = 79.047384716 Simplifying z = 79.047384716Subproblem 2
z + 78.2 = -157.247384716 Simplifying z + 78.2 = -157.247384716 Reorder the terms: 78.2 + z = -157.247384716 Solving 78.2 + z = -157.247384716 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-78.2' to each side of the equation. 78.2 + -78.2 + z = -157.247384716 + -78.2 Combine like terms: 78.2 + -78.2 = 0.0 0.0 + z = -157.247384716 + -78.2 z = -157.247384716 + -78.2 Combine like terms: -157.247384716 + -78.2 = -235.447384716 z = -235.447384716 Simplifying z = -235.447384716Solution
The solution to the problem is based on the solutions from the subproblems. z = {79.047384716, -235.447384716}
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