9x^2=45x-11

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Solution for 9x^2=45x-11 equation:


Simplifying
9x2 = 45x + -11

Reorder the terms:
9x2 = -11 + 45x

Solving
9x2 = -11 + 45x

Solving for variable 'x'.

Reorder the terms:
11 + -45x + 9x2 = -11 + 45x + 11 + -45x

Reorder the terms:
11 + -45x + 9x2 = -11 + 11 + 45x + -45x

Combine like terms: -11 + 11 = 0
11 + -45x + 9x2 = 0 + 45x + -45x
11 + -45x + 9x2 = 45x + -45x

Combine like terms: 45x + -45x = 0
11 + -45x + 9x2 = 0

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

Divide each side by '9'.
1.222222222 + -5x + x2 = 0

Move the constant term to the right:

Add '-1.222222222' to each side of the equation.
1.222222222 + -5x + -1.222222222 + x2 = 0 + -1.222222222

Reorder the terms:
1.222222222 + -1.222222222 + -5x + x2 = 0 + -1.222222222

Combine like terms: 1.222222222 + -1.222222222 = 0.000000000
0.000000000 + -5x + x2 = 0 + -1.222222222
-5x + x2 = 0 + -1.222222222

Combine like terms: 0 + -1.222222222 = -1.222222222
-5x + x2 = -1.222222222

The x term is -5x.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x + 6.25 + x2 = -1.222222222 + 6.25

Reorder the terms:
6.25 + -5x + x2 = -1.222222222 + 6.25

Combine like terms: -1.222222222 + 6.25 = 5.027777778
6.25 + -5x + x2 = 5.027777778

Factor a perfect square on the left side:
(x + -2.5)(x + -2.5) = 5.027777778

Calculate the square root of the right side: 2.242270675

Break this problem into two subproblems by setting 
(x + -2.5) equal to 2.242270675 and -2.242270675.

Subproblem 1

x + -2.5 = 2.242270675 Simplifying x + -2.5 = 2.242270675 Reorder the terms: -2.5 + x = 2.242270675 Solving -2.5 + x = 2.242270675 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = 2.242270675 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = 2.242270675 + 2.5 x = 2.242270675 + 2.5 Combine like terms: 2.242270675 + 2.5 = 4.742270675 x = 4.742270675 Simplifying x = 4.742270675

Subproblem 2

x + -2.5 = -2.242270675 Simplifying x + -2.5 = -2.242270675 Reorder the terms: -2.5 + x = -2.242270675 Solving -2.5 + x = -2.242270675 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = -2.242270675 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = -2.242270675 + 2.5 x = -2.242270675 + 2.5 Combine like terms: -2.242270675 + 2.5 = 0.257729325 x = 0.257729325 Simplifying x = 0.257729325

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.742270675, 0.257729325}

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