165=x(2x-1)

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Solution for 165=x(2x-1) equation:


Simplifying
165 = x(2x + -1)

Reorder the terms:
165 = x(-1 + 2x)
165 = (-1 * x + 2x * x)
165 = (-1x + 2x2)

Solving
165 = -1x + 2x2

Solving for variable 'x'.

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

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

Combine like terms: 2x2 + -2x2 = 0
165 + x + -2x2 = 0

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

Divide each side by '-2'.
-82.5 + -0.5x + x2 = 0

Move the constant term to the right:

Add '82.5' to each side of the equation.
-82.5 + -0.5x + 82.5 + x2 = 0 + 82.5

Reorder the terms:
-82.5 + 82.5 + -0.5x + x2 = 0 + 82.5

Combine like terms: -82.5 + 82.5 = 0.0
0.0 + -0.5x + x2 = 0 + 82.5
-0.5x + x2 = 0 + 82.5

Combine like terms: 0 + 82.5 = 82.5
-0.5x + x2 = 82.5

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-0.5x + 0.25 + x2 = 82.5 + 0.25

Reorder the terms:
0.25 + -0.5x + x2 = 82.5 + 0.25

Combine like terms: 82.5 + 0.25 = 82.75
0.25 + -0.5x + x2 = 82.75

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 82.75

Calculate the square root of the right side: 9.096702699

Break this problem into two subproblems by setting 
(x + 0.5) equal to 9.096702699 and -9.096702699.

Subproblem 1

x + 0.5 = 9.096702699 Simplifying x + 0.5 = 9.096702699 Reorder the terms: 0.5 + x = 9.096702699 Solving 0.5 + x = 9.096702699 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 9.096702699 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 9.096702699 + -0.5 x = 9.096702699 + -0.5 Combine like terms: 9.096702699 + -0.5 = 8.596702699 x = 8.596702699 Simplifying x = 8.596702699

Subproblem 2

x + 0.5 = -9.096702699 Simplifying x + 0.5 = -9.096702699 Reorder the terms: 0.5 + x = -9.096702699 Solving 0.5 + x = -9.096702699 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -9.096702699 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -9.096702699 + -0.5 x = -9.096702699 + -0.5 Combine like terms: -9.096702699 + -0.5 = -9.596702699 x = -9.596702699 Simplifying x = -9.596702699

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.596702699, -9.596702699}

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