-x^2+90x-54=0

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Solution for -x^2+90x-54=0 equation:


Simplifying
-1x2 + 90x + -54 = 0

Reorder the terms:
-54 + 90x + -1x2 = 0

Solving
-54 + 90x + -1x2 = 0

Solving for variable 'x'.

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

Divide each side by '-1'.
54 + -90x + x2 = 0

Move the constant term to the right:

Add '-54' to each side of the equation.
54 + -90x + -54 + x2 = 0 + -54

Reorder the terms:
54 + -54 + -90x + x2 = 0 + -54

Combine like terms: 54 + -54 = 0
0 + -90x + x2 = 0 + -54
-90x + x2 = 0 + -54

Combine like terms: 0 + -54 = -54
-90x + x2 = -54

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 = -54 + 2025

Reorder the terms:
2025 + -90x + x2 = -54 + 2025

Combine like terms: -54 + 2025 = 1971
2025 + -90x + x2 = 1971

Factor a perfect square on the left side:
(x + -45)(x + -45) = 1971

Calculate the square root of the right side: 44.395945761

Break this problem into two subproblems by setting 
(x + -45) equal to 44.395945761 and -44.395945761.

Subproblem 1

x + -45 = 44.395945761 Simplifying x + -45 = 44.395945761 Reorder the terms: -45 + x = 44.395945761 Solving -45 + x = 44.395945761 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 = 44.395945761 + 45 Combine like terms: -45 + 45 = 0 0 + x = 44.395945761 + 45 x = 44.395945761 + 45 Combine like terms: 44.395945761 + 45 = 89.395945761 x = 89.395945761 Simplifying x = 89.395945761

Subproblem 2

x + -45 = -44.395945761 Simplifying x + -45 = -44.395945761 Reorder the terms: -45 + x = -44.395945761 Solving -45 + x = -44.395945761 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 = -44.395945761 + 45 Combine like terms: -45 + 45 = 0 0 + x = -44.395945761 + 45 x = -44.395945761 + 45 Combine like terms: -44.395945761 + 45 = 0.604054239 x = 0.604054239 Simplifying x = 0.604054239

Solution

The solution to the problem is based on the solutions from the subproblems. x = {89.395945761, 0.604054239}

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