(4x^2)+80x+397=0

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Solution for (4x^2)+80x+397=0 equation:


Simplifying
(4x2) + 80x + 397 = 0

Reorder the terms:
397 + 80x + (4x2) = 0

Solving
397 + 80x + (4x2) = 0

Solving for variable 'x'.

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

Divide each side by '4'.
99.25 + 20x + x2 = 0

Move the constant term to the right:

Add '-99.25' to each side of the equation.
99.25 + 20x + -99.25 + x2 = 0 + -99.25

Reorder the terms:
99.25 + -99.25 + 20x + x2 = 0 + -99.25

Combine like terms: 99.25 + -99.25 = 0.00
0.00 + 20x + x2 = 0 + -99.25
20x + x2 = 0 + -99.25

Combine like terms: 0 + -99.25 = -99.25
20x + x2 = -99.25

The x term is 20x.  Take half its coefficient (10).
Square it (100) and add it to both sides.

Add '100' to each side of the equation.
20x + 100 + x2 = -99.25 + 100

Reorder the terms:
100 + 20x + x2 = -99.25 + 100

Combine like terms: -99.25 + 100 = 0.75
100 + 20x + x2 = 0.75

Factor a perfect square on the left side:
((x) + 10)((x) + 10) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
((x) + 10) equal to 0.866025404 and -0.866025404.

Subproblem 1

(x) + 10 = 0.866025404 Simplifying (x) + 10 = 0.866025404 x + 10 = 0.866025404 Reorder the terms: 10 + x = 0.866025404 Solving 10 + x = 0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-10' to each side of the equation. 10 + -10 + x = 0.866025404 + -10 Combine like terms: 10 + -10 = 0 0 + x = 0.866025404 + -10 x = 0.866025404 + -10 Combine like terms: 0.866025404 + -10 = -9.133974596 x = -9.133974596 Simplifying x = -9.133974596

Subproblem 2

(x) + 10 = -0.866025404 Simplifying (x) + 10 = -0.866025404 x + 10 = -0.866025404 Reorder the terms: 10 + x = -0.866025404 Solving 10 + x = -0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-10' to each side of the equation. 10 + -10 + x = -0.866025404 + -10 Combine like terms: 10 + -10 = 0 0 + x = -0.866025404 + -10 x = -0.866025404 + -10 Combine like terms: -0.866025404 + -10 = -10.866025404 x = -10.866025404 Simplifying x = -10.866025404

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-9.133974596, -10.866025404}

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