6x^2-19x=14

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Solution for 6x^2-19x=14 equation:


Simplifying
6x2 + -19x = 14

Reorder the terms:
-19x + 6x2 = 14

Solving
-19x + 6x2 = 14

Solving for variable 'x'.

Reorder the terms:
-14 + -19x + 6x2 = 14 + -14

Combine like terms: 14 + -14 = 0
-14 + -19x + 6x2 = 0

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

Divide each side by '6'.
-2.333333333 + -3.166666667x + x2 = 0

Move the constant term to the right:

Add '2.333333333' to each side of the equation.
-2.333333333 + -3.166666667x + 2.333333333 + x2 = 0 + 2.333333333

Reorder the terms:
-2.333333333 + 2.333333333 + -3.166666667x + x2 = 0 + 2.333333333

Combine like terms: -2.333333333 + 2.333333333 = 0.000000000
0.000000000 + -3.166666667x + x2 = 0 + 2.333333333
-3.166666667x + x2 = 0 + 2.333333333

Combine like terms: 0 + 2.333333333 = 2.333333333
-3.166666667x + x2 = 2.333333333

The x term is -3.166666667x.  Take half its coefficient (-1.583333334).
Square it (2.506944447) and add it to both sides.

Add '2.506944447' to each side of the equation.
-3.166666667x + 2.506944447 + x2 = 2.333333333 + 2.506944447

Reorder the terms:
2.506944447 + -3.166666667x + x2 = 2.333333333 + 2.506944447

Combine like terms: 2.333333333 + 2.506944447 = 4.84027778
2.506944447 + -3.166666667x + x2 = 4.84027778

Factor a perfect square on the left side:
(x + -1.583333334)(x + -1.583333334) = 4.84027778

Calculate the square root of the right side: 2.200063131

Break this problem into two subproblems by setting 
(x + -1.583333334) equal to 2.200063131 and -2.200063131.

Subproblem 1

x + -1.583333334 = 2.200063131 Simplifying x + -1.583333334 = 2.200063131 Reorder the terms: -1.583333334 + x = 2.200063131 Solving -1.583333334 + x = 2.200063131 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.583333334' to each side of the equation. -1.583333334 + 1.583333334 + x = 2.200063131 + 1.583333334 Combine like terms: -1.583333334 + 1.583333334 = 0.000000000 0.000000000 + x = 2.200063131 + 1.583333334 x = 2.200063131 + 1.583333334 Combine like terms: 2.200063131 + 1.583333334 = 3.783396465 x = 3.783396465 Simplifying x = 3.783396465

Subproblem 2

x + -1.583333334 = -2.200063131 Simplifying x + -1.583333334 = -2.200063131 Reorder the terms: -1.583333334 + x = -2.200063131 Solving -1.583333334 + x = -2.200063131 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.583333334' to each side of the equation. -1.583333334 + 1.583333334 + x = -2.200063131 + 1.583333334 Combine like terms: -1.583333334 + 1.583333334 = 0.000000000 0.000000000 + x = -2.200063131 + 1.583333334 x = -2.200063131 + 1.583333334 Combine like terms: -2.200063131 + 1.583333334 = -0.616729797 x = -0.616729797 Simplifying x = -0.616729797

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.783396465, -0.616729797}

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