z^2+5z=4

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Solution for z^2+5z=4 equation:


Simplifying
z2 + 5z = 4

Reorder the terms:
5z + z2 = 4

Solving
5z + z2 = 4

Solving for variable 'z'.

Reorder the terms:
-4 + 5z + z2 = 4 + -4

Combine like terms: 4 + -4 = 0
-4 + 5z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + 5z + 4 + z2 = 0 + 4

Reorder the terms:
-4 + 4 + 5z + z2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + 5z + z2 = 0 + 4
5z + z2 = 0 + 4

Combine like terms: 0 + 4 = 4
5z + z2 = 4

The z term is 5z.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5z + 6.25 + z2 = 4 + 6.25

Reorder the terms:
6.25 + 5z + z2 = 4 + 6.25

Combine like terms: 4 + 6.25 = 10.25
6.25 + 5z + z2 = 10.25

Factor a perfect square on the left side:
(z + 2.5)(z + 2.5) = 10.25

Calculate the square root of the right side: 3.201562119

Break this problem into two subproblems by setting 
(z + 2.5) equal to 3.201562119 and -3.201562119.

Subproblem 1

z + 2.5 = 3.201562119 Simplifying z + 2.5 = 3.201562119 Reorder the terms: 2.5 + z = 3.201562119 Solving 2.5 + z = 3.201562119 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + z = 3.201562119 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + z = 3.201562119 + -2.5 z = 3.201562119 + -2.5 Combine like terms: 3.201562119 + -2.5 = 0.701562119 z = 0.701562119 Simplifying z = 0.701562119

Subproblem 2

z + 2.5 = -3.201562119 Simplifying z + 2.5 = -3.201562119 Reorder the terms: 2.5 + z = -3.201562119 Solving 2.5 + z = -3.201562119 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + z = -3.201562119 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + z = -3.201562119 + -2.5 z = -3.201562119 + -2.5 Combine like terms: -3.201562119 + -2.5 = -5.701562119 z = -5.701562119 Simplifying z = -5.701562119

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.701562119, -5.701562119}

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