90x-x^2=100

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Solution for 90x-x^2=100 equation:


Simplifying
90x + -1x2 = 100

Solving
90x + -1x2 = 100

Solving for variable 'x'.

Reorder the terms:
-100 + 90x + -1x2 = 100 + -100

Combine like terms: 100 + -100 = 0
-100 + 90x + -1x2 = 0

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

Divide each side by '-1'.
100 + -90x + x2 = 0

Move the constant term to the right:

Add '-100' to each side of the equation.
100 + -90x + -100 + x2 = 0 + -100

Reorder the terms:
100 + -100 + -90x + x2 = 0 + -100

Combine like terms: 100 + -100 = 0
0 + -90x + x2 = 0 + -100
-90x + x2 = 0 + -100

Combine like terms: 0 + -100 = -100
-90x + x2 = -100

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 = -100 + 2025

Reorder the terms:
2025 + -90x + x2 = -100 + 2025

Combine like terms: -100 + 2025 = 1925
2025 + -90x + x2 = 1925

Factor a perfect square on the left side:
(x + -45)(x + -45) = 1925

Calculate the square root of the right side: 43.874821937

Break this problem into two subproblems by setting 
(x + -45) equal to 43.874821937 and -43.874821937.

Subproblem 1

x + -45 = 43.874821937 Simplifying x + -45 = 43.874821937 Reorder the terms: -45 + x = 43.874821937 Solving -45 + x = 43.874821937 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 = 43.874821937 + 45 Combine like terms: -45 + 45 = 0 0 + x = 43.874821937 + 45 x = 43.874821937 + 45 Combine like terms: 43.874821937 + 45 = 88.874821937 x = 88.874821937 Simplifying x = 88.874821937

Subproblem 2

x + -45 = -43.874821937 Simplifying x + -45 = -43.874821937 Reorder the terms: -45 + x = -43.874821937 Solving -45 + x = -43.874821937 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 = -43.874821937 + 45 Combine like terms: -45 + 45 = 0 0 + x = -43.874821937 + 45 x = -43.874821937 + 45 Combine like terms: -43.874821937 + 45 = 1.125178063 x = 1.125178063 Simplifying x = 1.125178063

Solution

The solution to the problem is based on the solutions from the subproblems. x = {88.874821937, 1.125178063}

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