3x^2-x-51=0

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Solution for 3x^2-x-51=0 equation:


Simplifying
3x2 + -1x + -51 = 0

Reorder the terms:
-51 + -1x + 3x2 = 0

Solving
-51 + -1x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
-17 + -0.3333333333x + x2 = 0

Move the constant term to the right:

Add '17' to each side of the equation.
-17 + -0.3333333333x + 17 + x2 = 0 + 17

Reorder the terms:
-17 + 17 + -0.3333333333x + x2 = 0 + 17

Combine like terms: -17 + 17 = 0
0 + -0.3333333333x + x2 = 0 + 17
-0.3333333333x + x2 = 0 + 17

Combine like terms: 0 + 17 = 17
-0.3333333333x + x2 = 17

The x term is -0.3333333333x.  Take half its coefficient (-0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
-0.3333333333x + 0.02777777779 + x2 = 17 + 0.02777777779

Reorder the terms:
0.02777777779 + -0.3333333333x + x2 = 17 + 0.02777777779

Combine like terms: 17 + 0.02777777779 = 17.02777777779
0.02777777779 + -0.3333333333x + x2 = 17.02777777779

Factor a perfect square on the left side:
(x + -0.1666666667)(x + -0.1666666667) = 17.02777777779

Calculate the square root of the right side: 4.126472801

Break this problem into two subproblems by setting 
(x + -0.1666666667) equal to 4.126472801 and -4.126472801.

Subproblem 1

x + -0.1666666667 = 4.126472801 Simplifying x + -0.1666666667 = 4.126472801 Reorder the terms: -0.1666666667 + x = 4.126472801 Solving -0.1666666667 + x = 4.126472801 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + x = 4.126472801 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + x = 4.126472801 + 0.1666666667 x = 4.126472801 + 0.1666666667 Combine like terms: 4.126472801 + 0.1666666667 = 4.2931394677 x = 4.2931394677 Simplifying x = 4.2931394677

Subproblem 2

x + -0.1666666667 = -4.126472801 Simplifying x + -0.1666666667 = -4.126472801 Reorder the terms: -0.1666666667 + x = -4.126472801 Solving -0.1666666667 + x = -4.126472801 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + x = -4.126472801 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + x = -4.126472801 + 0.1666666667 x = -4.126472801 + 0.1666666667 Combine like terms: -4.126472801 + 0.1666666667 = -3.9598061343 x = -3.9598061343 Simplifying x = -3.9598061343

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.2931394677, -3.9598061343}

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