15x^2-30x-1=0

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Solution for 15x^2-30x-1=0 equation:


Simplifying
15x2 + -30x + -1 = 0

Reorder the terms:
-1 + -30x + 15x2 = 0

Solving
-1 + -30x + 15x2 = 0

Solving for variable 'x'.

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

Divide each side by '15'.
-0.06666666667 + -2x + x2 = 0

Move the constant term to the right:

Add '0.06666666667' to each side of the equation.
-0.06666666667 + -2x + 0.06666666667 + x2 = 0 + 0.06666666667

Reorder the terms:
-0.06666666667 + 0.06666666667 + -2x + x2 = 0 + 0.06666666667

Combine like terms: -0.06666666667 + 0.06666666667 = 0.00000000000
0.00000000000 + -2x + x2 = 0 + 0.06666666667
-2x + x2 = 0 + 0.06666666667

Combine like terms: 0 + 0.06666666667 = 0.06666666667
-2x + x2 = 0.06666666667

The x term is -2x.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2x + 1 + x2 = 0.06666666667 + 1

Reorder the terms:
1 + -2x + x2 = 0.06666666667 + 1

Combine like terms: 0.06666666667 + 1 = 1.06666666667
1 + -2x + x2 = 1.06666666667

Factor a perfect square on the left side:
(x + -1)(x + -1) = 1.06666666667

Calculate the square root of the right side: 1.032795559

Break this problem into two subproblems by setting 
(x + -1) equal to 1.032795559 and -1.032795559.

Subproblem 1

x + -1 = 1.032795559 Simplifying x + -1 = 1.032795559 Reorder the terms: -1 + x = 1.032795559 Solving -1 + x = 1.032795559 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = 1.032795559 + 1 Combine like terms: -1 + 1 = 0 0 + x = 1.032795559 + 1 x = 1.032795559 + 1 Combine like terms: 1.032795559 + 1 = 2.032795559 x = 2.032795559 Simplifying x = 2.032795559

Subproblem 2

x + -1 = -1.032795559 Simplifying x + -1 = -1.032795559 Reorder the terms: -1 + x = -1.032795559 Solving -1 + x = -1.032795559 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = -1.032795559 + 1 Combine like terms: -1 + 1 = 0 0 + x = -1.032795559 + 1 x = -1.032795559 + 1 Combine like terms: -1.032795559 + 1 = -0.032795559 x = -0.032795559 Simplifying x = -0.032795559

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.032795559, -0.032795559}

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