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

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


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

Reorder the terms:
x(-3 + x) + x(x + 2) = 5x + -3
(-3 * x + x * x) + x(x + 2) = 5x + -3
(-3x + x2) + x(x + 2) = 5x + -3

Reorder the terms:
-3x + x2 + x(2 + x) = 5x + -3
-3x + x2 + (2 * x + x * x) = 5x + -3
-3x + x2 + (2x + x2) = 5x + -3

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

Combine like terms: -3x + 2x = -1x
-1x + x2 + x2 = 5x + -3

Combine like terms: x2 + x2 = 2x2
-1x + 2x2 = 5x + -3

Reorder the terms:
-1x + 2x2 = -3 + 5x

Solving
-1x + 2x2 = -3 + 5x

Solving for variable 'x'.

Reorder the terms:
3 + -1x + -5x + 2x2 = -3 + 5x + 3 + -5x

Combine like terms: -1x + -5x = -6x
3 + -6x + 2x2 = -3 + 5x + 3 + -5x

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

Combine like terms: -3 + 3 = 0
3 + -6x + 2x2 = 0 + 5x + -5x
3 + -6x + 2x2 = 5x + -5x

Combine like terms: 5x + -5x = 0
3 + -6x + 2x2 = 0

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

Divide each side by '2'.
1.5 + -3x + x2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + -3x + -1.5 + x2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + -3x + x2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + -3x + x2 = 0 + -1.5
-3x + x2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
-3x + x2 = -1.5

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = -1.5 + 2.25

Reorder the terms:
2.25 + -3x + x2 = -1.5 + 2.25

Combine like terms: -1.5 + 2.25 = 0.75
2.25 + -3x + x2 = 0.75

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(x + -1.5) equal to 0.866025404 and -0.866025404.

Subproblem 1

x + -1.5 = 0.866025404 Simplifying x + -1.5 = 0.866025404 Reorder the terms: -1.5 + x = 0.866025404 Solving -1.5 + x = 0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 0.866025404 + 1.5 x = 0.866025404 + 1.5 Combine like terms: 0.866025404 + 1.5 = 2.366025404 x = 2.366025404 Simplifying x = 2.366025404

Subproblem 2

x + -1.5 = -0.866025404 Simplifying x + -1.5 = -0.866025404 Reorder the terms: -1.5 + x = -0.866025404 Solving -1.5 + x = -0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -0.866025404 + 1.5 x = -0.866025404 + 1.5 Combine like terms: -0.866025404 + 1.5 = 0.633974596 x = 0.633974596 Simplifying x = 0.633974596

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.366025404, 0.633974596}

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