z^2+2z=74

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Solution for z^2+2z=74 equation:


Simplifying
z2 + 2z = 74

Reorder the terms:
2z + z2 = 74

Solving
2z + z2 = 74

Solving for variable 'z'.

Reorder the terms:
-74 + 2z + z2 = 74 + -74

Combine like terms: 74 + -74 = 0
-74 + 2z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '74' to each side of the equation.
-74 + 2z + 74 + z2 = 0 + 74

Reorder the terms:
-74 + 74 + 2z + z2 = 0 + 74

Combine like terms: -74 + 74 = 0
0 + 2z + z2 = 0 + 74
2z + z2 = 0 + 74

Combine like terms: 0 + 74 = 74
2z + z2 = 74

The z term is 2z.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2z + 1 + z2 = 74 + 1

Reorder the terms:
1 + 2z + z2 = 74 + 1

Combine like terms: 74 + 1 = 75
1 + 2z + z2 = 75

Factor a perfect square on the left side:
(z + 1)(z + 1) = 75

Calculate the square root of the right side: 8.660254038

Break this problem into two subproblems by setting 
(z + 1) equal to 8.660254038 and -8.660254038.

Subproblem 1

z + 1 = 8.660254038 Simplifying z + 1 = 8.660254038 Reorder the terms: 1 + z = 8.660254038 Solving 1 + z = 8.660254038 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + z = 8.660254038 + -1 Combine like terms: 1 + -1 = 0 0 + z = 8.660254038 + -1 z = 8.660254038 + -1 Combine like terms: 8.660254038 + -1 = 7.660254038 z = 7.660254038 Simplifying z = 7.660254038

Subproblem 2

z + 1 = -8.660254038 Simplifying z + 1 = -8.660254038 Reorder the terms: 1 + z = -8.660254038 Solving 1 + z = -8.660254038 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + z = -8.660254038 + -1 Combine like terms: 1 + -1 = 0 0 + z = -8.660254038 + -1 z = -8.660254038 + -1 Combine like terms: -8.660254038 + -1 = -9.660254038 z = -9.660254038 Simplifying z = -9.660254038

Solution

The solution to the problem is based on the solutions from the subproblems. z = {7.660254038, -9.660254038}

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