x^2-1=4x+1

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


Simplifying
x2 + -1 = 4x + 1

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

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

Solving
-1 + x2 = 1 + 4x

Solving for variable 'x'.

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

Combine like terms: -1 + -1 = -2
-2 + -4x + x2 = 1 + 4x + -1 + -4x

Reorder the terms:
-2 + -4x + x2 = 1 + -1 + 4x + -4x

Combine like terms: 1 + -1 = 0
-2 + -4x + x2 = 0 + 4x + -4x
-2 + -4x + x2 = 4x + -4x

Combine like terms: 4x + -4x = 0
-2 + -4x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -4x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + -4x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -4x + x2 = 0 + 2
-4x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
-4x + x2 = 2

The x term is -4x.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4x + 4 + x2 = 2 + 4

Reorder the terms:
4 + -4x + x2 = 2 + 4

Combine like terms: 2 + 4 = 6
4 + -4x + x2 = 6

Factor a perfect square on the left side:
(x + -2)(x + -2) = 6

Calculate the square root of the right side: 2.449489743

Break this problem into two subproblems by setting 
(x + -2) equal to 2.449489743 and -2.449489743.

Subproblem 1

x + -2 = 2.449489743 Simplifying x + -2 = 2.449489743 Reorder the terms: -2 + x = 2.449489743 Solving -2 + x = 2.449489743 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + x = 2.449489743 + 2 Combine like terms: -2 + 2 = 0 0 + x = 2.449489743 + 2 x = 2.449489743 + 2 Combine like terms: 2.449489743 + 2 = 4.449489743 x = 4.449489743 Simplifying x = 4.449489743

Subproblem 2

x + -2 = -2.449489743 Simplifying x + -2 = -2.449489743 Reorder the terms: -2 + x = -2.449489743 Solving -2 + x = -2.449489743 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + x = -2.449489743 + 2 Combine like terms: -2 + 2 = 0 0 + x = -2.449489743 + 2 x = -2.449489743 + 2 Combine like terms: -2.449489743 + 2 = -0.449489743 x = -0.449489743 Simplifying x = -0.449489743

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.449489743, -0.449489743}

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