z^2-19z+69=0

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Solution for z^2-19z+69=0 equation:


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 = 14.109772229

Subproblem 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 = 4.890227771

Solution

The solution to the problem is based on the solutions from the subproblems. z = {14.109772229, 4.890227771}

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