48=12z+z^2

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Solution for 48=12z+z^2 equation:


Simplifying
48 = 12z + z2

Solving
48 = 12z + z2

Solving for variable 'z'.

Reorder the terms:
48 + -12z + -1z2 = 12z + -12z + z2 + -1z2

Combine like terms: 12z + -12z = 0
48 + -12z + -1z2 = 0 + z2 + -1z2
48 + -12z + -1z2 = z2 + -1z2

Combine like terms: z2 + -1z2 = 0
48 + -12z + -1z2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-48 + 12z + z2 = 0

Move the constant term to the right:

Add '48' to each side of the equation.
-48 + 12z + 48 + z2 = 0 + 48

Reorder the terms:
-48 + 48 + 12z + z2 = 0 + 48

Combine like terms: -48 + 48 = 0
0 + 12z + z2 = 0 + 48
12z + z2 = 0 + 48

Combine like terms: 0 + 48 = 48
12z + z2 = 48

The z term is 12z.  Take half its coefficient (6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
12z + 36 + z2 = 48 + 36

Reorder the terms:
36 + 12z + z2 = 48 + 36

Combine like terms: 48 + 36 = 84
36 + 12z + z2 = 84

Factor a perfect square on the left side:
(z + 6)(z + 6) = 84

Calculate the square root of the right side: 9.16515139

Break this problem into two subproblems by setting 
(z + 6) equal to 9.16515139 and -9.16515139.

Subproblem 1

z + 6 = 9.16515139 Simplifying z + 6 = 9.16515139 Reorder the terms: 6 + z = 9.16515139 Solving 6 + z = 9.16515139 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + z = 9.16515139 + -6 Combine like terms: 6 + -6 = 0 0 + z = 9.16515139 + -6 z = 9.16515139 + -6 Combine like terms: 9.16515139 + -6 = 3.16515139 z = 3.16515139 Simplifying z = 3.16515139

Subproblem 2

z + 6 = -9.16515139 Simplifying z + 6 = -9.16515139 Reorder the terms: 6 + z = -9.16515139 Solving 6 + z = -9.16515139 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + z = -9.16515139 + -6 Combine like terms: 6 + -6 = 0 0 + z = -9.16515139 + -6 z = -9.16515139 + -6 Combine like terms: -9.16515139 + -6 = -15.16515139 z = -15.16515139 Simplifying z = -15.16515139

Solution

The solution to the problem is based on the solutions from the subproblems. z = {3.16515139, -15.16515139}

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