16x^2-x-1=0

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Solution for 16x^2-x-1=0 equation:


Simplifying
16x2 + -1x + -1 = 0

Reorder the terms:
-1 + -1x + 16x2 = 0

Solving
-1 + -1x + 16x2 = 0

Solving for variable 'x'.

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

Divide each side by '16'.
-0.0625 + -0.0625x + x2 = 0

Move the constant term to the right:

Add '0.0625' to each side of the equation.
-0.0625 + -0.0625x + 0.0625 + x2 = 0 + 0.0625

Reorder the terms:
-0.0625 + 0.0625 + -0.0625x + x2 = 0 + 0.0625

Combine like terms: -0.0625 + 0.0625 = 0.0000
0.0000 + -0.0625x + x2 = 0 + 0.0625
-0.0625x + x2 = 0 + 0.0625

Combine like terms: 0 + 0.0625 = 0.0625
-0.0625x + x2 = 0.0625

The x term is -0.0625x.  Take half its coefficient (-0.03125).
Square it (0.0009765625) and add it to both sides.

Add '0.0009765625' to each side of the equation.
-0.0625x + 0.0009765625 + x2 = 0.0625 + 0.0009765625

Reorder the terms:
0.0009765625 + -0.0625x + x2 = 0.0625 + 0.0009765625

Combine like terms: 0.0625 + 0.0009765625 = 0.0634765625
0.0009765625 + -0.0625x + x2 = 0.0634765625

Factor a perfect square on the left side:
(x + -0.03125)(x + -0.03125) = 0.0634765625

Calculate the square root of the right side: 0.251945555

Break this problem into two subproblems by setting 
(x + -0.03125) equal to 0.251945555 and -0.251945555.

Subproblem 1

x + -0.03125 = 0.251945555 Simplifying x + -0.03125 = 0.251945555 Reorder the terms: -0.03125 + x = 0.251945555 Solving -0.03125 + x = 0.251945555 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.03125' to each side of the equation. -0.03125 + 0.03125 + x = 0.251945555 + 0.03125 Combine like terms: -0.03125 + 0.03125 = 0.00000 0.00000 + x = 0.251945555 + 0.03125 x = 0.251945555 + 0.03125 Combine like terms: 0.251945555 + 0.03125 = 0.283195555 x = 0.283195555 Simplifying x = 0.283195555

Subproblem 2

x + -0.03125 = -0.251945555 Simplifying x + -0.03125 = -0.251945555 Reorder the terms: -0.03125 + x = -0.251945555 Solving -0.03125 + x = -0.251945555 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.03125' to each side of the equation. -0.03125 + 0.03125 + x = -0.251945555 + 0.03125 Combine like terms: -0.03125 + 0.03125 = 0.00000 0.00000 + x = -0.251945555 + 0.03125 x = -0.251945555 + 0.03125 Combine like terms: -0.251945555 + 0.03125 = -0.220695555 x = -0.220695555 Simplifying x = -0.220695555

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.283195555, -0.220695555}

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