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Simplifying z2 + -23z + 84 = 0 Reorder the terms: 84 + -23z + z2 = 0 Solving 84 + -23z + z2 = 0 Solving for variable 'z'. Begin completing the square. Move the constant term to the right: Add '-84' to each side of the equation. 84 + -23z + -84 + z2 = 0 + -84 Reorder the terms: 84 + -84 + -23z + z2 = 0 + -84 Combine like terms: 84 + -84 = 0 0 + -23z + z2 = 0 + -84 -23z + z2 = 0 + -84 Combine like terms: 0 + -84 = -84 -23z + z2 = -84 The z term is -23z. Take half its coefficient (-11.5). Square it (132.25) and add it to both sides. Add '132.25' to each side of the equation. -23z + 132.25 + z2 = -84 + 132.25 Reorder the terms: 132.25 + -23z + z2 = -84 + 132.25 Combine like terms: -84 + 132.25 = 48.25 132.25 + -23z + z2 = 48.25 Factor a perfect square on the left side: (z + -11.5)(z + -11.5) = 48.25 Calculate the square root of the right side: 6.946221995 Break this problem into two subproblems by setting (z + -11.5) equal to 6.946221995 and -6.946221995.Subproblem 1
z + -11.5 = 6.946221995 Simplifying z + -11.5 = 6.946221995 Reorder the terms: -11.5 + z = 6.946221995 Solving -11.5 + z = 6.946221995 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '11.5' to each side of the equation. -11.5 + 11.5 + z = 6.946221995 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + z = 6.946221995 + 11.5 z = 6.946221995 + 11.5 Combine like terms: 6.946221995 + 11.5 = 18.446221995 z = 18.446221995 Simplifying z = 18.446221995Subproblem 2
z + -11.5 = -6.946221995 Simplifying z + -11.5 = -6.946221995 Reorder the terms: -11.5 + z = -6.946221995 Solving -11.5 + z = -6.946221995 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '11.5' to each side of the equation. -11.5 + 11.5 + z = -6.946221995 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + z = -6.946221995 + 11.5 z = -6.946221995 + 11.5 Combine like terms: -6.946221995 + 11.5 = 4.553778005 z = 4.553778005 Simplifying z = 4.553778005Solution
The solution to the problem is based on the solutions from the subproblems. z = {18.446221995, 4.553778005}
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