z^2-13z=32

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Solution for z^2-13z=32 equation:


Simplifying
z2 + -13z = 32

Reorder the terms:
-13z + z2 = 32

Solving
-13z + z2 = 32

Solving for variable 'z'.

Reorder the terms:
-32 + -13z + z2 = 32 + -32

Combine like terms: 32 + -32 = 0
-32 + -13z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '32' to each side of the equation.
-32 + -13z + 32 + z2 = 0 + 32

Reorder the terms:
-32 + 32 + -13z + z2 = 0 + 32

Combine like terms: -32 + 32 = 0
0 + -13z + z2 = 0 + 32
-13z + z2 = 0 + 32

Combine like terms: 0 + 32 = 32
-13z + z2 = 32

The z term is -13z.  Take half its coefficient (-6.5).
Square it (42.25) and add it to both sides.

Add '42.25' to each side of the equation.
-13z + 42.25 + z2 = 32 + 42.25

Reorder the terms:
42.25 + -13z + z2 = 32 + 42.25

Combine like terms: 32 + 42.25 = 74.25
42.25 + -13z + z2 = 74.25

Factor a perfect square on the left side:
(z + -6.5)(z + -6.5) = 74.25

Calculate the square root of the right side: 8.61684397

Break this problem into two subproblems by setting 
(z + -6.5) equal to 8.61684397 and -8.61684397.

Subproblem 1

z + -6.5 = 8.61684397 Simplifying z + -6.5 = 8.61684397 Reorder the terms: -6.5 + z = 8.61684397 Solving -6.5 + z = 8.61684397 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + z = 8.61684397 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + z = 8.61684397 + 6.5 z = 8.61684397 + 6.5 Combine like terms: 8.61684397 + 6.5 = 15.11684397 z = 15.11684397 Simplifying z = 15.11684397

Subproblem 2

z + -6.5 = -8.61684397 Simplifying z + -6.5 = -8.61684397 Reorder the terms: -6.5 + z = -8.61684397 Solving -6.5 + z = -8.61684397 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '6.5' to each side of the equation. -6.5 + 6.5 + z = -8.61684397 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + z = -8.61684397 + 6.5 z = -8.61684397 + 6.5 Combine like terms: -8.61684397 + 6.5 = -2.11684397 z = -2.11684397 Simplifying z = -2.11684397

Solution

The solution to the problem is based on the solutions from the subproblems. z = {15.11684397, -2.11684397}

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