12x^2-15x-1=0

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Solution for 12x^2-15x-1=0 equation:


Simplifying
12x2 + -15x + -1 = 0

Reorder the terms:
-1 + -15x + 12x2 = 0

Solving
-1 + -15x + 12x2 = 0

Solving for variable 'x'.

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

Divide each side by '12'.
-0.08333333333 + -1.25x + x2 = 0

Move the constant term to the right:

Add '0.08333333333' to each side of the equation.
-0.08333333333 + -1.25x + 0.08333333333 + x2 = 0 + 0.08333333333

Reorder the terms:
-0.08333333333 + 0.08333333333 + -1.25x + x2 = 0 + 0.08333333333

Combine like terms: -0.08333333333 + 0.08333333333 = 0.00000000000
0.00000000000 + -1.25x + x2 = 0 + 0.08333333333
-1.25x + x2 = 0 + 0.08333333333

Combine like terms: 0 + 0.08333333333 = 0.08333333333
-1.25x + x2 = 0.08333333333

The x term is -1.25x.  Take half its coefficient (-0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
-1.25x + 0.390625 + x2 = 0.08333333333 + 0.390625

Reorder the terms:
0.390625 + -1.25x + x2 = 0.08333333333 + 0.390625

Combine like terms: 0.08333333333 + 0.390625 = 0.47395833333
0.390625 + -1.25x + x2 = 0.47395833333

Factor a perfect square on the left side:
(x + -0.625)(x + -0.625) = 0.47395833333

Calculate the square root of the right side: 0.688446318

Break this problem into two subproblems by setting 
(x + -0.625) equal to 0.688446318 and -0.688446318.

Subproblem 1

x + -0.625 = 0.688446318 Simplifying x + -0.625 = 0.688446318 Reorder the terms: -0.625 + x = 0.688446318 Solving -0.625 + x = 0.688446318 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + x = 0.688446318 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + x = 0.688446318 + 0.625 x = 0.688446318 + 0.625 Combine like terms: 0.688446318 + 0.625 = 1.313446318 x = 1.313446318 Simplifying x = 1.313446318

Subproblem 2

x + -0.625 = -0.688446318 Simplifying x + -0.625 = -0.688446318 Reorder the terms: -0.625 + x = -0.688446318 Solving -0.625 + x = -0.688446318 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + x = -0.688446318 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + x = -0.688446318 + 0.625 x = -0.688446318 + 0.625 Combine like terms: -0.688446318 + 0.625 = -0.063446318 x = -0.063446318 Simplifying x = -0.063446318

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.313446318, -0.063446318}

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