3z+z^2-8=0

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Solution for 3z+z^2-8=0 equation:


Simplifying
3z + z2 + -8 = 0

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

Solving
-8 + 3z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

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

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

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

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

The z term is 3z.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3z + 2.25 + z2 = 8 + 2.25

Reorder the terms:
2.25 + 3z + z2 = 8 + 2.25

Combine like terms: 8 + 2.25 = 10.25
2.25 + 3z + z2 = 10.25

Factor a perfect square on the left side:
(z + 1.5)(z + 1.5) = 10.25

Calculate the square root of the right side: 3.201562119

Break this problem into two subproblems by setting 
(z + 1.5) equal to 3.201562119 and -3.201562119.

Subproblem 1

z + 1.5 = 3.201562119 Simplifying z + 1.5 = 3.201562119 Reorder the terms: 1.5 + z = 3.201562119 Solving 1.5 + z = 3.201562119 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + z = 3.201562119 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + z = 3.201562119 + -1.5 z = 3.201562119 + -1.5 Combine like terms: 3.201562119 + -1.5 = 1.701562119 z = 1.701562119 Simplifying z = 1.701562119

Subproblem 2

z + 1.5 = -3.201562119 Simplifying z + 1.5 = -3.201562119 Reorder the terms: 1.5 + z = -3.201562119 Solving 1.5 + z = -3.201562119 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + z = -3.201562119 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + z = -3.201562119 + -1.5 z = -3.201562119 + -1.5 Combine like terms: -3.201562119 + -1.5 = -4.701562119 z = -4.701562119 Simplifying z = -4.701562119

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.701562119, -4.701562119}

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