4x+1=x^2-5

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


Simplifying
4x + 1 = x2 + -5

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

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

Solving
1 + 4x = -5 + x2

Solving for variable 'x'.

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

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

Reorder the terms:
6 + 4x + -1x2 = -5 + 5 + x2 + -1x2

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

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

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-6 + -4x + x2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + -4x + 6 + x2 = 0 + 6

Reorder the terms:
-6 + 6 + -4x + x2 = 0 + 6

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

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

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 = 6 + 4

Reorder the terms:
4 + -4x + x2 = 6 + 4

Combine like terms: 6 + 4 = 10
4 + -4x + x2 = 10

Factor a perfect square on the left side:
(x + -2)(x + -2) = 10

Calculate the square root of the right side: 3.16227766

Break this problem into two subproblems by setting 
(x + -2) equal to 3.16227766 and -3.16227766.

Subproblem 1

x + -2 = 3.16227766 Simplifying x + -2 = 3.16227766 Reorder the terms: -2 + x = 3.16227766 Solving -2 + x = 3.16227766 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 = 3.16227766 + 2 Combine like terms: -2 + 2 = 0 0 + x = 3.16227766 + 2 x = 3.16227766 + 2 Combine like terms: 3.16227766 + 2 = 5.16227766 x = 5.16227766 Simplifying x = 5.16227766

Subproblem 2

x + -2 = -3.16227766 Simplifying x + -2 = -3.16227766 Reorder the terms: -2 + x = -3.16227766 Solving -2 + x = -3.16227766 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 = -3.16227766 + 2 Combine like terms: -2 + 2 = 0 0 + x = -3.16227766 + 2 x = -3.16227766 + 2 Combine like terms: -3.16227766 + 2 = -1.16227766 x = -1.16227766 Simplifying x = -1.16227766

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.16227766, -1.16227766}

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