x(20x-45)=90

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Solution for x(20x-45)=90 equation: 


Simplifying
x(20x + -45) = 90

Reorder the terms:
x(-45 + 20x) = 90
(-45 * x + 20x * x) = 90
(-45x + 20x2) = 90

Solving
-45x + 20x2 = 90

Solving for variable 'x'.

Reorder the terms:
-90 + -45x + 20x2 = 90 + -90

Combine like terms: 90 + -90 = 0
-90 + -45x + 20x2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-18 + -9x + 4x2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(-18 + -9x + 4x2)' equal to zero and attempt to solve:

Simplifying
-18 + -9x + 4x2 = 0

Solving
-18 + -9x + 4x2 = 0

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

Divide each side by '4'.
-4.5 + -2.25x + x2 = 0

Move the constant term to the right:

Add '4.5' to each side of the equation.
-4.5 + -2.25x + 4.5 + x2 = 0 + 4.5

Reorder the terms:
-4.5 + 4.5 + -2.25x + x2 = 0 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + -2.25x + x2 = 0 + 4.5
-2.25x + x2 = 0 + 4.5

Combine like terms: 0 + 4.5 = 4.5
-2.25x + x2 = 4.5

The x term is -2.25x.  Take half its coefficient (-1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
-2.25x + 1.265625 + x2 = 4.5 + 1.265625

Reorder the terms:
1.265625 + -2.25x + x2 = 4.5 + 1.265625

Combine like terms: 4.5 + 1.265625 = 5.765625
1.265625 + -2.25x + x2 = 5.765625

Factor a perfect square on the left side:
(x + -1.125)(x + -1.125) = 5.765625

Calculate the square root of the right side: 2.401171589

Break this problem into two subproblems by setting 
(x + -1.125) equal to 2.401171589 and -2.401171589.


Subproblem 1
x + -1.125 = 2.401171589

Simplifying
x + -1.125 = 2.401171589

Reorder the terms:
-1.125 + x = 2.401171589

Solving
-1.125 + x = 2.401171589

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '1.125' to each side of the equation.
-1.125 + 1.125 + x = 2.401171589 + 1.125

Combine like terms: -1.125 + 1.125 = 0.000
0.000 + x = 2.401171589 + 1.125
x = 2.401171589 + 1.125

Combine like terms: 2.401171589 + 1.125 = 3.526171589
x = 3.526171589

Simplifying
x = 3.526171589


Subproblem 2
x + -1.125 = -2.401171589

Simplifying
x + -1.125 = -2.401171589

Reorder the terms:
-1.125 + x = -2.401171589

Solving
-1.125 + x = -2.401171589

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '1.125' to each side of the equation.
-1.125 + 1.125 + x = -2.401171589 + 1.125

Combine like terms: -1.125 + 1.125 = 0.000
0.000 + x = -2.401171589 + 1.125
x = -2.401171589 + 1.125

Combine like terms: -2.401171589 + 1.125 = -1.276171589
x = -1.276171589

Simplifying
x = -1.276171589


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.526171589, -1.276171589}
Solution
x = {3.526171589, -1.276171589}

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