2x(x-3)=3x(4-x)+5

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Solution for 2x(x-3)=3x(4-x)+5 equation:


Simplifying
2x(x + -3) = 3x(4 + -1x) + 5

Reorder the terms:
2x(-3 + x) = 3x(4 + -1x) + 5
(-3 * 2x + x * 2x) = 3x(4 + -1x) + 5
(-6x + 2x2) = 3x(4 + -1x) + 5
-6x + 2x2 = (4 * 3x + -1x * 3x) + 5
-6x + 2x2 = (12x + -3x2) + 5

Reorder the terms:
-6x + 2x2 = 5 + 12x + -3x2

Solving
-6x + 2x2 = 5 + 12x + -3x2

Solving for variable 'x'.

Reorder the terms:
-5 + -6x + -12x + 2x2 + 3x2 = 5 + 12x + -3x2 + -5 + -12x + 3x2

Combine like terms: -6x + -12x = -18x
-5 + -18x + 2x2 + 3x2 = 5 + 12x + -3x2 + -5 + -12x + 3x2

Combine like terms: 2x2 + 3x2 = 5x2
-5 + -18x + 5x2 = 5 + 12x + -3x2 + -5 + -12x + 3x2

Reorder the terms:
-5 + -18x + 5x2 = 5 + -5 + 12x + -12x + -3x2 + 3x2

Combine like terms: 5 + -5 = 0
-5 + -18x + 5x2 = 0 + 12x + -12x + -3x2 + 3x2
-5 + -18x + 5x2 = 12x + -12x + -3x2 + 3x2

Combine like terms: 12x + -12x = 0
-5 + -18x + 5x2 = 0 + -3x2 + 3x2
-5 + -18x + 5x2 = -3x2 + 3x2

Combine like terms: -3x2 + 3x2 = 0
-5 + -18x + 5x2 = 0

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

Divide each side by '5'.
-1 + -3.6x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -3.6x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + -3.6x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -3.6x + x2 = 0 + 1
-3.6x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
-3.6x + x2 = 1

The x term is -3.6x.  Take half its coefficient (-1.8).
Square it (3.24) and add it to both sides.

Add '3.24' to each side of the equation.
-3.6x + 3.24 + x2 = 1 + 3.24

Reorder the terms:
3.24 + -3.6x + x2 = 1 + 3.24

Combine like terms: 1 + 3.24 = 4.24
3.24 + -3.6x + x2 = 4.24

Factor a perfect square on the left side:
(x + -1.8)(x + -1.8) = 4.24

Calculate the square root of the right side: 2.059126028

Break this problem into two subproblems by setting 
(x + -1.8) equal to 2.059126028 and -2.059126028.

Subproblem 1

x + -1.8 = 2.059126028 Simplifying x + -1.8 = 2.059126028 Reorder the terms: -1.8 + x = 2.059126028 Solving -1.8 + x = 2.059126028 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.8' to each side of the equation. -1.8 + 1.8 + x = 2.059126028 + 1.8 Combine like terms: -1.8 + 1.8 = 0.0 0.0 + x = 2.059126028 + 1.8 x = 2.059126028 + 1.8 Combine like terms: 2.059126028 + 1.8 = 3.859126028 x = 3.859126028 Simplifying x = 3.859126028

Subproblem 2

x + -1.8 = -2.059126028 Simplifying x + -1.8 = -2.059126028 Reorder the terms: -1.8 + x = -2.059126028 Solving -1.8 + x = -2.059126028 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.8' to each side of the equation. -1.8 + 1.8 + x = -2.059126028 + 1.8 Combine like terms: -1.8 + 1.8 = 0.0 0.0 + x = -2.059126028 + 1.8 x = -2.059126028 + 1.8 Combine like terms: -2.059126028 + 1.8 = -0.259126028 x = -0.259126028 Simplifying x = -0.259126028

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.859126028, -0.259126028}

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