x(2x-9)=1

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Solution for x(2x-9)=1 equation:


Simplifying
x(2x + -9) = 1

Reorder the terms:
x(-9 + 2x) = 1
(-9 * x + 2x * x) = 1
(-9x + 2x2) = 1

Solving
-9x + 2x2 = 1

Solving for variable 'x'.

Reorder the terms:
-1 + -9x + 2x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -9x + 2x2 = 0

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

Divide each side by '2'.
-0.5 + -4.5x + x2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -4.5x + 0.5 + x2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -4.5x + x2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -4.5x + x2 = 0 + 0.5
-4.5x + x2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-4.5x + x2 = 0.5

The x term is -4.5x.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5x + 5.0625 + x2 = 0.5 + 5.0625

Reorder the terms:
5.0625 + -4.5x + x2 = 0.5 + 5.0625

Combine like terms: 0.5 + 5.0625 = 5.5625
5.0625 + -4.5x + x2 = 5.5625

Factor a perfect square on the left side:
(x + -2.25)(x + -2.25) = 5.5625

Calculate the square root of the right side: 2.358495283

Break this problem into two subproblems by setting 
(x + -2.25) equal to 2.358495283 and -2.358495283.

Subproblem 1

x + -2.25 = 2.358495283 Simplifying x + -2.25 = 2.358495283 Reorder the terms: -2.25 + x = 2.358495283 Solving -2.25 + x = 2.358495283 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + x = 2.358495283 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + x = 2.358495283 + 2.25 x = 2.358495283 + 2.25 Combine like terms: 2.358495283 + 2.25 = 4.608495283 x = 4.608495283 Simplifying x = 4.608495283

Subproblem 2

x + -2.25 = -2.358495283 Simplifying x + -2.25 = -2.358495283 Reorder the terms: -2.25 + x = -2.358495283 Solving -2.25 + x = -2.358495283 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + x = -2.358495283 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + x = -2.358495283 + 2.25 x = -2.358495283 + 2.25 Combine like terms: -2.358495283 + 2.25 = -0.108495283 x = -0.108495283 Simplifying x = -0.108495283

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.608495283, -0.108495283}

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