2z^2+z=13

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Solution for 2z^2+z=13 equation:


Simplifying
2z2 + z = 13

Reorder the terms:
z + 2z2 = 13

Solving
z + 2z2 = 13

Solving for variable 'z'.

Reorder the terms:
-13 + z + 2z2 = 13 + -13

Combine like terms: 13 + -13 = 0
-13 + z + 2z2 = 0

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

Divide each side by '2'.
-6.5 + 0.5z + z2 = 0

Move the constant term to the right:

Add '6.5' to each side of the equation.
-6.5 + 0.5z + 6.5 + z2 = 0 + 6.5

Reorder the terms:
-6.5 + 6.5 + 0.5z + z2 = 0 + 6.5

Combine like terms: -6.5 + 6.5 = 0.0
0.0 + 0.5z + z2 = 0 + 6.5
0.5z + z2 = 0 + 6.5

Combine like terms: 0 + 6.5 = 6.5
0.5z + z2 = 6.5

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.
0.5z + 0.25 + z2 = 6.5 + 0.25

Reorder the terms:
0.25 + 0.5z + z2 = 6.5 + 0.25

Combine like terms: 6.5 + 0.25 = 6.75
0.25 + 0.5z + z2 = 6.75

Factor a perfect square on the left side:
(z + 0.5)(z + 0.5) = 6.75

Calculate the square root of the right side: 2.598076211

Break this problem into two subproblems by setting 
(z + 0.5) equal to 2.598076211 and -2.598076211.

Subproblem 1

z + 0.5 = 2.598076211 Simplifying z + 0.5 = 2.598076211 Reorder the terms: 0.5 + z = 2.598076211 Solving 0.5 + z = 2.598076211 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 = 2.598076211 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = 2.598076211 + -0.5 z = 2.598076211 + -0.5 Combine like terms: 2.598076211 + -0.5 = 2.098076211 z = 2.098076211 Simplifying z = 2.098076211

Subproblem 2

z + 0.5 = -2.598076211 Simplifying z + 0.5 = -2.598076211 Reorder the terms: 0.5 + z = -2.598076211 Solving 0.5 + z = -2.598076211 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 = -2.598076211 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = -2.598076211 + -0.5 z = -2.598076211 + -0.5 Combine like terms: -2.598076211 + -0.5 = -3.098076211 z = -3.098076211 Simplifying z = -3.098076211

Solution

The solution to the problem is based on the solutions from the subproblems. z = {2.098076211, -3.098076211}

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