0=6x^2-15x-7

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Solution for 0=6x^2-15x-7 equation:


Simplifying
0 = 6x2 + -15x + -7

Reorder the terms:
0 = -7 + -15x + 6x2

Solving
0 = -7 + -15x + 6x2

Solving for variable 'x'.

Combine like terms: 0 + 7 = 7
7 + 15x + -6x2 = -7 + -15x + 6x2 + 7 + 15x + -6x2

Reorder the terms:
7 + 15x + -6x2 = -7 + 7 + -15x + 15x + 6x2 + -6x2

Combine like terms: -7 + 7 = 0
7 + 15x + -6x2 = 0 + -15x + 15x + 6x2 + -6x2
7 + 15x + -6x2 = -15x + 15x + 6x2 + -6x2

Combine like terms: -15x + 15x = 0
7 + 15x + -6x2 = 0 + 6x2 + -6x2
7 + 15x + -6x2 = 6x2 + -6x2

Combine like terms: 6x2 + -6x2 = 0
7 + 15x + -6x2 = 0

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

Divide each side by '-6'.
-1.166666667 + -2.5x + x2 = 0

Move the constant term to the right:

Add '1.166666667' to each side of the equation.
-1.166666667 + -2.5x + 1.166666667 + x2 = 0 + 1.166666667

Reorder the terms:
-1.166666667 + 1.166666667 + -2.5x + x2 = 0 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + -2.5x + x2 = 0 + 1.166666667
-2.5x + x2 = 0 + 1.166666667

Combine like terms: 0 + 1.166666667 = 1.166666667
-2.5x + x2 = 1.166666667

The x term is -2.5x.  Take half its coefficient (-1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
-2.5x + 1.5625 + x2 = 1.166666667 + 1.5625

Reorder the terms:
1.5625 + -2.5x + x2 = 1.166666667 + 1.5625

Combine like terms: 1.166666667 + 1.5625 = 2.729166667
1.5625 + -2.5x + x2 = 2.729166667

Factor a perfect square on the left side:
(x + -1.25)(x + -1.25) = 2.729166667

Calculate the square root of the right side: 1.652018967

Break this problem into two subproblems by setting 
(x + -1.25) equal to 1.652018967 and -1.652018967.

Subproblem 1

x + -1.25 = 1.652018967 Simplifying x + -1.25 = 1.652018967 Reorder the terms: -1.25 + x = 1.652018967 Solving -1.25 + x = 1.652018967 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = 1.652018967 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = 1.652018967 + 1.25 x = 1.652018967 + 1.25 Combine like terms: 1.652018967 + 1.25 = 2.902018967 x = 2.902018967 Simplifying x = 2.902018967

Subproblem 2

x + -1.25 = -1.652018967 Simplifying x + -1.25 = -1.652018967 Reorder the terms: -1.25 + x = -1.652018967 Solving -1.25 + x = -1.652018967 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = -1.652018967 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = -1.652018967 + 1.25 x = -1.652018967 + 1.25 Combine like terms: -1.652018967 + 1.25 = -0.402018967 x = -0.402018967 Simplifying x = -0.402018967

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.902018967, -0.402018967}

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