x^2+8x+22=198-4x-.25x^2

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Solution for x^2+8x+22=198-4x-.25x^2 equation:


Simplifying
x2 + 8x + 22 = 198 + -4x + -0.25x2

Reorder the terms:
22 + 8x + x2 = 198 + -4x + -0.25x2

Solving
22 + 8x + x2 = 198 + -4x + -0.25x2

Solving for variable 'x'.

Reorder the terms:
22 + -198 + 8x + 4x + x2 + 0.25x2 = 198 + -4x + -0.25x2 + -198 + 4x + 0.25x2

Combine like terms: 22 + -198 = -176
-176 + 8x + 4x + x2 + 0.25x2 = 198 + -4x + -0.25x2 + -198 + 4x + 0.25x2

Combine like terms: 8x + 4x = 12x
-176 + 12x + x2 + 0.25x2 = 198 + -4x + -0.25x2 + -198 + 4x + 0.25x2

Combine like terms: x2 + 0.25x2 = 1.25x2
-176 + 12x + 1.25x2 = 198 + -4x + -0.25x2 + -198 + 4x + 0.25x2

Reorder the terms:
-176 + 12x + 1.25x2 = 198 + -198 + -4x + 4x + -0.25x2 + 0.25x2

Combine like terms: 198 + -198 = 0
-176 + 12x + 1.25x2 = 0 + -4x + 4x + -0.25x2 + 0.25x2
-176 + 12x + 1.25x2 = -4x + 4x + -0.25x2 + 0.25x2

Combine like terms: -4x + 4x = 0
-176 + 12x + 1.25x2 = 0 + -0.25x2 + 0.25x2
-176 + 12x + 1.25x2 = -0.25x2 + 0.25x2

Combine like terms: -0.25x2 + 0.25x2 = 0.00
-176 + 12x + 1.25x2 = 0.00

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

Divide each side by '1.25'.
-140.8 + 9.6x + x2 = 0

Move the constant term to the right:

Add '140.8' to each side of the equation.
-140.8 + 9.6x + 140.8 + x2 = 0 + 140.8

Reorder the terms:
-140.8 + 140.8 + 9.6x + x2 = 0 + 140.8

Combine like terms: -140.8 + 140.8 = 0.0
0.0 + 9.6x + x2 = 0 + 140.8
9.6x + x2 = 0 + 140.8

Combine like terms: 0 + 140.8 = 140.8
9.6x + x2 = 140.8

The x term is 9.6x.  Take half its coefficient (4.8).
Square it (23.04) and add it to both sides.

Add '23.04' to each side of the equation.
9.6x + 23.04 + x2 = 140.8 + 23.04

Reorder the terms:
23.04 + 9.6x + x2 = 140.8 + 23.04

Combine like terms: 140.8 + 23.04 = 163.84
23.04 + 9.6x + x2 = 163.84

Factor a perfect square on the left side:
(x + 4.8)(x + 4.8) = 163.84

Calculate the square root of the right side: 12.8

Break this problem into two subproblems by setting 
(x + 4.8) equal to 12.8 and -12.8.

Subproblem 1

x + 4.8 = 12.8 Simplifying x + 4.8 = 12.8 Reorder the terms: 4.8 + x = 12.8 Solving 4.8 + x = 12.8 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.8' to each side of the equation. 4.8 + -4.8 + x = 12.8 + -4.8 Combine like terms: 4.8 + -4.8 = 0.0 0.0 + x = 12.8 + -4.8 x = 12.8 + -4.8 Combine like terms: 12.8 + -4.8 = 8 x = 8 Simplifying x = 8

Subproblem 2

x + 4.8 = -12.8 Simplifying x + 4.8 = -12.8 Reorder the terms: 4.8 + x = -12.8 Solving 4.8 + x = -12.8 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.8' to each side of the equation. 4.8 + -4.8 + x = -12.8 + -4.8 Combine like terms: 4.8 + -4.8 = 0.0 0.0 + x = -12.8 + -4.8 x = -12.8 + -4.8 Combine like terms: -12.8 + -4.8 = -17.6 x = -17.6 Simplifying x = -17.6

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8, -17.6}

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