z^2-14z=-19

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Solution for z^2-14z=-19 equation:


Simplifying
z2 + -14z = -19

Reorder the terms:
-14z + z2 = -19

Solving
-14z + z2 = -19

Solving for variable 'z'.

Reorder the terms:
19 + -14z + z2 = -19 + 19

Combine like terms: -19 + 19 = 0
19 + -14z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-19' to each side of the equation.
19 + -14z + -19 + z2 = 0 + -19

Reorder the terms:
19 + -19 + -14z + z2 = 0 + -19

Combine like terms: 19 + -19 = 0
0 + -14z + z2 = 0 + -19
-14z + z2 = 0 + -19

Combine like terms: 0 + -19 = -19
-14z + z2 = -19

The z term is -14z.  Take half its coefficient (-7).
Square it (49) and add it to both sides.

Add '49' to each side of the equation.
-14z + 49 + z2 = -19 + 49

Reorder the terms:
49 + -14z + z2 = -19 + 49

Combine like terms: -19 + 49 = 30
49 + -14z + z2 = 30

Factor a perfect square on the left side:
(z + -7)(z + -7) = 30

Calculate the square root of the right side: 5.477225575

Break this problem into two subproblems by setting 
(z + -7) equal to 5.477225575 and -5.477225575.

Subproblem 1

z + -7 = 5.477225575 Simplifying z + -7 = 5.477225575 Reorder the terms: -7 + z = 5.477225575 Solving -7 + z = 5.477225575 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + z = 5.477225575 + 7 Combine like terms: -7 + 7 = 0 0 + z = 5.477225575 + 7 z = 5.477225575 + 7 Combine like terms: 5.477225575 + 7 = 12.477225575 z = 12.477225575 Simplifying z = 12.477225575

Subproblem 2

z + -7 = -5.477225575 Simplifying z + -7 = -5.477225575 Reorder the terms: -7 + z = -5.477225575 Solving -7 + z = -5.477225575 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + z = -5.477225575 + 7 Combine like terms: -7 + 7 = 0 0 + z = -5.477225575 + 7 z = -5.477225575 + 7 Combine like terms: -5.477225575 + 7 = 1.522774425 z = 1.522774425 Simplifying z = 1.522774425

Solution

The solution to the problem is based on the solutions from the subproblems. z = {12.477225575, 1.522774425}

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