(x^2+1)=10(x+1)+75

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Solution for (x^2+1)=10(x+1)+75 equation:


Simplifying
(x2 + 1) = 10(x + 1) + 75

Reorder the terms:
(1 + x2) = 10(x + 1) + 75

Remove parenthesis around (1 + x2)
1 + x2 = 10(x + 1) + 75

Reorder the terms:
1 + x2 = 10(1 + x) + 75
1 + x2 = (1 * 10 + x * 10) + 75
1 + x2 = (10 + 10x) + 75

Reorder the terms:
1 + x2 = 10 + 75 + 10x

Combine like terms: 10 + 75 = 85
1 + x2 = 85 + 10x

Solving
1 + x2 = 85 + 10x

Solving for variable 'x'.

Reorder the terms:
1 + -85 + -10x + x2 = 85 + 10x + -85 + -10x

Combine like terms: 1 + -85 = -84
-84 + -10x + x2 = 85 + 10x + -85 + -10x

Reorder the terms:
-84 + -10x + x2 = 85 + -85 + 10x + -10x

Combine like terms: 85 + -85 = 0
-84 + -10x + x2 = 0 + 10x + -10x
-84 + -10x + x2 = 10x + -10x

Combine like terms: 10x + -10x = 0
-84 + -10x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '84' to each side of the equation.
-84 + -10x + 84 + x2 = 0 + 84

Reorder the terms:
-84 + 84 + -10x + x2 = 0 + 84

Combine like terms: -84 + 84 = 0
0 + -10x + x2 = 0 + 84
-10x + x2 = 0 + 84

Combine like terms: 0 + 84 = 84
-10x + x2 = 84

The x term is -10x.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10x + 25 + x2 = 84 + 25

Reorder the terms:
25 + -10x + x2 = 84 + 25

Combine like terms: 84 + 25 = 109
25 + -10x + x2 = 109

Factor a perfect square on the left side:
(x + -5)(x + -5) = 109

Calculate the square root of the right side: 10.440306509

Break this problem into two subproblems by setting 
(x + -5) equal to 10.440306509 and -10.440306509.

Subproblem 1

x + -5 = 10.440306509 Simplifying x + -5 = 10.440306509 Reorder the terms: -5 + x = 10.440306509 Solving -5 + x = 10.440306509 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x = 10.440306509 + 5 Combine like terms: -5 + 5 = 0 0 + x = 10.440306509 + 5 x = 10.440306509 + 5 Combine like terms: 10.440306509 + 5 = 15.440306509 x = 15.440306509 Simplifying x = 15.440306509

Subproblem 2

x + -5 = -10.440306509 Simplifying x + -5 = -10.440306509 Reorder the terms: -5 + x = -10.440306509 Solving -5 + x = -10.440306509 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x = -10.440306509 + 5 Combine like terms: -5 + 5 = 0 0 + x = -10.440306509 + 5 x = -10.440306509 + 5 Combine like terms: -10.440306509 + 5 = -5.440306509 x = -5.440306509 Simplifying x = -5.440306509

Solution

The solution to the problem is based on the solutions from the subproblems. x = {15.440306509, -5.440306509}

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