6x^2-5x-15=0

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Solution for 6x^2-5x-15=0 equation:


Simplifying
6x2 + -5x + -15 = 0

Reorder the terms:
-15 + -5x + 6x2 = 0

Solving
-15 + -5x + 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 + -0.8333333333x + x2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + -0.8333333333x + 2.5 + x2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + -0.8333333333x + x2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + -0.8333333333x + x2 = 0 + 2.5
-0.8333333333x + x2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
-0.8333333333x + x2 = 2.5

The x term is -0.8333333333x.  Take half its coefficient (-0.4166666667).
Square it (0.1736111111) and add it to both sides.

Add '0.1736111111' to each side of the equation.
-0.8333333333x + 0.1736111111 + x2 = 2.5 + 0.1736111111

Reorder the terms:
0.1736111111 + -0.8333333333x + x2 = 2.5 + 0.1736111111

Combine like terms: 2.5 + 0.1736111111 = 2.6736111111
0.1736111111 + -0.8333333333x + x2 = 2.6736111111

Factor a perfect square on the left side:
(x + -0.4166666667)(x + -0.4166666667) = 2.6736111111

Calculate the square root of the right side: 1.635118073

Break this problem into two subproblems by setting 
(x + -0.4166666667) equal to 1.635118073 and -1.635118073.

Subproblem 1

x + -0.4166666667 = 1.635118073 Simplifying x + -0.4166666667 = 1.635118073 Reorder the terms: -0.4166666667 + x = 1.635118073 Solving -0.4166666667 + x = 1.635118073 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.4166666667 + x = 1.635118073 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + x = 1.635118073 + 0.4166666667 x = 1.635118073 + 0.4166666667 Combine like terms: 1.635118073 + 0.4166666667 = 2.0517847397 x = 2.0517847397 Simplifying x = 2.0517847397

Subproblem 2

x + -0.4166666667 = -1.635118073 Simplifying x + -0.4166666667 = -1.635118073 Reorder the terms: -0.4166666667 + x = -1.635118073 Solving -0.4166666667 + x = -1.635118073 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.4166666667 + x = -1.635118073 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + x = -1.635118073 + 0.4166666667 x = -1.635118073 + 0.4166666667 Combine like terms: -1.635118073 + 0.4166666667 = -1.2184514063 x = -1.2184514063 Simplifying x = -1.2184514063

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.0517847397, -1.2184514063}

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