x^2+5x+1.5x+7.5=65

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Solution for x^2+5x+1.5x+7.5=65 equation: 


Simplifying
x2 + 5x + 1.5x + 7.5 = 65

Reorder the terms:
7.5 + 5x + 1.5x + x2 = 65

Combine like terms: 5x + 1.5x = 6.5x
7.5 + 6.5x + x2 = 65

Solving
7.5 + 6.5x + x2 = 65

Solving for variable 'x'.

Reorder the terms:
7.5 + -65 + 6.5x + x2 = 65 + -65

Combine like terms: 7.5 + -65 = -57.5
-57.5 + 6.5x + x2 = 65 + -65

Combine like terms: 65 + -65 = 0
-57.5 + 6.5x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '57.5' to each side of the equation.
-57.5 + 6.5x + 57.5 + x2 = 0 + 57.5

Reorder the terms:
-57.5 + 57.5 + 6.5x + x2 = 0 + 57.5

Combine like terms: -57.5 + 57.5 = 0.0
0.0 + 6.5x + x2 = 0 + 57.5
6.5x + x2 = 0 + 57.5

Combine like terms: 0 + 57.5 = 57.5
6.5x + x2 = 57.5

The x term is 6.5x.  Take half its coefficient (3.25).
Square it (10.5625) and add it to both sides.

Add '10.5625' to each side of the equation.
6.5x + 10.5625 + x2 = 57.5 + 10.5625

Reorder the terms:
10.5625 + 6.5x + x2 = 57.5 + 10.5625

Combine like terms: 57.5 + 10.5625 = 68.0625
10.5625 + 6.5x + x2 = 68.0625

Factor a perfect square on the left side:
(x + 3.25)(x + 3.25) = 68.0625

Calculate the square root of the right side: 8.25

Break this problem into two subproblems by setting 
(x + 3.25) equal to 8.25 and -8.25.


Subproblem 1
x + 3.25 = 8.25

Simplifying
x + 3.25 = 8.25

Reorder the terms:
3.25 + x = 8.25

Solving
3.25 + x = 8.25

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-3.25' to each side of the equation.
3.25 + -3.25 + x = 8.25 + -3.25

Combine like terms: 3.25 + -3.25 = 0.00
0.00 + x = 8.25 + -3.25
x = 8.25 + -3.25

Combine like terms: 8.25 + -3.25 = 5
x = 5

Simplifying
x = 5


Subproblem 2
x + 3.25 = -8.25

Simplifying
x + 3.25 = -8.25

Reorder the terms:
3.25 + x = -8.25

Solving
3.25 + x = -8.25

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-3.25' to each side of the equation.
3.25 + -3.25 + x = -8.25 + -3.25

Combine like terms: 3.25 + -3.25 = 0.00
0.00 + x = -8.25 + -3.25
x = -8.25 + -3.25

Combine like terms: -8.25 + -3.25 = -11.5
x = -11.5

Simplifying
x = -11.5


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5, -11.5}

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