7.5=2x^2+7x

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Solution for 7.5=2x^2+7x equation: 


Simplifying
7.5 = 2x2 + 7x

Reorder the terms:
7.5 = 7x + 2x2

Solving
7.5 = 7x + 2x2

Solving for variable 'x'.

Reorder the terms:
7.5 + -7x + -2x2 = 7x + -7x + 2x2 + -2x2

Combine like terms: 7x + -7x = 0
7.5 + -7x + -2x2 = 0 + 2x2 + -2x2
7.5 + -7x + -2x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
7.5 + -7x + -2x2 = 0

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

Divide each side by '-2'.
-3.75 + 3.5x + x2 = 0

Move the constant term to the right:

Add '3.75' to each side of the equation.
-3.75 + 3.5x + 3.75 + x2 = 0 + 3.75

Reorder the terms:
-3.75 + 3.75 + 3.5x + x2 = 0 + 3.75

Combine like terms: -3.75 + 3.75 = 0.00
0.00 + 3.5x + x2 = 0 + 3.75
3.5x + x2 = 0 + 3.75

Combine like terms: 0 + 3.75 = 3.75
3.5x + x2 = 3.75

The x term is 3.5x.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5x + 3.0625 + x2 = 3.75 + 3.0625

Reorder the terms:
3.0625 + 3.5x + x2 = 3.75 + 3.0625

Combine like terms: 3.75 + 3.0625 = 6.8125
3.0625 + 3.5x + x2 = 6.8125

Factor a perfect square on the left side:
(x + 1.75)(x + 1.75) = 6.8125

Calculate the square root of the right side: 2.610076627

Break this problem into two subproblems by setting 
(x + 1.75) equal to 2.610076627 and -2.610076627.


Subproblem 1
x + 1.75 = 2.610076627

Simplifying
x + 1.75 = 2.610076627

Reorder the terms:
1.75 + x = 2.610076627

Solving
1.75 + x = 2.610076627

Solving for variable 'x'.

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

Add '-1.75' to each side of the equation.
1.75 + -1.75 + x = 2.610076627 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + x = 2.610076627 + -1.75
x = 2.610076627 + -1.75

Combine like terms: 2.610076627 + -1.75 = 0.860076627
x = 0.860076627

Simplifying
x = 0.860076627


Subproblem 2
x + 1.75 = -2.610076627

Simplifying
x + 1.75 = -2.610076627

Reorder the terms:
1.75 + x = -2.610076627

Solving
1.75 + x = -2.610076627

Solving for variable 'x'.

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

Add '-1.75' to each side of the equation.
1.75 + -1.75 + x = -2.610076627 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + x = -2.610076627 + -1.75
x = -2.610076627 + -1.75

Combine like terms: -2.610076627 + -1.75 = -4.360076627
x = -4.360076627

Simplifying
x = -4.360076627


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.860076627, -4.360076627}

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