-2x^2+45x-108=0

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Solution for -2x^2+45x-108=0 equation:


Simplifying
-2x2 + 45x + -108 = 0

Reorder the terms:
-108 + 45x + -2x2 = 0

Solving
-108 + 45x + -2x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
54 + -22.5x + x2 = 0

Move the constant term to the right:

Add '-54' to each side of the equation.
54 + -22.5x + -54 + x2 = 0 + -54

Reorder the terms:
54 + -54 + -22.5x + x2 = 0 + -54

Combine like terms: 54 + -54 = 0
0 + -22.5x + x2 = 0 + -54
-22.5x + x2 = 0 + -54

Combine like terms: 0 + -54 = -54
-22.5x + x2 = -54

The x term is -22.5x.  Take half its coefficient (-11.25).
Square it (126.5625) and add it to both sides.

Add '126.5625' to each side of the equation.
-22.5x + 126.5625 + x2 = -54 + 126.5625

Reorder the terms:
126.5625 + -22.5x + x2 = -54 + 126.5625

Combine like terms: -54 + 126.5625 = 72.5625
126.5625 + -22.5x + x2 = 72.5625

Factor a perfect square on the left side:
(x + -11.25)(x + -11.25) = 72.5625

Calculate the square root of the right side: 8.518362519

Break this problem into two subproblems by setting 
(x + -11.25) equal to 8.518362519 and -8.518362519.

Subproblem 1

x + -11.25 = 8.518362519 Simplifying x + -11.25 = 8.518362519 Reorder the terms: -11.25 + x = 8.518362519 Solving -11.25 + x = 8.518362519 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '11.25' to each side of the equation. -11.25 + 11.25 + x = 8.518362519 + 11.25 Combine like terms: -11.25 + 11.25 = 0.00 0.00 + x = 8.518362519 + 11.25 x = 8.518362519 + 11.25 Combine like terms: 8.518362519 + 11.25 = 19.768362519 x = 19.768362519 Simplifying x = 19.768362519

Subproblem 2

x + -11.25 = -8.518362519 Simplifying x + -11.25 = -8.518362519 Reorder the terms: -11.25 + x = -8.518362519 Solving -11.25 + x = -8.518362519 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '11.25' to each side of the equation. -11.25 + 11.25 + x = -8.518362519 + 11.25 Combine like terms: -11.25 + 11.25 = 0.00 0.00 + x = -8.518362519 + 11.25 x = -8.518362519 + 11.25 Combine like terms: -8.518362519 + 11.25 = 2.731637481 x = 2.731637481 Simplifying x = 2.731637481

Solution

The solution to the problem is based on the solutions from the subproblems. x = {19.768362519, 2.731637481}

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