x^2+(9x)(9-x)=45

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Solution for x^2+(9x)(9-x)=45 equation:


Simplifying
x2 + (9x)(9 + -1x) = 45

Remove parenthesis around (9x)
x2 + 9x(9 + -1x) = 45
x2 + (9 * 9x + -1x * 9x) = 45
x2 + (81x + -9x2) = 45

Reorder the terms:
81x + x2 + -9x2 = 45

Combine like terms: x2 + -9x2 = -8x2
81x + -8x2 = 45

Solving
81x + -8x2 = 45

Solving for variable 'x'.

Reorder the terms:
-45 + 81x + -8x2 = 45 + -45

Combine like terms: 45 + -45 = 0
-45 + 81x + -8x2 = 0

Begin completing the square.  Divide all terms by
-8 the coefficient of the squared term: 

Divide each side by '-8'.
5.625 + -10.125x + x2 = 0

Move the constant term to the right:

Add '-5.625' to each side of the equation.
5.625 + -10.125x + -5.625 + x2 = 0 + -5.625

Reorder the terms:
5.625 + -5.625 + -10.125x + x2 = 0 + -5.625

Combine like terms: 5.625 + -5.625 = 0.000
0.000 + -10.125x + x2 = 0 + -5.625
-10.125x + x2 = 0 + -5.625

Combine like terms: 0 + -5.625 = -5.625
-10.125x + x2 = -5.625

The x term is -10.125x.  Take half its coefficient (-5.0625).
Square it (25.62890625) and add it to both sides.

Add '25.62890625' to each side of the equation.
-10.125x + 25.62890625 + x2 = -5.625 + 25.62890625

Reorder the terms:
25.62890625 + -10.125x + x2 = -5.625 + 25.62890625

Combine like terms: -5.625 + 25.62890625 = 20.00390625
25.62890625 + -10.125x + x2 = 20.00390625

Factor a perfect square on the left side:
(x + -5.0625)(x + -5.0625) = 20.00390625

Calculate the square root of the right side: 4.472572666

Break this problem into two subproblems by setting 
(x + -5.0625) equal to 4.472572666 and -4.472572666.

Subproblem 1

x + -5.0625 = 4.472572666 Simplifying x + -5.0625 = 4.472572666 Reorder the terms: -5.0625 + x = 4.472572666 Solving -5.0625 + x = 4.472572666 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.0625' to each side of the equation. -5.0625 + 5.0625 + x = 4.472572666 + 5.0625 Combine like terms: -5.0625 + 5.0625 = 0.0000 0.0000 + x = 4.472572666 + 5.0625 x = 4.472572666 + 5.0625 Combine like terms: 4.472572666 + 5.0625 = 9.535072666 x = 9.535072666 Simplifying x = 9.535072666

Subproblem 2

x + -5.0625 = -4.472572666 Simplifying x + -5.0625 = -4.472572666 Reorder the terms: -5.0625 + x = -4.472572666 Solving -5.0625 + x = -4.472572666 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.0625' to each side of the equation. -5.0625 + 5.0625 + x = -4.472572666 + 5.0625 Combine like terms: -5.0625 + 5.0625 = 0.0000 0.0000 + x = -4.472572666 + 5.0625 x = -4.472572666 + 5.0625 Combine like terms: -4.472572666 + 5.0625 = 0.589927334 x = 0.589927334 Simplifying x = 0.589927334

Solution

The solution to the problem is based on the solutions from the subproblems. x = {9.535072666, 0.589927334}

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