x^2-5=2x-1

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


Simplifying
x2 + -5 = 2x + -1

Reorder the terms:
-5 + x2 = 2x + -1

Reorder the terms:
-5 + x2 = -1 + 2x

Solving
-5 + x2 = -1 + 2x

Solving for variable 'x'.

Reorder the terms:
-5 + 1 + -2x + x2 = -1 + 2x + 1 + -2x

Combine like terms: -5 + 1 = -4
-4 + -2x + x2 = -1 + 2x + 1 + -2x

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

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

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

Begin completing the square.

Move the constant term to the right:

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

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

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

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

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 = 4 + 1

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

Combine like terms: 4 + 1 = 5
1 + -2x + x2 = 5

Factor a perfect square on the left side:
(x + -1)(x + -1) = 5

Calculate the square root of the right side: 2.236067978

Break this problem into two subproblems by setting 
(x + -1) equal to 2.236067978 and -2.236067978.

Subproblem 1

x + -1 = 2.236067978 Simplifying x + -1 = 2.236067978 Reorder the terms: -1 + x = 2.236067978 Solving -1 + x = 2.236067978 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 = 2.236067978 + 1 Combine like terms: -1 + 1 = 0 0 + x = 2.236067978 + 1 x = 2.236067978 + 1 Combine like terms: 2.236067978 + 1 = 3.236067978 x = 3.236067978 Simplifying x = 3.236067978

Subproblem 2

x + -1 = -2.236067978 Simplifying x + -1 = -2.236067978 Reorder the terms: -1 + x = -2.236067978 Solving -1 + x = -2.236067978 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 = -2.236067978 + 1 Combine like terms: -1 + 1 = 0 0 + x = -2.236067978 + 1 x = -2.236067978 + 1 Combine like terms: -2.236067978 + 1 = -1.236067978 x = -1.236067978 Simplifying x = -1.236067978

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.236067978, -1.236067978}

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