8z^2+z-2=0

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


Simplifying
8z2 + z + -2 = 0

Reorder the terms:
-2 + z + 8z2 = 0

Solving
-2 + z + 8z2 = 0

Solving for variable 'z'.

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

Divide each side by '8'.
-0.25 + 0.125z + z2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + 0.125z + 0.25 + z2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + 0.125z + z2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + 0.125z + z2 = 0 + 0.25
0.125z + z2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
0.125z + z2 = 0.25

The z term is z.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.125z + 0.25 + z2 = 0.25 + 0.25

Reorder the terms:
0.25 + 0.125z + z2 = 0.25 + 0.25

Combine like terms: 0.25 + 0.25 = 0.5
0.25 + 0.125z + z2 = 0.5

Factor a perfect square on the left side:
(z + 0.5)(z + 0.5) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(z + 0.5) equal to 0.707106781 and -0.707106781.

Subproblem 1

z + 0.5 = 0.707106781 Simplifying z + 0.5 = 0.707106781 Reorder the terms: 0.5 + z = 0.707106781 Solving 0.5 + z = 0.707106781 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = 0.707106781 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = 0.707106781 + -0.5 z = 0.707106781 + -0.5 Combine like terms: 0.707106781 + -0.5 = 0.207106781 z = 0.207106781 Simplifying z = 0.207106781

Subproblem 2

z + 0.5 = -0.707106781 Simplifying z + 0.5 = -0.707106781 Reorder the terms: 0.5 + z = -0.707106781 Solving 0.5 + z = -0.707106781 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = -0.707106781 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = -0.707106781 + -0.5 z = -0.707106781 + -0.5 Combine like terms: -0.707106781 + -0.5 = -1.207106781 z = -1.207106781 Simplifying z = -1.207106781

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.207106781, -1.207106781}

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