(-2x+9)-(x^2+x-7)=1

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Solution for (-2x+9)-(x^2+x-7)=1 equation: 


Simplifying
(-2x + 9) + -1(x2 + x + -7) = 1

Reorder the terms:
(9 + -2x) + -1(x2 + x + -7) = 1

Remove parenthesis around (9 + -2x)
9 + -2x + -1(x2 + x + -7) = 1

Reorder the terms:
9 + -2x + -1(-7 + x + x2) = 1
9 + -2x + (-7 * -1 + x * -1 + x2 * -1) = 1
9 + -2x + (7 + -1x + -1x2) = 1

Reorder the terms:
9 + 7 + -2x + -1x + -1x2 = 1

Combine like terms: 9 + 7 = 16
16 + -2x + -1x + -1x2 = 1

Combine like terms: -2x + -1x = -3x
16 + -3x + -1x2 = 1

Solving
16 + -3x + -1x2 = 1

Solving for variable 'x'.

Reorder the terms:
16 + -1 + -3x + -1x2 = 1 + -1

Combine like terms: 16 + -1 = 15
15 + -3x + -1x2 = 1 + -1

Combine like terms: 1 + -1 = 0
15 + -3x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-15 + 3x + x2 = 0

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + 3x + 15 + x2 = 0 + 15

Reorder the terms:
-15 + 15 + 3x + x2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + 3x + x2 = 0 + 15
3x + x2 = 0 + 15

Combine like terms: 0 + 15 = 15
3x + x2 = 15

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 = 15 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 15 + 2.25

Combine like terms: 15 + 2.25 = 17.25
2.25 + 3x + x2 = 17.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 17.25

Calculate the square root of the right side: 4.153311931

Break this problem into two subproblems by setting 
(x + 1.5) equal to 4.153311931 and -4.153311931.


Subproblem 1
x + 1.5 = 4.153311931

Simplifying
x + 1.5 = 4.153311931

Reorder the terms:
1.5 + x = 4.153311931

Solving
1.5 + x = 4.153311931

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 = 4.153311931 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = 4.153311931 + -1.5
x = 4.153311931 + -1.5

Combine like terms: 4.153311931 + -1.5 = 2.653311931
x = 2.653311931

Simplifying
x = 2.653311931


Subproblem 2
x + 1.5 = -4.153311931

Simplifying
x + 1.5 = -4.153311931

Reorder the terms:
1.5 + x = -4.153311931

Solving
1.5 + x = -4.153311931

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 = -4.153311931 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = -4.153311931 + -1.5
x = -4.153311931 + -1.5

Combine like terms: -4.153311931 + -1.5 = -5.653311931
x = -5.653311931

Simplifying
x = -5.653311931


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.653311931, -5.653311931}

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