z^2-7z=1

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Solution for z^2-7z=1 equation:


Simplifying
z2 + -7z = 1

Reorder the terms:
-7z + z2 = 1

Solving
-7z + z2 = 1

Solving for variable 'z'.

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

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

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -7z + 1 + z2 = 0 + 1

Reorder the terms:
-1 + 1 + -7z + z2 = 0 + 1

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

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

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 = 1 + 12.25

Reorder the terms:
12.25 + -7z + z2 = 1 + 12.25

Combine like terms: 1 + 12.25 = 13.25
12.25 + -7z + z2 = 13.25

Factor a perfect square on the left side:
(z + -3.5)(z + -3.5) = 13.25

Calculate the square root of the right side: 3.640054945

Break this problem into two subproblems by setting 
(z + -3.5) equal to 3.640054945 and -3.640054945.

Subproblem 1

z + -3.5 = 3.640054945 Simplifying z + -3.5 = 3.640054945 Reorder the terms: -3.5 + z = 3.640054945 Solving -3.5 + z = 3.640054945 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 = 3.640054945 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + z = 3.640054945 + 3.5 z = 3.640054945 + 3.5 Combine like terms: 3.640054945 + 3.5 = 7.140054945 z = 7.140054945 Simplifying z = 7.140054945

Subproblem 2

z + -3.5 = -3.640054945 Simplifying z + -3.5 = -3.640054945 Reorder the terms: -3.5 + z = -3.640054945 Solving -3.5 + z = -3.640054945 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 = -3.640054945 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + z = -3.640054945 + 3.5 z = -3.640054945 + 3.5 Combine like terms: -3.640054945 + 3.5 = -0.140054945 z = -0.140054945 Simplifying z = -0.140054945

Solution

The solution to the problem is based on the solutions from the subproblems. z = {7.140054945, -0.140054945}

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