90*10*z=8100+100+z^2

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Solution for 90*10*z=8100+100+z^2 equation:


Simplifying
90 * 10z = 8100 + 100 + z2

Multiply 90 * 10
900z = 8100 + 100 + z2

Combine like terms: 8100 + 100 = 8200
900z = 8200 + z2

Solving
900z = 8200 + z2

Solving for variable 'z'.

Reorder the terms:
-8200 + 900z + -1z2 = 8200 + z2 + -8200 + -1z2

Reorder the terms:
-8200 + 900z + -1z2 = 8200 + -8200 + z2 + -1z2

Combine like terms: 8200 + -8200 = 0
-8200 + 900z + -1z2 = 0 + z2 + -1z2
-8200 + 900z + -1z2 = z2 + -1z2

Combine like terms: z2 + -1z2 = 0
-8200 + 900z + -1z2 = 0

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

Divide each side by '-1'.
8200 + -900z + z2 = 0

Move the constant term to the right:

Add '-8200' to each side of the equation.
8200 + -900z + -8200 + z2 = 0 + -8200

Reorder the terms:
8200 + -8200 + -900z + z2 = 0 + -8200

Combine like terms: 8200 + -8200 = 0
0 + -900z + z2 = 0 + -8200
-900z + z2 = 0 + -8200

Combine like terms: 0 + -8200 = -8200
-900z + z2 = -8200

The z term is -900z.  Take half its coefficient (-450).
Square it (202500) and add it to both sides.

Add '202500' to each side of the equation.
-900z + 202500 + z2 = -8200 + 202500

Reorder the terms:
202500 + -900z + z2 = -8200 + 202500

Combine like terms: -8200 + 202500 = 194300
202500 + -900z + z2 = 194300

Factor a perfect square on the left side:
(z + -450)(z + -450) = 194300

Calculate the square root of the right side: 440.794736811

Break this problem into two subproblems by setting 
(z + -450) equal to 440.794736811 and -440.794736811.

Subproblem 1

z + -450 = 440.794736811 Simplifying z + -450 = 440.794736811 Reorder the terms: -450 + z = 440.794736811 Solving -450 + z = 440.794736811 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '450' to each side of the equation. -450 + 450 + z = 440.794736811 + 450 Combine like terms: -450 + 450 = 0 0 + z = 440.794736811 + 450 z = 440.794736811 + 450 Combine like terms: 440.794736811 + 450 = 890.794736811 z = 890.794736811 Simplifying z = 890.794736811

Subproblem 2

z + -450 = -440.794736811 Simplifying z + -450 = -440.794736811 Reorder the terms: -450 + z = -440.794736811 Solving -450 + z = -440.794736811 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '450' to each side of the equation. -450 + 450 + z = -440.794736811 + 450 Combine like terms: -450 + 450 = 0 0 + z = -440.794736811 + 450 z = -440.794736811 + 450 Combine like terms: -440.794736811 + 450 = 9.205263189 z = 9.205263189 Simplifying z = 9.205263189

Solution

The solution to the problem is based on the solutions from the subproblems. z = {890.794736811, 9.205263189}

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