10x-5=x^2+1

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Solution for 10x-5=x^2+1 equation:


Simplifying
10x + -5 = x2 + 1

Reorder the terms:
-5 + 10x = x2 + 1

Reorder the terms:
-5 + 10x = 1 + x2

Solving
-5 + 10x = 1 + x2

Solving for variable 'x'.

Reorder the terms:
-5 + -1 + 10x + -1x2 = 1 + x2 + -1 + -1x2

Combine like terms: -5 + -1 = -6
-6 + 10x + -1x2 = 1 + x2 + -1 + -1x2

Reorder the terms:
-6 + 10x + -1x2 = 1 + -1 + x2 + -1x2

Combine like terms: 1 + -1 = 0
-6 + 10x + -1x2 = 0 + x2 + -1x2
-6 + 10x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-6 + 10x + -1x2 = 0

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

Divide each side by '-1'.
6 + -10x + x2 = 0

Move the constant term to the right:

Add '-6' to each side of the equation.
6 + -10x + -6 + x2 = 0 + -6

Reorder the terms:
6 + -6 + -10x + x2 = 0 + -6

Combine like terms: 6 + -6 = 0
0 + -10x + x2 = 0 + -6
-10x + x2 = 0 + -6

Combine like terms: 0 + -6 = -6
-10x + x2 = -6

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 = -6 + 25

Reorder the terms:
25 + -10x + x2 = -6 + 25

Combine like terms: -6 + 25 = 19
25 + -10x + x2 = 19

Factor a perfect square on the left side:
(x + -5)(x + -5) = 19

Calculate the square root of the right side: 4.358898944

Break this problem into two subproblems by setting 
(x + -5) equal to 4.358898944 and -4.358898944.

Subproblem 1

x + -5 = 4.358898944 Simplifying x + -5 = 4.358898944 Reorder the terms: -5 + x = 4.358898944 Solving -5 + x = 4.358898944 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 = 4.358898944 + 5 Combine like terms: -5 + 5 = 0 0 + x = 4.358898944 + 5 x = 4.358898944 + 5 Combine like terms: 4.358898944 + 5 = 9.358898944 x = 9.358898944 Simplifying x = 9.358898944

Subproblem 2

x + -5 = -4.358898944 Simplifying x + -5 = -4.358898944 Reorder the terms: -5 + x = -4.358898944 Solving -5 + x = -4.358898944 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 = -4.358898944 + 5 Combine like terms: -5 + 5 = 0 0 + x = -4.358898944 + 5 x = -4.358898944 + 5 Combine like terms: -4.358898944 + 5 = 0.641101056 x = 0.641101056 Simplifying x = 0.641101056

Solution

The solution to the problem is based on the solutions from the subproblems. x = {9.358898944, 0.641101056}

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