X+7=X^2-4

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Solution for X+7=X^2-4 equation:


Simplifying
X + 7 = X2 + -4

Reorder the terms:
7 + X = X2 + -4

Reorder the terms:
7 + X = -4 + X2

Solving
7 + X = -4 + X2

Solving for variable 'X'.

Reorder the terms:
7 + 4 + X + -1X2 = -4 + X2 + 4 + -1X2

Combine like terms: 7 + 4 = 11
11 + X + -1X2 = -4 + X2 + 4 + -1X2

Reorder the terms:
11 + X + -1X2 = -4 + 4 + X2 + -1X2

Combine like terms: -4 + 4 = 0
11 + X + -1X2 = 0 + X2 + -1X2
11 + X + -1X2 = X2 + -1X2

Combine like terms: X2 + -1X2 = 0
11 + X + -1X2 = 0

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

Divide each side by '-1'.
-11 + -1X + X2 = 0

Move the constant term to the right:

Add '11' to each side of the equation.
-11 + -1X + 11 + X2 = 0 + 11

Reorder the terms:
-11 + 11 + -1X + X2 = 0 + 11

Combine like terms: -11 + 11 = 0
0 + -1X + X2 = 0 + 11
-1X + X2 = 0 + 11

Combine like terms: 0 + 11 = 11
-1X + X2 = 11

The X term is X.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1X + 0.25 + X2 = 11 + 0.25

Reorder the terms:
0.25 + -1X + X2 = 11 + 0.25

Combine like terms: 11 + 0.25 = 11.25
0.25 + -1X + X2 = 11.25

Factor a perfect square on the left side:
(X + 0.5)(X + 0.5) = 11.25

Calculate the square root of the right side: 3.354101966

Break this problem into two subproblems by setting 
(X + 0.5) equal to 3.354101966 and -3.354101966.

Subproblem 1

X + 0.5 = 3.354101966 Simplifying X + 0.5 = 3.354101966 Reorder the terms: 0.5 + X = 3.354101966 Solving 0.5 + X = 3.354101966 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + X = 3.354101966 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + X = 3.354101966 + -0.5 X = 3.354101966 + -0.5 Combine like terms: 3.354101966 + -0.5 = 2.854101966 X = 2.854101966 Simplifying X = 2.854101966

Subproblem 2

X + 0.5 = -3.354101966 Simplifying X + 0.5 = -3.354101966 Reorder the terms: 0.5 + X = -3.354101966 Solving 0.5 + X = -3.354101966 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + X = -3.354101966 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + X = -3.354101966 + -0.5 X = -3.354101966 + -0.5 Combine like terms: -3.354101966 + -0.5 = -3.854101966 X = -3.854101966 Simplifying X = -3.854101966

Solution

The solution to the problem is based on the solutions from the subproblems. X = {2.854101966, -3.854101966}

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