(2x+0.3)(0.5x+2)+(-3x+1)(0.5x+2)=0

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Solution for (2x+0.3)(0.5x+2)+(-3x+1)(0.5x+2)=0 equation:


Simplifying
(2x + 0.3)(0.5x + 2) + (-3x + 1)(0.5x + 2) = 0

Reorder the terms:
(0.3 + 2x)(0.5x + 2) + (-3x + 1)(0.5x + 2) = 0

Reorder the terms:
(0.3 + 2x)(2 + 0.5x) + (-3x + 1)(0.5x + 2) = 0

Multiply (0.3 + 2x) * (2 + 0.5x)
(0.3(2 + 0.5x) + 2x * (2 + 0.5x)) + (-3x + 1)(0.5x + 2) = 0
((2 * 0.3 + 0.5x * 0.3) + 2x * (2 + 0.5x)) + (-3x + 1)(0.5x + 2) = 0
((0.6 + 0.15x) + 2x * (2 + 0.5x)) + (-3x + 1)(0.5x + 2) = 0
(0.6 + 0.15x + (2 * 2x + 0.5x * 2x)) + (-3x + 1)(0.5x + 2) = 0
(0.6 + 0.15x + (4x + 1x2)) + (-3x + 1)(0.5x + 2) = 0

Combine like terms: 0.15x + 4x = 4.15x
(0.6 + 4.15x + 1x2) + (-3x + 1)(0.5x + 2) = 0

Reorder the terms:
0.6 + 4.15x + 1x2 + (1 + -3x)(0.5x + 2) = 0

Reorder the terms:
0.6 + 4.15x + 1x2 + (1 + -3x)(2 + 0.5x) = 0

Multiply (1 + -3x) * (2 + 0.5x)
0.6 + 4.15x + 1x2 + (1(2 + 0.5x) + -3x * (2 + 0.5x)) = 0
0.6 + 4.15x + 1x2 + ((2 * 1 + 0.5x * 1) + -3x * (2 + 0.5x)) = 0
0.6 + 4.15x + 1x2 + ((2 + 0.5x) + -3x * (2 + 0.5x)) = 0
0.6 + 4.15x + 1x2 + (2 + 0.5x + (2 * -3x + 0.5x * -3x)) = 0
0.6 + 4.15x + 1x2 + (2 + 0.5x + (-6x + -1.5x2)) = 0

Combine like terms: 0.5x + -6x = -5.5x
0.6 + 4.15x + 1x2 + (2 + -5.5x + -1.5x2) = 0

Reorder the terms:
0.6 + 2 + 4.15x + -5.5x + 1x2 + -1.5x2 = 0

Combine like terms: 0.6 + 2 = 2.6
2.6 + 4.15x + -5.5x + 1x2 + -1.5x2 = 0

Combine like terms: 4.15x + -5.5x = -1.35x
2.6 + -1.35x + 1x2 + -1.5x2 = 0

Combine like terms: 1x2 + -1.5x2 = -0.5x2
2.6 + -1.35x + -0.5x2 = 0

Solving
2.6 + -1.35x + -0.5x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
-0.5 the coefficient of the squared term: 

Divide each side by '-0.5'.
-5.2 + 2.7x + x2 = 0

Move the constant term to the right:

Add '5.2' to each side of the equation.
-5.2 + 2.7x + 5.2 + x2 = 0 + 5.2

Reorder the terms:
-5.2 + 5.2 + 2.7x + x2 = 0 + 5.2

Combine like terms: -5.2 + 5.2 = 0.0
0.0 + 2.7x + x2 = 0 + 5.2
2.7x + x2 = 0 + 5.2

Combine like terms: 0 + 5.2 = 5.2
2.7x + x2 = 5.2

The x term is 2.7x.  Take half its coefficient (1.35).
Square it (1.8225) and add it to both sides.

Add '1.8225' to each side of the equation.
2.7x + 1.8225 + x2 = 5.2 + 1.8225

Reorder the terms:
1.8225 + 2.7x + x2 = 5.2 + 1.8225

Combine like terms: 5.2 + 1.8225 = 7.0225
1.8225 + 2.7x + x2 = 7.0225

Factor a perfect square on the left side:
(x + 1.35)(x + 1.35) = 7.0225

Calculate the square root of the right side: 2.65

Break this problem into two subproblems by setting 
(x + 1.35) equal to 2.65 and -2.65.

Subproblem 1

x + 1.35 = 2.65 Simplifying x + 1.35 = 2.65 Reorder the terms: 1.35 + x = 2.65 Solving 1.35 + x = 2.65 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.35' to each side of the equation. 1.35 + -1.35 + x = 2.65 + -1.35 Combine like terms: 1.35 + -1.35 = 0.00 0.00 + x = 2.65 + -1.35 x = 2.65 + -1.35 Combine like terms: 2.65 + -1.35 = 1.3 x = 1.3 Simplifying x = 1.3

Subproblem 2

x + 1.35 = -2.65 Simplifying x + 1.35 = -2.65 Reorder the terms: 1.35 + x = -2.65 Solving 1.35 + x = -2.65 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.35' to each side of the equation. 1.35 + -1.35 + x = -2.65 + -1.35 Combine like terms: 1.35 + -1.35 = 0.00 0.00 + x = -2.65 + -1.35 x = -2.65 + -1.35 Combine like terms: -2.65 + -1.35 = -4 x = -4 Simplifying x = -4

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.3, -4}

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