z^2+7z=12

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Solution for z^2+7z=12 equation:


Simplifying
z2 + 7z = 12

Reorder the terms:
7z + z2 = 12

Solving
7z + z2 = 12

Solving for variable 'z'.

Reorder the terms:
-12 + 7z + z2 = 12 + -12

Combine like terms: 12 + -12 = 0
-12 + 7z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '12' to each side of the equation.
-12 + 7z + 12 + z2 = 0 + 12

Reorder the terms:
-12 + 12 + 7z + z2 = 0 + 12

Combine like terms: -12 + 12 = 0
0 + 7z + z2 = 0 + 12
7z + z2 = 0 + 12

Combine like terms: 0 + 12 = 12
7z + z2 = 12

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 = 12 + 12.25

Reorder the terms:
12.25 + 7z + z2 = 12 + 12.25

Combine like terms: 12 + 12.25 = 24.25
12.25 + 7z + z2 = 24.25

Factor a perfect square on the left side:
(z + 3.5)(z + 3.5) = 24.25

Calculate the square root of the right side: 4.924428901

Break this problem into two subproblems by setting 
(z + 3.5) equal to 4.924428901 and -4.924428901.

Subproblem 1

z + 3.5 = 4.924428901 Simplifying z + 3.5 = 4.924428901 Reorder the terms: 3.5 + z = 4.924428901 Solving 3.5 + z = 4.924428901 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 = 4.924428901 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = 4.924428901 + -3.5 z = 4.924428901 + -3.5 Combine like terms: 4.924428901 + -3.5 = 1.424428901 z = 1.424428901 Simplifying z = 1.424428901

Subproblem 2

z + 3.5 = -4.924428901 Simplifying z + 3.5 = -4.924428901 Reorder the terms: 3.5 + z = -4.924428901 Solving 3.5 + z = -4.924428901 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 = -4.924428901 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = -4.924428901 + -3.5 z = -4.924428901 + -3.5 Combine like terms: -4.924428901 + -3.5 = -8.424428901 z = -8.424428901 Simplifying z = -8.424428901

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.424428901, -8.424428901}

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