z^2-7z-30=z

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Solution for z^2-7z-30=z equation:


Simplifying
z2 + -7z + -30 = z

Reorder the terms:
-30 + -7z + z2 = z

Solving
-30 + -7z + z2 = z

Solving for variable 'z'.

Reorder the terms:
-30 + -7z + -1z + z2 = z + -1z

Combine like terms: -7z + -1z = -8z
-30 + -8z + z2 = z + -1z

Combine like terms: z + -1z = 0
-30 + -8z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '30' to each side of the equation.
-30 + -8z + 30 + z2 = 0 + 30

Reorder the terms:
-30 + 30 + -8z + z2 = 0 + 30

Combine like terms: -30 + 30 = 0
0 + -8z + z2 = 0 + 30
-8z + z2 = 0 + 30

Combine like terms: 0 + 30 = 30
-8z + z2 = 30

The z term is -8z.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8z + 16 + z2 = 30 + 16

Reorder the terms:
16 + -8z + z2 = 30 + 16

Combine like terms: 30 + 16 = 46
16 + -8z + z2 = 46

Factor a perfect square on the left side:
(z + -4)(z + -4) = 46

Calculate the square root of the right side: 6.782329983

Break this problem into two subproblems by setting 
(z + -4) equal to 6.782329983 and -6.782329983.

Subproblem 1

z + -4 = 6.782329983 Simplifying z + -4 = 6.782329983 Reorder the terms: -4 + z = 6.782329983 Solving -4 + z = 6.782329983 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + z = 6.782329983 + 4 Combine like terms: -4 + 4 = 0 0 + z = 6.782329983 + 4 z = 6.782329983 + 4 Combine like terms: 6.782329983 + 4 = 10.782329983 z = 10.782329983 Simplifying z = 10.782329983

Subproblem 2

z + -4 = -6.782329983 Simplifying z + -4 = -6.782329983 Reorder the terms: -4 + z = -6.782329983 Solving -4 + z = -6.782329983 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + z = -6.782329983 + 4 Combine like terms: -4 + 4 = 0 0 + z = -6.782329983 + 4 z = -6.782329983 + 4 Combine like terms: -6.782329983 + 4 = -2.782329983 z = -2.782329983 Simplifying z = -2.782329983

Solution

The solution to the problem is based on the solutions from the subproblems. z = {10.782329983, -2.782329983}

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