7x^2+61x+97=0

Solution for 7x^2+61x+97=0 equation:

Simplifying
7x2 + 61x + 97 = 0

Reorder the terms:
97 + 61x + 7x2 = 0

Solving
97 + 61x + 7x2 = 0

Solving for variable 'x'.

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

Divide each side by '7'.
13.85714286 + 8.714285714x + x2 = 0

Move the constant term to the right:

Add '-13.85714286' to each side of the equation.
13.85714286 + 8.714285714x + -13.85714286 + x2 = 0 + -13.85714286

Reorder the terms:
13.85714286 + -13.85714286 + 8.714285714x + x2 = 0 + -13.85714286

Combine like terms: 13.85714286 + -13.85714286 = 0.00000000
0.00000000 + 8.714285714x + x2 = 0 + -13.85714286
8.714285714x + x2 = 0 + -13.85714286

Combine like terms: 0 + -13.85714286 = -13.85714286
8.714285714x + x2 = -13.85714286

The x term is 8.714285714x.  Take half its coefficient (4.357142857).
Square it (18.98469388) and add it to both sides.

Add '18.98469388' to each side of the equation.
8.714285714x + 18.98469388 + x2 = -13.85714286 + 18.98469388

Reorder the terms:
18.98469388 + 8.714285714x + x2 = -13.85714286 + 18.98469388

Combine like terms: -13.85714286 + 18.98469388 = 5.12755102
18.98469388 + 8.714285714x + x2 = 5.12755102

Factor a perfect square on the left side:
(x + 4.357142857)(x + 4.357142857) = 5.12755102

Calculate the square root of the right side: 2.264409641

Break this problem into two subproblems by setting 
(x + 4.357142857) equal to 2.264409641 and -2.264409641.

Subproblem 1
x + 4.357142857 = 2.264409641

Simplifying
x + 4.357142857 = 2.264409641

Reorder the terms:
4.357142857 + x = 2.264409641

Solving
4.357142857 + x = 2.264409641

Solving for variable 'x'.

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

Add '-4.357142857' to each side of the equation.
4.357142857 + -4.357142857 + x = 2.264409641 + -4.357142857

Combine like terms: 4.357142857 + -4.357142857 = 0.000000000
0.000000000 + x = 2.264409641 + -4.357142857
x = 2.264409641 + -4.357142857

Combine like terms: 2.264409641 + -4.357142857 = -2.092733216
x = -2.092733216

Simplifying
x = -2.092733216

Subproblem 2
x + 4.357142857 = -2.264409641

Simplifying
x + 4.357142857 = -2.264409641

Reorder the terms:
4.357142857 + x = -2.264409641

Solving
4.357142857 + x = -2.264409641

Solving for variable 'x'.

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

Add '-4.357142857' to each side of the equation.
4.357142857 + -4.357142857 + x = -2.264409641 + -4.357142857

Combine like terms: 4.357142857 + -4.357142857 = 0.000000000
0.000000000 + x = -2.264409641 + -4.357142857
x = -2.264409641 + -4.357142857

Combine like terms: -2.264409641 + -4.357142857 = -6.621552498
x = -6.621552498

Simplifying
x = -6.621552498

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-2.092733216, -6.621552498}