x+49=x^2-14

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Solution for x+49=x^2-14 equation:


Simplifying
x + 49 = x2 + -14

Reorder the terms:
49 + x = x2 + -14

Reorder the terms:
49 + x = -14 + x2

Solving
49 + x = -14 + x2

Solving for variable 'x'.

Reorder the terms:
49 + 14 + x + -1x2 = -14 + x2 + 14 + -1x2

Combine like terms: 49 + 14 = 63
63 + x + -1x2 = -14 + x2 + 14 + -1x2

Reorder the terms:
63 + x + -1x2 = -14 + 14 + x2 + -1x2

Combine like terms: -14 + 14 = 0
63 + x + -1x2 = 0 + x2 + -1x2
63 + x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
63 + x + -1x2 = 0

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

Divide each side by '-1'.
-63 + -1x + x2 = 0

Move the constant term to the right:

Add '63' to each side of the equation.
-63 + -1x + 63 + x2 = 0 + 63

Reorder the terms:
-63 + 63 + -1x + x2 = 0 + 63

Combine like terms: -63 + 63 = 0
0 + -1x + x2 = 0 + 63
-1x + x2 = 0 + 63

Combine like terms: 0 + 63 = 63
-1x + x2 = 63

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 = 63 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 63 + 0.25

Combine like terms: 63 + 0.25 = 63.25
0.25 + -1x + x2 = 63.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 63.25

Calculate the square root of the right side: 7.95298686

Break this problem into two subproblems by setting 
(x + 0.5) equal to 7.95298686 and -7.95298686.

Subproblem 1

x + 0.5 = 7.95298686 Simplifying x + 0.5 = 7.95298686 Reorder the terms: 0.5 + x = 7.95298686 Solving 0.5 + x = 7.95298686 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 = 7.95298686 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 7.95298686 + -0.5 x = 7.95298686 + -0.5 Combine like terms: 7.95298686 + -0.5 = 7.45298686 x = 7.45298686 Simplifying x = 7.45298686

Subproblem 2

x + 0.5 = -7.95298686 Simplifying x + 0.5 = -7.95298686 Reorder the terms: 0.5 + x = -7.95298686 Solving 0.5 + x = -7.95298686 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 = -7.95298686 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -7.95298686 + -0.5 x = -7.95298686 + -0.5 Combine like terms: -7.95298686 + -0.5 = -8.45298686 x = -8.45298686 Simplifying x = -8.45298686

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.45298686, -8.45298686}

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