4x^2-x-1=5

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Solution for 4x^2-x-1=5 equation:


Simplifying
4x2 + -1x + -1 = 5

Reorder the terms:
-1 + -1x + 4x2 = 5

Solving
-1 + -1x + 4x2 = 5

Solving for variable 'x'.

Reorder the terms:
-1 + -5 + -1x + 4x2 = 5 + -5

Combine like terms: -1 + -5 = -6
-6 + -1x + 4x2 = 5 + -5

Combine like terms: 5 + -5 = 0
-6 + -1x + 4x2 = 0

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

Divide each side by '4'.
-1.5 + -0.25x + x2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -0.25x + 1.5 + x2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -0.25x + x2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -0.25x + x2 = 0 + 1.5
-0.25x + x2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-0.25x + x2 = 1.5

The x term is -0.25x.  Take half its coefficient (-0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
-0.25x + 0.015625 + x2 = 1.5 + 0.015625

Reorder the terms:
0.015625 + -0.25x + x2 = 1.5 + 0.015625

Combine like terms: 1.5 + 0.015625 = 1.515625
0.015625 + -0.25x + x2 = 1.515625

Factor a perfect square on the left side:
(x + -0.125)(x + -0.125) = 1.515625

Calculate the square root of the right side: 1.231107225

Break this problem into two subproblems by setting 
(x + -0.125) equal to 1.231107225 and -1.231107225.

Subproblem 1

x + -0.125 = 1.231107225 Simplifying x + -0.125 = 1.231107225 Reorder the terms: -0.125 + x = 1.231107225 Solving -0.125 + x = 1.231107225 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + x = 1.231107225 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + x = 1.231107225 + 0.125 x = 1.231107225 + 0.125 Combine like terms: 1.231107225 + 0.125 = 1.356107225 x = 1.356107225 Simplifying x = 1.356107225

Subproblem 2

x + -0.125 = -1.231107225 Simplifying x + -0.125 = -1.231107225 Reorder the terms: -0.125 + x = -1.231107225 Solving -0.125 + x = -1.231107225 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + x = -1.231107225 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + x = -1.231107225 + 0.125 x = -1.231107225 + 0.125 Combine like terms: -1.231107225 + 0.125 = -1.106107225 x = -1.106107225 Simplifying x = -1.106107225

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.356107225, -1.106107225}

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