8.5=x^2+2x+6

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Solution for 8.5=x^2+2x+6 equation:


Simplifying
8.5 = x2 + 2x + 6

Reorder the terms:
8.5 = 6 + 2x + x2

Solving
8.5 = 6 + 2x + x2

Solving for variable 'x'.

Combine like terms: 8.5 + -6 = 2.5
2.5 + -2x + -1x2 = 6 + 2x + x2 + -6 + -2x + -1x2

Reorder the terms:
2.5 + -2x + -1x2 = 6 + -6 + 2x + -2x + x2 + -1x2

Combine like terms: 6 + -6 = 0
2.5 + -2x + -1x2 = 0 + 2x + -2x + x2 + -1x2
2.5 + -2x + -1x2 = 2x + -2x + x2 + -1x2

Combine like terms: 2x + -2x = 0
2.5 + -2x + -1x2 = 0 + x2 + -1x2
2.5 + -2x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
2.5 + -2x + -1x2 = 0

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

Divide each side by '-1'.
-2.5 + 2x + x2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + 2x + 2.5 + x2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + 2x + x2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + 2x + x2 = 0 + 2.5
2x + x2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
2x + x2 = 2.5

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 = 2.5 + 1

Reorder the terms:
1 + 2x + x2 = 2.5 + 1

Combine like terms: 2.5 + 1 = 3.5
1 + 2x + x2 = 3.5

Factor a perfect square on the left side:
(x + 1)(x + 1) = 3.5

Calculate the square root of the right side: 1.870828693

Break this problem into two subproblems by setting 
(x + 1) equal to 1.870828693 and -1.870828693.

Subproblem 1

x + 1 = 1.870828693 Simplifying x + 1 = 1.870828693 Reorder the terms: 1 + x = 1.870828693 Solving 1 + x = 1.870828693 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 = 1.870828693 + -1 Combine like terms: 1 + -1 = 0 0 + x = 1.870828693 + -1 x = 1.870828693 + -1 Combine like terms: 1.870828693 + -1 = 0.870828693 x = 0.870828693 Simplifying x = 0.870828693

Subproblem 2

x + 1 = -1.870828693 Simplifying x + 1 = -1.870828693 Reorder the terms: 1 + x = -1.870828693 Solving 1 + x = -1.870828693 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 = -1.870828693 + -1 Combine like terms: 1 + -1 = 0 0 + x = -1.870828693 + -1 x = -1.870828693 + -1 Combine like terms: -1.870828693 + -1 = -2.870828693 x = -2.870828693 Simplifying x = -2.870828693

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.870828693, -2.870828693}

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