2(z+4)z=5

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Solution for 2(z+4)z=5 equation:


Simplifying
2(z + 4) * z = 5

Reorder the terms:
2(4 + z) * z = 5

Reorder the terms for easier multiplication:
2z(4 + z) = 5
(4 * 2z + z * 2z) = 5
(8z + 2z2) = 5

Solving
8z + 2z2 = 5

Solving for variable 'z'.

Reorder the terms:
-5 + 8z + 2z2 = 5 + -5

Combine like terms: 5 + -5 = 0
-5 + 8z + 2z2 = 0

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

Divide each side by '2'.
-2.5 + 4z + z2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + 4z + 2.5 + z2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + 4z + z2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + 4z + z2 = 0 + 2.5
4z + z2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
4z + z2 = 2.5

The z term is 4z.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4z + 4 + z2 = 2.5 + 4

Reorder the terms:
4 + 4z + z2 = 2.5 + 4

Combine like terms: 2.5 + 4 = 6.5
4 + 4z + z2 = 6.5

Factor a perfect square on the left side:
(z + 2)(z + 2) = 6.5

Calculate the square root of the right side: 2.549509757

Break this problem into two subproblems by setting 
(z + 2) equal to 2.549509757 and -2.549509757.

Subproblem 1

z + 2 = 2.549509757 Simplifying z + 2 = 2.549509757 Reorder the terms: 2 + z = 2.549509757 Solving 2 + z = 2.549509757 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + z = 2.549509757 + -2 Combine like terms: 2 + -2 = 0 0 + z = 2.549509757 + -2 z = 2.549509757 + -2 Combine like terms: 2.549509757 + -2 = 0.549509757 z = 0.549509757 Simplifying z = 0.549509757

Subproblem 2

z + 2 = -2.549509757 Simplifying z + 2 = -2.549509757 Reorder the terms: 2 + z = -2.549509757 Solving 2 + z = -2.549509757 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + z = -2.549509757 + -2 Combine like terms: 2 + -2 = 0 0 + z = -2.549509757 + -2 z = -2.549509757 + -2 Combine like terms: -2.549509757 + -2 = -4.549509757 z = -4.549509757 Simplifying z = -4.549509757

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.549509757, -4.549509757}

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