z^2-6z+7z-48=0

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Solution for z^2-6z+7z-48=0 equation:


Simplifying
z2 + -6z + 7z + -48 = 0

Reorder the terms:
-48 + -6z + 7z + z2 = 0

Combine like terms: -6z + 7z = 1z
-48 + 1z + z2 = 0

Solving
-48 + 1z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '48' to each side of the equation.
-48 + 1z + 48 + z2 = 0 + 48

Reorder the terms:
-48 + 48 + 1z + z2 = 0 + 48

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

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

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.
z + 0.25 + z2 = 48 + 0.25

Reorder the terms:
0.25 + z + z2 = 48 + 0.25

Combine like terms: 48 + 0.25 = 48.25
0.25 + z + z2 = 48.25

Factor a perfect square on the left side:
(z + 0.5)(z + 0.5) = 48.25

Calculate the square root of the right side: 6.946221995

Break this problem into two subproblems by setting 
(z + 0.5) equal to 6.946221995 and -6.946221995.

Subproblem 1

z + 0.5 = 6.946221995 Simplifying z + 0.5 = 6.946221995 Reorder the terms: 0.5 + z = 6.946221995 Solving 0.5 + z = 6.946221995 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 = 6.946221995 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = 6.946221995 + -0.5 z = 6.946221995 + -0.5 Combine like terms: 6.946221995 + -0.5 = 6.446221995 z = 6.446221995 Simplifying z = 6.446221995

Subproblem 2

z + 0.5 = -6.946221995 Simplifying z + 0.5 = -6.946221995 Reorder the terms: 0.5 + z = -6.946221995 Solving 0.5 + z = -6.946221995 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 = -6.946221995 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = -6.946221995 + -0.5 z = -6.946221995 + -0.5 Combine like terms: -6.946221995 + -0.5 = -7.446221995 z = -7.446221995 Simplifying z = -7.446221995

Solution

The solution to the problem is based on the solutions from the subproblems. z = {6.446221995, -7.446221995}

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