2x(x-3)=1+4x

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Solution for 2x(x-3)=1+4x equation:


Simplifying
2x(x + -3) = 1 + 4x

Reorder the terms:
2x(-3 + x) = 1 + 4x
(-3 * 2x + x * 2x) = 1 + 4x
(-6x + 2x2) = 1 + 4x

Solving
-6x + 2x2 = 1 + 4x

Solving for variable 'x'.

Reorder the terms:
-1 + -6x + -4x + 2x2 = 1 + 4x + -1 + -4x

Combine like terms: -6x + -4x = -10x
-1 + -10x + 2x2 = 1 + 4x + -1 + -4x

Reorder the terms:
-1 + -10x + 2x2 = 1 + -1 + 4x + -4x

Combine like terms: 1 + -1 = 0
-1 + -10x + 2x2 = 0 + 4x + -4x
-1 + -10x + 2x2 = 4x + -4x

Combine like terms: 4x + -4x = 0
-1 + -10x + 2x2 = 0

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

Divide each side by '2'.
-0.5 + -5x + x2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -5x + 0.5 + x2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -5x + x2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -5x + x2 = 0 + 0.5
-5x + x2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-5x + x2 = 0.5

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 = 0.5 + 6.25

Reorder the terms:
6.25 + -5x + x2 = 0.5 + 6.25

Combine like terms: 0.5 + 6.25 = 6.75
6.25 + -5x + x2 = 6.75

Factor a perfect square on the left side:
(x + -2.5)(x + -2.5) = 6.75

Calculate the square root of the right side: 2.598076211

Break this problem into two subproblems by setting 
(x + -2.5) equal to 2.598076211 and -2.598076211.

Subproblem 1

x + -2.5 = 2.598076211 Simplifying x + -2.5 = 2.598076211 Reorder the terms: -2.5 + x = 2.598076211 Solving -2.5 + x = 2.598076211 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 = 2.598076211 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = 2.598076211 + 2.5 x = 2.598076211 + 2.5 Combine like terms: 2.598076211 + 2.5 = 5.098076211 x = 5.098076211 Simplifying x = 5.098076211

Subproblem 2

x + -2.5 = -2.598076211 Simplifying x + -2.5 = -2.598076211 Reorder the terms: -2.5 + x = -2.598076211 Solving -2.5 + x = -2.598076211 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 = -2.598076211 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = -2.598076211 + 2.5 x = -2.598076211 + 2.5 Combine like terms: -2.598076211 + 2.5 = -0.098076211 x = -0.098076211 Simplifying x = -0.098076211

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.098076211, -0.098076211}

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