z^2-16z=-19

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Solution for z^2-16z=-19 equation:


Simplifying
z2 + -16z = -19

Reorder the terms:
-16z + z2 = -19

Solving
-16z + z2 = -19

Solving for variable 'z'.

Reorder the terms:
19 + -16z + z2 = -19 + 19

Combine like terms: -19 + 19 = 0
19 + -16z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-19' to each side of the equation.
19 + -16z + -19 + z2 = 0 + -19

Reorder the terms:
19 + -19 + -16z + z2 = 0 + -19

Combine like terms: 19 + -19 = 0
0 + -16z + z2 = 0 + -19
-16z + z2 = 0 + -19

Combine like terms: 0 + -19 = -19
-16z + z2 = -19

The z term is -16z.  Take half its coefficient (-8).
Square it (64) and add it to both sides.

Add '64' to each side of the equation.
-16z + 64 + z2 = -19 + 64

Reorder the terms:
64 + -16z + z2 = -19 + 64

Combine like terms: -19 + 64 = 45
64 + -16z + z2 = 45

Factor a perfect square on the left side:
(z + -8)(z + -8) = 45

Calculate the square root of the right side: 6.708203933

Break this problem into two subproblems by setting 
(z + -8) equal to 6.708203933 and -6.708203933.

Subproblem 1

z + -8 = 6.708203933 Simplifying z + -8 = 6.708203933 Reorder the terms: -8 + z = 6.708203933 Solving -8 + z = 6.708203933 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + z = 6.708203933 + 8 Combine like terms: -8 + 8 = 0 0 + z = 6.708203933 + 8 z = 6.708203933 + 8 Combine like terms: 6.708203933 + 8 = 14.708203933 z = 14.708203933 Simplifying z = 14.708203933

Subproblem 2

z + -8 = -6.708203933 Simplifying z + -8 = -6.708203933 Reorder the terms: -8 + z = -6.708203933 Solving -8 + z = -6.708203933 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + z = -6.708203933 + 8 Combine like terms: -8 + 8 = 0 0 + z = -6.708203933 + 8 z = -6.708203933 + 8 Combine like terms: -6.708203933 + 8 = 1.291796067 z = 1.291796067 Simplifying z = 1.291796067

Solution

The solution to the problem is based on the solutions from the subproblems. z = {14.708203933, 1.291796067}

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