8z-z^2=3

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Solution for 8z-z^2=3 equation:


Simplifying
8z + -1z2 = 3

Solving
8z + -1z2 = 3

Solving for variable 'z'.

Reorder the terms:
-3 + 8z + -1z2 = 3 + -3

Combine like terms: 3 + -3 = 0
-3 + 8z + -1z2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
3 + -8z + z2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -8z + -3 + z2 = 0 + -3

Reorder the terms:
3 + -3 + -8z + z2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -8z + z2 = 0 + -3
-8z + z2 = 0 + -3

Combine like terms: 0 + -3 = -3
-8z + z2 = -3

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 = -3 + 16

Reorder the terms:
16 + -8z + z2 = -3 + 16

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

Factor a perfect square on the left side:
(z + -4)(z + -4) = 13

Calculate the square root of the right side: 3.605551275

Break this problem into two subproblems by setting 
(z + -4) equal to 3.605551275 and -3.605551275.

Subproblem 1

z + -4 = 3.605551275 Simplifying z + -4 = 3.605551275 Reorder the terms: -4 + z = 3.605551275 Solving -4 + z = 3.605551275 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 = 3.605551275 + 4 Combine like terms: -4 + 4 = 0 0 + z = 3.605551275 + 4 z = 3.605551275 + 4 Combine like terms: 3.605551275 + 4 = 7.605551275 z = 7.605551275 Simplifying z = 7.605551275

Subproblem 2

z + -4 = -3.605551275 Simplifying z + -4 = -3.605551275 Reorder the terms: -4 + z = -3.605551275 Solving -4 + z = -3.605551275 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 = -3.605551275 + 4 Combine like terms: -4 + 4 = 0 0 + z = -3.605551275 + 4 z = -3.605551275 + 4 Combine like terms: -3.605551275 + 4 = 0.394448725 z = 0.394448725 Simplifying z = 0.394448725

Solution

The solution to the problem is based on the solutions from the subproblems. z = {7.605551275, 0.394448725}

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