314.019418x^2+24673.20799626468x+1=0

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Solution for 314.019418x^2+24673.20799626468x+1=0 equation:


Simplifying
314.019418x2 + 24673.20799626468x + 1 = 0

Reorder the terms:
1 + 24673.20799626468x + 314.019418x2 = 0

Solving
1 + 24673.20799626468x + 314.019418x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
314.019418 the coefficient of the squared term: 

Divide each side by '314.019418'.
0.003184516443 + 78.57223656x + x2 = 0

Move the constant term to the right:

Add '-0.003184516443' to each side of the equation.
0.003184516443 + 78.57223656x + -0.003184516443 + x2 = 0 + -0.003184516443

Reorder the terms:
0.003184516443 + -0.003184516443 + 78.57223656x + x2 = 0 + -0.003184516443

Combine like terms: 0.003184516443 + -0.003184516443 = 0.000000000000
0.000000000000 + 78.57223656x + x2 = 0 + -0.003184516443
78.57223656x + x2 = 0 + -0.003184516443

Combine like terms: 0 + -0.003184516443 = -0.003184516443
78.57223656x + x2 = -0.003184516443

The x term is 78.57223656x.  Take half its coefficient (39.28611828).
Square it (1543.399090) and add it to both sides.

Add '1543.399090' to each side of the equation.
78.57223656x + 1543.399090 + x2 = -0.003184516443 + 1543.399090

Reorder the terms:
1543.399090 + 78.57223656x + x2 = -0.003184516443 + 1543.399090

Combine like terms: -0.003184516443 + 1543.399090 = 1543.395905483557
1543.399090 + 78.57223656x + x2 = 1543.395905483557

Factor a perfect square on the left side:
(x + 39.28611828)(x + 39.28611828) = 1543.395905483557

Calculate the square root of the right side: 39.286077756

Break this problem into two subproblems by setting 
(x + 39.28611828) equal to 39.286077756 and -39.286077756.

Subproblem 1

x + 39.28611828 = 39.286077756 Simplifying x + 39.28611828 = 39.286077756 Reorder the terms: 39.28611828 + x = 39.286077756 Solving 39.28611828 + x = 39.286077756 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-39.28611828' to each side of the equation. 39.28611828 + -39.28611828 + x = 39.286077756 + -39.28611828 Combine like terms: 39.28611828 + -39.28611828 = 0.00000000 0.00000000 + x = 39.286077756 + -39.28611828 x = 39.286077756 + -39.28611828 Combine like terms: 39.286077756 + -39.28611828 = -0.000040524 x = -0.000040524 Simplifying x = -0.000040524

Subproblem 2

x + 39.28611828 = -39.286077756 Simplifying x + 39.28611828 = -39.286077756 Reorder the terms: 39.28611828 + x = -39.286077756 Solving 39.28611828 + x = -39.286077756 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-39.28611828' to each side of the equation. 39.28611828 + -39.28611828 + x = -39.286077756 + -39.28611828 Combine like terms: 39.28611828 + -39.28611828 = 0.00000000 0.00000000 + x = -39.286077756 + -39.28611828 x = -39.286077756 + -39.28611828 Combine like terms: -39.286077756 + -39.28611828 = -78.572196036 x = -78.572196036 Simplifying x = -78.572196036

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.000040524, -78.572196036}

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