-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 = 1543.402274516443
1543.399090 + -78.57223656x + x2 = 1543.402274516443

Factor a perfect square on the left side:
(x + -39.28611828)(x + -39.28611828) = 1543.402274516443

Calculate the square root of the right side: 39.286158816

Break this problem into two subproblems by setting 
(x + -39.28611828) equal to 39.286158816 and -39.286158816.

Subproblem 1

x + -39.28611828 = 39.286158816 Simplifying x + -39.28611828 = 39.286158816 Reorder the terms: -39.28611828 + x = 39.286158816 Solving -39.28611828 + x = 39.286158816 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 = 39.286158816 + 39.28611828 Combine like terms: -39.28611828 + 39.28611828 = 0.00000000 0.00000000 + x = 39.286158816 + 39.28611828 x = 39.286158816 + 39.28611828 Combine like terms: 39.286158816 + 39.28611828 = 78.572277096 x = 78.572277096 Simplifying x = 78.572277096

Subproblem 2

x + -39.28611828 = -39.286158816 Simplifying x + -39.28611828 = -39.286158816 Reorder the terms: -39.28611828 + x = -39.286158816 Solving -39.28611828 + x = -39.286158816 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 = -39.286158816 + 39.28611828 Combine like terms: -39.28611828 + 39.28611828 = 0.00000000 0.00000000 + x = -39.286158816 + 39.28611828 x = -39.286158816 + 39.28611828 Combine like terms: -39.286158816 + 39.28611828 = -0.000040536 x = -0.000040536 Simplifying x = -0.000040536

Solution

The solution to the problem is based on the solutions from the subproblems. x = {78.572277096, -0.000040536}

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