4x^2-31x-257=0

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Solution for 4x^2-31x-257=0 equation:


Simplifying
4x2 + -31x + -257 = 0

Reorder the terms:
-257 + -31x + 4x2 = 0

Solving
-257 + -31x + 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'.
-64.25 + -7.75x + x2 = 0

Move the constant term to the right:

Add '64.25' to each side of the equation.
-64.25 + -7.75x + 64.25 + x2 = 0 + 64.25

Reorder the terms:
-64.25 + 64.25 + -7.75x + x2 = 0 + 64.25

Combine like terms: -64.25 + 64.25 = 0.00
0.00 + -7.75x + x2 = 0 + 64.25
-7.75x + x2 = 0 + 64.25

Combine like terms: 0 + 64.25 = 64.25
-7.75x + x2 = 64.25

The x term is -7.75x.  Take half its coefficient (-3.875).
Square it (15.015625) and add it to both sides.

Add '15.015625' to each side of the equation.
-7.75x + 15.015625 + x2 = 64.25 + 15.015625

Reorder the terms:
15.015625 + -7.75x + x2 = 64.25 + 15.015625

Combine like terms: 64.25 + 15.015625 = 79.265625
15.015625 + -7.75x + x2 = 79.265625

Factor a perfect square on the left side:
(x + -3.875)(x + -3.875) = 79.265625

Calculate the square root of the right side: 8.903124452

Break this problem into two subproblems by setting 
(x + -3.875) equal to 8.903124452 and -8.903124452.

Subproblem 1

x + -3.875 = 8.903124452 Simplifying x + -3.875 = 8.903124452 Reorder the terms: -3.875 + x = 8.903124452 Solving -3.875 + x = 8.903124452 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.875' to each side of the equation. -3.875 + 3.875 + x = 8.903124452 + 3.875 Combine like terms: -3.875 + 3.875 = 0.000 0.000 + x = 8.903124452 + 3.875 x = 8.903124452 + 3.875 Combine like terms: 8.903124452 + 3.875 = 12.778124452 x = 12.778124452 Simplifying x = 12.778124452

Subproblem 2

x + -3.875 = -8.903124452 Simplifying x + -3.875 = -8.903124452 Reorder the terms: -3.875 + x = -8.903124452 Solving -3.875 + x = -8.903124452 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.875' to each side of the equation. -3.875 + 3.875 + x = -8.903124452 + 3.875 Combine like terms: -3.875 + 3.875 = 0.000 0.000 + x = -8.903124452 + 3.875 x = -8.903124452 + 3.875 Combine like terms: -8.903124452 + 3.875 = -5.028124452 x = -5.028124452 Simplifying x = -5.028124452

Solution

The solution to the problem is based on the solutions from the subproblems. x = {12.778124452, -5.028124452}

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