x^2+5x-1972=0

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Solution for x^2+5x-1972=0 equation: 


Simplifying
x2 + 5x + -1972 = 0

Reorder the terms:
-1972 + 5x + x2 = 0

Solving
-1972 + 5x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '1972' to each side of the equation.
-1972 + 5x + 1972 + x2 = 0 + 1972

Reorder the terms:
-1972 + 1972 + 5x + x2 = 0 + 1972

Combine like terms: -1972 + 1972 = 0
0 + 5x + x2 = 0 + 1972
5x + x2 = 0 + 1972

Combine like terms: 0 + 1972 = 1972
5x + x2 = 1972

The x term is 5x.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5x + 6.25 + x2 = 1972 + 6.25

Reorder the terms:
6.25 + 5x + x2 = 1972 + 6.25

Combine like terms: 1972 + 6.25 = 1978.25
6.25 + 5x + x2 = 1978.25

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 1978.25

Calculate the square root of the right side: 44.477522413

Break this problem into two subproblems by setting 
(x + 2.5) equal to 44.477522413 and -44.477522413.


Subproblem 1
x + 2.5 = 44.477522413

Simplifying
x + 2.5 = 44.477522413

Reorder the terms:
2.5 + x = 44.477522413

Solving
2.5 + x = 44.477522413

Solving for variable 'x'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + x = 44.477522413 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = 44.477522413 + -2.5
x = 44.477522413 + -2.5

Combine like terms: 44.477522413 + -2.5 = 41.977522413
x = 41.977522413

Simplifying
x = 41.977522413


Subproblem 2
x + 2.5 = -44.477522413

Simplifying
x + 2.5 = -44.477522413

Reorder the terms:
2.5 + x = -44.477522413

Solving
2.5 + x = -44.477522413

Solving for variable 'x'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + x = -44.477522413 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = -44.477522413 + -2.5
x = -44.477522413 + -2.5

Combine like terms: -44.477522413 + -2.5 = -46.977522413
x = -46.977522413

Simplifying
x = -46.977522413


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {41.977522413, -46.977522413}

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