18x-x^2=40.5

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Solution for 18x-x^2=40.5 equation:


Simplifying
18x + -1x2 = 40.5

Solving
18x + -1x2 = 40.5

Solving for variable 'x'.

Reorder the terms:
-40.5 + 18x + -1x2 = 40.5 + -40.5

Combine like terms: 40.5 + -40.5 = 0.0
-40.5 + 18x + -1x2 = 0.0

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

Divide each side by '-1'.
40.5 + -18x + x2 = 0.0

Move the constant term to the right:

Add '-40.5' to each side of the equation.
40.5 + -18x + -40.5 + x2 = 0.0 + -40.5

Reorder the terms:
40.5 + -40.5 + -18x + x2 = 0.0 + -40.5

Combine like terms: 40.5 + -40.5 = 0.0
0.0 + -18x + x2 = 0.0 + -40.5
-18x + x2 = 0.0 + -40.5

Combine like terms: 0.0 + -40.5 = -40.5
-18x + x2 = -40.5

The x term is -18x.  Take half its coefficient (-9).
Square it (81) and add it to both sides.

Add '81' to each side of the equation.
-18x + 81 + x2 = -40.5 + 81

Reorder the terms:
81 + -18x + x2 = -40.5 + 81

Combine like terms: -40.5 + 81 = 40.5
81 + -18x + x2 = 40.5

Factor a perfect square on the left side:
(x + -9)(x + -9) = 40.5

Calculate the square root of the right side: 6.363961031

Break this problem into two subproblems by setting 
(x + -9) equal to 6.363961031 and -6.363961031.

Subproblem 1

x + -9 = 6.363961031 Simplifying x + -9 = 6.363961031 Reorder the terms: -9 + x = 6.363961031 Solving -9 + x = 6.363961031 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + x = 6.363961031 + 9 Combine like terms: -9 + 9 = 0 0 + x = 6.363961031 + 9 x = 6.363961031 + 9 Combine like terms: 6.363961031 + 9 = 15.363961031 x = 15.363961031 Simplifying x = 15.363961031

Subproblem 2

x + -9 = -6.363961031 Simplifying x + -9 = -6.363961031 Reorder the terms: -9 + x = -6.363961031 Solving -9 + x = -6.363961031 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + x = -6.363961031 + 9 Combine like terms: -9 + 9 = 0 0 + x = -6.363961031 + 9 x = -6.363961031 + 9 Combine like terms: -6.363961031 + 9 = 2.636038969 x = 2.636038969 Simplifying x = 2.636038969

Solution

The solution to the problem is based on the solutions from the subproblems. x = {15.363961031, 2.636038969}

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