z^2-14=z+7

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Solution for z^2-14=z+7 equation:


Simplifying
z2 + -14 = z + 7

Reorder the terms:
-14 + z2 = z + 7

Reorder the terms:
-14 + z2 = 7 + z

Solving
-14 + z2 = 7 + z

Solving for variable 'z'.

Reorder the terms:
-14 + -7 + -1z + z2 = 7 + z + -7 + -1z

Combine like terms: -14 + -7 = -21
-21 + -1z + z2 = 7 + z + -7 + -1z

Reorder the terms:
-21 + -1z + z2 = 7 + -7 + z + -1z

Combine like terms: 7 + -7 = 0
-21 + -1z + z2 = 0 + z + -1z
-21 + -1z + z2 = z + -1z

Combine like terms: z + -1z = 0
-21 + -1z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + -1z + 21 + z2 = 0 + 21

Reorder the terms:
-21 + 21 + -1z + z2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + -1z + z2 = 0 + 21
-1z + z2 = 0 + 21

Combine like terms: 0 + 21 = 21
-1z + z2 = 21

The z term is -1z.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1z + 0.25 + z2 = 21 + 0.25

Reorder the terms:
0.25 + -1z + z2 = 21 + 0.25

Combine like terms: 21 + 0.25 = 21.25
0.25 + -1z + z2 = 21.25

Factor a perfect square on the left side:
(z + -0.5)(z + -0.5) = 21.25

Calculate the square root of the right side: 4.609772229

Break this problem into two subproblems by setting 
(z + -0.5) equal to 4.609772229 and -4.609772229.

Subproblem 1

z + -0.5 = 4.609772229 Simplifying z + -0.5 = 4.609772229 Reorder the terms: -0.5 + z = 4.609772229 Solving -0.5 + z = 4.609772229 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + z = 4.609772229 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z = 4.609772229 + 0.5 z = 4.609772229 + 0.5 Combine like terms: 4.609772229 + 0.5 = 5.109772229 z = 5.109772229 Simplifying z = 5.109772229

Subproblem 2

z + -0.5 = -4.609772229 Simplifying z + -0.5 = -4.609772229 Reorder the terms: -0.5 + z = -4.609772229 Solving -0.5 + z = -4.609772229 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + z = -4.609772229 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z = -4.609772229 + 0.5 z = -4.609772229 + 0.5 Combine like terms: -4.609772229 + 0.5 = -4.109772229 z = -4.109772229 Simplifying z = -4.109772229

Solution

The solution to the problem is based on the solutions from the subproblems. z = {5.109772229, -4.109772229}

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