z^2-8z=-13

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Solution for z^2-8z=-13 equation:


Simplifying
z2 + -8z = -13

Reorder the terms:
-8z + z2 = -13

Solving
-8z + z2 = -13

Solving for variable 'z'.

Reorder the terms:
13 + -8z + z2 = -13 + 13

Combine like terms: -13 + 13 = 0
13 + -8z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-13' to each side of the equation.
13 + -8z + -13 + z2 = 0 + -13

Reorder the terms:
13 + -13 + -8z + z2 = 0 + -13

Combine like terms: 13 + -13 = 0
0 + -8z + z2 = 0 + -13
-8z + z2 = 0 + -13

Combine like terms: 0 + -13 = -13
-8z + z2 = -13

The z term is -8z.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8z + 16 + z2 = -13 + 16

Reorder the terms:
16 + -8z + z2 = -13 + 16

Combine like terms: -13 + 16 = 3
16 + -8z + z2 = 3

Factor a perfect square on the left side:
(z + -4)(z + -4) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(z + -4) equal to 1.732050808 and -1.732050808.

Subproblem 1

z + -4 = 1.732050808 Simplifying z + -4 = 1.732050808 Reorder the terms: -4 + z = 1.732050808 Solving -4 + z = 1.732050808 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + z = 1.732050808 + 4 Combine like terms: -4 + 4 = 0 0 + z = 1.732050808 + 4 z = 1.732050808 + 4 Combine like terms: 1.732050808 + 4 = 5.732050808 z = 5.732050808 Simplifying z = 5.732050808

Subproblem 2

z + -4 = -1.732050808 Simplifying z + -4 = -1.732050808 Reorder the terms: -4 + z = -1.732050808 Solving -4 + z = -1.732050808 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + z = -1.732050808 + 4 Combine like terms: -4 + 4 = 0 0 + z = -1.732050808 + 4 z = -1.732050808 + 4 Combine like terms: -1.732050808 + 4 = 2.267949192 z = 2.267949192 Simplifying z = 2.267949192

Solution

The solution to the problem is based on the solutions from the subproblems. z = {5.732050808, 2.267949192}

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