6x^2-19x-15=0

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Solution for 6x^2-19x-15=0 equation:


Simplifying
6x2 + -19x + -15 = 0

Reorder the terms:
-15 + -19x + 6x2 = 0

Solving
-15 + -19x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
-2.5 + -3.166666667x + x2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + -3.166666667x + 2.5 + x2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + -3.166666667x + x2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + -3.166666667x + x2 = 0 + 2.5
-3.166666667x + x2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
-3.166666667x + x2 = 2.5

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 = 2.5 + 2.506944447

Reorder the terms:
2.506944447 + -3.166666667x + x2 = 2.5 + 2.506944447

Combine like terms: 2.5 + 2.506944447 = 5.006944447
2.506944447 + -3.166666667x + x2 = 5.006944447

Factor a perfect square on the left side:
(x + -1.583333334)(x + -1.583333334) = 5.006944447

Calculate the square root of the right side: 2.237620264

Break this problem into two subproblems by setting 
(x + -1.583333334) equal to 2.237620264 and -2.237620264.

Subproblem 1

x + -1.583333334 = 2.237620264 Simplifying x + -1.583333334 = 2.237620264 Reorder the terms: -1.583333334 + x = 2.237620264 Solving -1.583333334 + x = 2.237620264 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 = 2.237620264 + 1.583333334 Combine like terms: -1.583333334 + 1.583333334 = 0.000000000 0.000000000 + x = 2.237620264 + 1.583333334 x = 2.237620264 + 1.583333334 Combine like terms: 2.237620264 + 1.583333334 = 3.820953598 x = 3.820953598 Simplifying x = 3.820953598

Subproblem 2

x + -1.583333334 = -2.237620264 Simplifying x + -1.583333334 = -2.237620264 Reorder the terms: -1.583333334 + x = -2.237620264 Solving -1.583333334 + x = -2.237620264 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 = -2.237620264 + 1.583333334 Combine like terms: -1.583333334 + 1.583333334 = 0.000000000 0.000000000 + x = -2.237620264 + 1.583333334 x = -2.237620264 + 1.583333334 Combine like terms: -2.237620264 + 1.583333334 = -0.65428693 x = -0.65428693 Simplifying x = -0.65428693

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.820953598, -0.65428693}

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