3z(z+7)=21

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Solution for 3z(z+7)=21 equation: 


Simplifying
3z(z + 7) = 21

Reorder the terms:
3z(7 + z) = 21
(7 * 3z + z * 3z) = 21
(21z + 3z2) = 21

Solving
21z + 3z2 = 21

Solving for variable 'z'.

Reorder the terms:
-21 + 21z + 3z2 = 21 + -21

Combine like terms: 21 + -21 = 0
-21 + 21z + 3z2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-7 + 7z + z2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-7 + 7z + z2)' equal to zero and attempt to solve:

Simplifying
-7 + 7z + z2 = 0

Solving
-7 + 7z + z2 = 0

Begin completing the square.

Move the constant term to the right:

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

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

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

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

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 = 7 + 12.25

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

Combine like terms: 7 + 12.25 = 19.25
12.25 + 7z + z2 = 19.25

Factor a perfect square on the left side:
(z + 3.5)(z + 3.5) = 19.25

Calculate the square root of the right side: 4.387482194

Break this problem into two subproblems by setting 
(z + 3.5) equal to 4.387482194 and -4.387482194.


Subproblem 1
z + 3.5 = 4.387482194

Simplifying
z + 3.5 = 4.387482194

Reorder the terms:
3.5 + z = 4.387482194

Solving
3.5 + z = 4.387482194

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 = 4.387482194 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + z = 4.387482194 + -3.5
z = 4.387482194 + -3.5

Combine like terms: 4.387482194 + -3.5 = 0.887482194
z = 0.887482194

Simplifying
z = 0.887482194


Subproblem 2
z + 3.5 = -4.387482194

Simplifying
z + 3.5 = -4.387482194

Reorder the terms:
3.5 + z = -4.387482194

Solving
3.5 + z = -4.387482194

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 = -4.387482194 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + z = -4.387482194 + -3.5
z = -4.387482194 + -3.5

Combine like terms: -4.387482194 + -3.5 = -7.887482194
z = -7.887482194

Simplifying
z = -7.887482194


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.887482194, -7.887482194}
Solution
z = {0.887482194, -7.887482194}

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