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Simplifying z2 + -19z + 69 = 0 Reorder the terms: 69 + -19z + z2 = 0 Solving 69 + -19z + z2 = 0 Solving for variable 'z'. Begin completing the square. Move the constant term to the right: Add '-69' to each side of the equation. 69 + -19z + -69 + z2 = 0 + -69 Reorder the terms: 69 + -69 + -19z + z2 = 0 + -69 Combine like terms: 69 + -69 = 0 0 + -19z + z2 = 0 + -69 -19z + z2 = 0 + -69 Combine like terms: 0 + -69 = -69 -19z + z2 = -69 The z term is -19z. Take half its coefficient (-9.5). Square it (90.25) and add it to both sides. Add '90.25' to each side of the equation. -19z + 90.25 + z2 = -69 + 90.25 Reorder the terms: 90.25 + -19z + z2 = -69 + 90.25 Combine like terms: -69 + 90.25 = 21.25 90.25 + -19z + z2 = 21.25 Factor a perfect square on the left side: (z + -9.5)(z + -9.5) = 21.25 Calculate the square root of the right side: 4.609772229 Break this problem into two subproblems by setting (z + -9.5) equal to 4.609772229 and -4.609772229.Subproblem 1
z + -9.5 = 4.609772229 Simplifying z + -9.5 = 4.609772229 Reorder the terms: -9.5 + z = 4.609772229 Solving -9.5 + z = 4.609772229 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '9.5' to each side of the equation. -9.5 + 9.5 + z = 4.609772229 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + z = 4.609772229 + 9.5 z = 4.609772229 + 9.5 Combine like terms: 4.609772229 + 9.5 = 14.109772229 z = 14.109772229 Simplifying z = 14.109772229Subproblem 2
z + -9.5 = -4.609772229 Simplifying z + -9.5 = -4.609772229 Reorder the terms: -9.5 + z = -4.609772229 Solving -9.5 + z = -4.609772229 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '9.5' to each side of the equation. -9.5 + 9.5 + z = -4.609772229 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + z = -4.609772229 + 9.5 z = -4.609772229 + 9.5 Combine like terms: -4.609772229 + 9.5 = 4.890227771 z = 4.890227771 Simplifying z = 4.890227771Solution
The solution to the problem is based on the solutions from the subproblems. z = {14.109772229, 4.890227771}
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