z^2+7z=4

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Solution for z^2+7z=4 equation:


Simplifying
z2 + 7z = 4

Reorder the terms:
7z + z2 = 4

Solving
7z + z2 = 4

Solving for variable 'z'.

Reorder the terms:
-4 + 7z + z2 = 4 + -4

Combine like terms: 4 + -4 = 0
-4 + 7z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + 7z + 4 + z2 = 0 + 4

Reorder the terms:
-4 + 4 + 7z + z2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + 7z + z2 = 0 + 4
7z + z2 = 0 + 4

Combine like terms: 0 + 4 = 4
7z + z2 = 4

The z term is 7z.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7z + 12.25 + z2 = 4 + 12.25

Reorder the terms:
12.25 + 7z + z2 = 4 + 12.25

Combine like terms: 4 + 12.25 = 16.25
12.25 + 7z + z2 = 16.25

Factor a perfect square on the left side:
(z + 3.5)(z + 3.5) = 16.25

Calculate the square root of the right side: 4.031128874

Break this problem into two subproblems by setting 
(z + 3.5) equal to 4.031128874 and -4.031128874.

Subproblem 1

z + 3.5 = 4.031128874 Simplifying z + 3.5 = 4.031128874 Reorder the terms: 3.5 + z = 4.031128874 Solving 3.5 + z = 4.031128874 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + z = 4.031128874 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = 4.031128874 + -3.5 z = 4.031128874 + -3.5 Combine like terms: 4.031128874 + -3.5 = 0.531128874 z = 0.531128874 Simplifying z = 0.531128874

Subproblem 2

z + 3.5 = -4.031128874 Simplifying z + 3.5 = -4.031128874 Reorder the terms: 3.5 + z = -4.031128874 Solving 3.5 + z = -4.031128874 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + z = -4.031128874 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = -4.031128874 + -3.5 z = -4.031128874 + -3.5 Combine like terms: -4.031128874 + -3.5 = -7.531128874 z = -7.531128874 Simplifying z = -7.531128874

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.531128874, -7.531128874}

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