0=x^2-90x+1

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Solution for 0=x^2-90x+1 equation:


Simplifying
0 = x2 + -90x + 1

Reorder the terms:
0 = 1 + -90x + x2

Solving
0 = 1 + -90x + x2

Solving for variable 'x'.

Combine like terms: 0 + -1 = -1
-1 + 90x + -1x2 = 1 + -90x + x2 + -1 + 90x + -1x2

Reorder the terms:
-1 + 90x + -1x2 = 1 + -1 + -90x + 90x + x2 + -1x2

Combine like terms: 1 + -1 = 0
-1 + 90x + -1x2 = 0 + -90x + 90x + x2 + -1x2
-1 + 90x + -1x2 = -90x + 90x + x2 + -1x2

Combine like terms: -90x + 90x = 0
-1 + 90x + -1x2 = 0 + x2 + -1x2
-1 + 90x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-1 + 90x + -1x2 = 0

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

Divide each side by '-1'.
1 + -90x + x2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -90x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + -90x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -90x + x2 = 0 + -1
-90x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
-90x + x2 = -1

The x term is -90x.  Take half its coefficient (-45).
Square it (2025) and add it to both sides.

Add '2025' to each side of the equation.
-90x + 2025 + x2 = -1 + 2025

Reorder the terms:
2025 + -90x + x2 = -1 + 2025

Combine like terms: -1 + 2025 = 2024
2025 + -90x + x2 = 2024

Factor a perfect square on the left side:
(x + -45)(x + -45) = 2024

Calculate the square root of the right side: 44.988887517

Break this problem into two subproblems by setting 
(x + -45) equal to 44.988887517 and -44.988887517.

Subproblem 1

x + -45 = 44.988887517 Simplifying x + -45 = 44.988887517 Reorder the terms: -45 + x = 44.988887517 Solving -45 + x = 44.988887517 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '45' to each side of the equation. -45 + 45 + x = 44.988887517 + 45 Combine like terms: -45 + 45 = 0 0 + x = 44.988887517 + 45 x = 44.988887517 + 45 Combine like terms: 44.988887517 + 45 = 89.988887517 x = 89.988887517 Simplifying x = 89.988887517

Subproblem 2

x + -45 = -44.988887517 Simplifying x + -45 = -44.988887517 Reorder the terms: -45 + x = -44.988887517 Solving -45 + x = -44.988887517 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '45' to each side of the equation. -45 + 45 + x = -44.988887517 + 45 Combine like terms: -45 + 45 = 0 0 + x = -44.988887517 + 45 x = -44.988887517 + 45 Combine like terms: -44.988887517 + 45 = 0.011112483 x = 0.011112483 Simplifying x = 0.011112483

Solution

The solution to the problem is based on the solutions from the subproblems. x = {89.988887517, 0.011112483}

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