z^2+14z=-31

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Solution for z^2+14z=-31 equation:


Simplifying
z2 + 14z = -31

Reorder the terms:
14z + z2 = -31

Solving
14z + z2 = -31

Solving for variable 'z'.

Reorder the terms:
31 + 14z + z2 = -31 + 31

Combine like terms: -31 + 31 = 0
31 + 14z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-31' to each side of the equation.
31 + 14z + -31 + z2 = 0 + -31

Reorder the terms:
31 + -31 + 14z + z2 = 0 + -31

Combine like terms: 31 + -31 = 0
0 + 14z + z2 = 0 + -31
14z + z2 = 0 + -31

Combine like terms: 0 + -31 = -31
14z + z2 = -31

The z term is 14z.  Take half its coefficient (7).
Square it (49) and add it to both sides.

Add '49' to each side of the equation.
14z + 49 + z2 = -31 + 49

Reorder the terms:
49 + 14z + z2 = -31 + 49

Combine like terms: -31 + 49 = 18
49 + 14z + z2 = 18

Factor a perfect square on the left side:
(z + 7)(z + 7) = 18

Calculate the square root of the right side: 4.242640687

Break this problem into two subproblems by setting 
(z + 7) equal to 4.242640687 and -4.242640687.

Subproblem 1

z + 7 = 4.242640687 Simplifying z + 7 = 4.242640687 Reorder the terms: 7 + z = 4.242640687 Solving 7 + z = 4.242640687 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + z = 4.242640687 + -7 Combine like terms: 7 + -7 = 0 0 + z = 4.242640687 + -7 z = 4.242640687 + -7 Combine like terms: 4.242640687 + -7 = -2.757359313 z = -2.757359313 Simplifying z = -2.757359313

Subproblem 2

z + 7 = -4.242640687 Simplifying z + 7 = -4.242640687 Reorder the terms: 7 + z = -4.242640687 Solving 7 + z = -4.242640687 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + z = -4.242640687 + -7 Combine like terms: 7 + -7 = 0 0 + z = -4.242640687 + -7 z = -4.242640687 + -7 Combine like terms: -4.242640687 + -7 = -11.242640687 z = -11.242640687 Simplifying z = -11.242640687

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-2.757359313, -11.242640687}

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