x^2-x-1991=0

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Solution for x^2-x-1991=0 equation:


Simplifying
x2 + -1x + -1991 = 0

Reorder the terms:
-1991 + -1x + x2 = 0

Solving
-1991 + -1x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '1991' to each side of the equation.
-1991 + -1x + 1991 + x2 = 0 + 1991

Reorder the terms:
-1991 + 1991 + -1x + x2 = 0 + 1991

Combine like terms: -1991 + 1991 = 0
0 + -1x + x2 = 0 + 1991
-1x + x2 = 0 + 1991

Combine like terms: 0 + 1991 = 1991
-1x + x2 = 1991

The x term is -1x.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x + 0.25 + x2 = 1991 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 1991 + 0.25

Combine like terms: 1991 + 0.25 = 1991.25
0.25 + -1x + x2 = 1991.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 1991.25

Calculate the square root of the right side: 44.623424342

Break this problem into two subproblems by setting 
(x + -0.5) equal to 44.623424342 and -44.623424342.

Subproblem 1

x + -0.5 = 44.623424342 Simplifying x + -0.5 = 44.623424342 Reorder the terms: -0.5 + x = 44.623424342 Solving -0.5 + x = 44.623424342 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 44.623424342 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 44.623424342 + 0.5 x = 44.623424342 + 0.5 Combine like terms: 44.623424342 + 0.5 = 45.123424342 x = 45.123424342 Simplifying x = 45.123424342

Subproblem 2

x + -0.5 = -44.623424342 Simplifying x + -0.5 = -44.623424342 Reorder the terms: -0.5 + x = -44.623424342 Solving -0.5 + x = -44.623424342 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -44.623424342 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -44.623424342 + 0.5 x = -44.623424342 + 0.5 Combine like terms: -44.623424342 + 0.5 = -44.123424342 x = -44.123424342 Simplifying x = -44.123424342

Solution

The solution to the problem is based on the solutions from the subproblems. x = {45.123424342, -44.123424342}

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